Futures And Spot Commodity Prices, Options, And Forward ... · Di erent traders have di erent...

Futures And Spot Commodity Prices,

Options, And Forward interest rates:

Model And Empirical Analysis

YU MIAO

(B.Sc., NANJING UNIVERSITY)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF PHYSICS

NATIONAL UNIVERSITY OF SINGAPORE

2017

Supervisor:

Dr. Wang Qinghai

Examiners:

Professor. Wang Jiansheng

Associate Professor. Tan Mengchwan

Professor. Emmanuel Haven, Memorial University of

Newfoundland

Declaration

I hereby declare that the thesis is my original work and it has been written by me in its

entirety. I have duly acknowledged all the sources which have been used in the thesis.

This thesis has also not been submitted for any degree in any university previously.

YU MIAO

JULY 2017

i

Acknowledgements

I would like to thank all people who have supported me during my graduate study.

First and foremost, I would like to express my sincerest gratitude and greatest respect to

Professor Belal E Baaquie and my supervisor Dr. Wang Qinghai. My main work is under

the guidance of Professor Baaquie. Without his encouragement, supervision, and support,

the dissertation is definitely not possible. His deep and distinctive view of quantum physics

guided me to think of the world in the language of quantum field theory. Besides the research

field, the opinion and experiences for the life that he shares with me will affect my whole life.

I thank my supervisor Dr. Wang Qinghai for vast advices and guidance for my research and

my PhD thesis.

My sincere thanks also goes to Du Xin, Jiten Bhanap, and Cao Yang for their useful

discussion and collaboration. I thank National University of Singapore and Department of

Physics for the financial support.

Last but not least, I would like to thank my parents, my father Yu Jianxin and my mother

Yu Xiuying, and my wife Tian Maoshan for their unconditional support and precious love.

iii

Summary

Financial market is a platform to improve the efficiency and convenience for all traders in the

world. Different traders have different purposes for their trading and there are three main

types of traders including hedgers, speculators and arbitrageurs, due to three vital purposes.

Whether you aim to hedge, speculate or arbitrage in the market, quantitative analysis is a

strong tool to help you make a better decision. Although recent quantitative study using

statistics and stochastic process as the basic method has already made huge achievements in

financial domain, there are still great numbers of unsolved financial fields which may require

new theories to fill in. In this thesis, quantum field theory is chosen as a new method to analyse

financial instruments. In Chapter 2, a new correlation factor among different commodities

is defined by a new microeconomics theory based on quantum field theory. In Chapter 3,

a new theory of commodity futures with two-Dimension quantum field is proposed and the

theory may fill the gap of the futures theory for financial market. In Chapter 4, an indicator is

designed by comparing option pricing with stochastic method and quantum method to reflect

the market instability. In Chapter 5, a new theory of pricing bond option is raised up to give

contract makers a better reference to price the bond option contract.

v

vi

Publication List

[1] B.E. Baaquie, Miao Yu∗, and Xin Du. ”Multiple Commodities in Statistical Microe-

conomics: Model and Market.” Physica A: Statistical Mechanics and its Applications 462

(2016): 912-929.

[2] B.E. Baaquie and Miao Yu∗. Statistical Field Theory of Futures Commodity Prices.

Submitting paper.

[3] B.E. Baaquie, Miao Yu∗. ”Option Price and Market Instability.” Physica A: Statistical

Mechanics and its Applications 471 (2017): 512-535.

[4] B.E. Baaquie, Miao Yu∗ and Jitendra Bhanap. Risky Forward Interest Rates and

Swaptions: Quantum Finance Model and Empirical Results. Submitting paper.

* Corresponding author

Contents

Declaration i

Acknowledgements iii

Summary v

List of Tables xiv

List of Figures xxi

List of Symbols xxii

1 Introduction 1

§ 1.1 Review of financial market and financial modeling . . . . . . . . . . . . . . . . 1

§ 1.2 Introduction of financial instrument . . . . . . . . . . . . . . . . . . . . . . . . 3

§ 1.2.1 Commodity price and Futures . . . . . . . . . . . . . . . . . . . . . . . 3

§ 1.2.2 Option price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

§ 1.2.3 Interest rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

vii

CONTENTS viii

§ 1.2.4 Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

§ 1.2.5 One type of swaption: bond option . . . . . . . . . . . . . . . . . . . . 7

§ 1.3 Introduction of financial models . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

§ 1.3.1 Lagrangian model based on supply and demand . . . . . . . . . . . . . 8

§ 1.3.2 Black-Scholes model for option pricing . . . . . . . . . . . . . . . . . . 10

§ 1.3.3 HJM Model for forward interest rate . . . . . . . . . . . . . . . . . . . 12

2 Multiple Commodities in Statistical Microeconomics: Model and Market 14

§ 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

§ 2.2 The microeconomic action functional . . . . . . . . . . . . . . . . . . . . . . . . 16

§ 2.3 Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

§ 2.3.1 Expansion of Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

§ 2.3.2 Auto-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

§ 2.3.3 Cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

§ 2.3.4 Nonlinear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

§ 2.4 Market data and model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

§ 2.5 Fitting with Market Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

§ 2.6 Fits for GII , GIJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

§ 2.6.1 Two commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

§ 2.6.2 Three commodities fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

§ 2.6.3 Four commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

CONTENTS ix

§ 2.6.4 Six commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

§ 2.7 Comparison of single and multiple commodities fit . . . . . . . . . . . . . . . . 44

§ 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

§ 2.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

§ 2.9.1 Derivation of D(0)IJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

§ 2.9.2 Consistency check for D(0)IJ . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Statistical Field Theory of Futures Commodity Prices 53

§ 3.1 Futures commodity prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

§ 3.2 Single commodity; Gaussian approximation . . . . . . . . . . . . . . . . . . . . 57

§ 3.3 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

§ 3.4 Propagator for spot prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

§ 3.4.1 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

§ 3.4.2 Special case γ = γ2 = γ1 . . . . . . . . . . . . . . . . . . . . . . . . . . 65

§ 3.4.3 Limit of γ → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

§ 3.5 Contour map of G(t, ξ; 0, 0) and α . . . . . . . . . . . . . . . . . . . . . . . . . 67

§ 3.6 Spot rate G(t, t; t′, t′): empirical and model . . . . . . . . . . . . . . . . . . . . 68

§ 3.7 Spot-futures G(t, ξ; 0, 0): empirical and model . . . . . . . . . . . . . . . . . . . 70

§ 3.8 Algorithm for empirical GE(z+, z−) . . . . . . . . . . . . . . . . . . . . . . . . . 72

§ 3.9 Binning of empirical D(k)E (a, b, c) . . . . . . . . . . . . . . . . . . . . . . . . . . 76

§ 3.10 Empirical results for GE(z+; z−) . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS x

§ 3.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

§ 3.12 Appendix I(τ, θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

§ 3.12.1 Appendix: Algorithm for binning the propagator . . . . . . . . . . . . . 82

4 Option Price and Market Instability 85

§ 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

§ 4.2 Quantum finance formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

§ 4.3 Transition amplitude K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

§ 4.4 BY Model option price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

§ 4.4.1 Martingale condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

§ 4.4.2 BY Option: market time . . . . . . . . . . . . . . . . . . . . . . . . . . 94

§ 4.5 Mapping BY Model to data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

§ 4.6 Calibration of the BY Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

§ 4.7 Fitting Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

§ 4.8 Global crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

§ 4.9 Result for five major FX options . . . . . . . . . . . . . . . . . . . . . . . . . . 110

§ 4.9.1 Euro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

§ 4.9.2 Australia Dollar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

§ 4.9.3 Swiss Franc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

§ 4.9.4 British Pound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

§ 4.9.5 Japanese Yen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

CONTENTS xi

§ 4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

§ 4.11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

§ 4.11.1 Classical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Risky Forward Interest Rates and Swaptions: Quantum Finance Model and

Empirical Results 121

§ 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

§ 5.2 Quantum finance model of forward interest rates . . . . . . . . . . . . . . . . . 123

§ 5.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

§ 5.4 Stiff propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

§ 5.5 Market correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

§ 5.6 Empirical volatility and propagators . . . . . . . . . . . . . . . . . . . . . . . . 135

§ 5.6.1 Stand-alone Singapore rates . . . . . . . . . . . . . . . . . . . . . . . . 137

§ 5.7 Calibration of US and Singapore models . . . . . . . . . . . . . . . . . . . . . . 138

§ 5.8 Determination of ∆(θ, θ′): Coupling of US-Singapore rates . . . . . . . . . . . . 140

§ 5.8.1 Malaysian forward interest rates . . . . . . . . . . . . . . . . . . . . . . 143

§ 5.9 Summary of Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . 145

§ 5.10 Interest rate swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

§ 5.10.1 US swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

§ 5.10.2 Singapore swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

§ 5.10.3 Malaysian swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

CONTENTS xii

§ 5.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

§ 5.12 Appendix 1: Risky coupon bond option . . . . . . . . . . . . . . . . . . . . . . 156

§ 5.13 Appendix 2: Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

§ 5.14 Appendix 3: Black’s Model for Swaption . . . . . . . . . . . . . . . . . . . . . . 167

§ 5.14.1 Par value of fixed payments . . . . . . . . . . . . . . . . . . . . . . . . 169

§ 5.15 Appendix 4: Zero coupon bonds from coupon bonds . . . . . . . . . . . . . . . 169

§ 5.16 Appendix 5: Forward interest rates and zero coupon bonds . . . . . . . . . . . 173

List of Tables

2.1 Number and Type of Commodity . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Gold-Silver. η = 0.7; λ = 0.1004 . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Crude oil-Heating oil-Brent oil. η = 0.7; λ = 0.775 . . . . . . . . . . . . . . . . 35

2.4 Orangejuice-Cattle-Soybean. η = 0.7;λ = 1.132 . . . . . . . . . . . . . . . . . 36

2.5 Gold-Silver-Platinum. η = 0.7; λ = 0.344 . . . . . . . . . . . . . . . . . . . . . 38

2.6 Crude oil-Platinum-Cocoa. η = 0.70; λ = 0.54 . . . . . . . . . . . . . . . . . . 39

2.7 Gold-Silver-Crude oil-Natural gas. η = 0.70; λ = 0.260. . . . . . . . . . . . . . 41

2.8 Gold-Silver-Crude oil-Natural gas-Soybean oil-Cattle. η = 0.70; λ = 0.699. . . 42

2.9 Comparison of Single-Commodity fit(S-) with Multiple-Commodities fit(M-).

Group 1 is Gold-Silver-Crude oil (GSC) and Group 2 is Crude Oil-Platinum-

Cocoa (CPC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 Calibration: Spot-Futures Correlations . . . . . . . . . . . . . . . . . . . . . . 72

3.2 Crude oil Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1 Fitting Parameters for Group 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . 95

xiii

LIST OF TABLES xiv

4.2 Parameters for EURUSD Fitting . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1 Model’s parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

List of Figures

1.1 Option payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Supply and Demand in economics . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Supply and Demand as potential . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Random paths of the security . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 α = 0.1, β = 0.15, φ = 30, θ = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 α = 0.1, β = 0.15, φ = 20, θ = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Matrix of ∆ij for 18 commodities. Note that for all pairs, |∆IJ | < 0.08 but one

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Silver and Gold with η = 0.7; λ = 0.1004 . . . . . . . . . . . . . . . . . . . . . 34

2.5 Crude oil-Heating oil-Brent oil (a)Autocorrelation and (b)Crosscorrelation with

η = 0.7; λ = 0.775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Orange juice-Cattle-Soybean (a)Autocorrelation and (b)Crosscorrelation with

η = 0.7;λ = 1.132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Gold-Silver-Platinum (a)Autocorrelation and (b)Crosscorrelation with η = 0.7;

λ = 0.344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

xv

LIST OF FIGURES xvi

2.8 Crude oil-Platinum-Cocoa (a)Autocorrelation and (b)Crosscorrelation with η =

0.70; λ = 0.54. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.9 Gold-Silver-Crude oil-Natural gas (a)Autocorrelation and (b)Crosscorrelation

with η = 0.70; λ = 0.260. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.10 Gold-Silver-Crude oil-Natural gas-Soybean oil-Cattle (a)Autocorrelation and

(b)Crosscorrelation with η = 0.70; λ = 0.699. . . . . . . . . . . . . . . . . . . . 42

3.1 Points on the boundary are calendar time (t, t); (t′, t′) and points (t, ξ); (t′, ξ′)

are in future time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Theoretical plot of G(z+; z−) as a function of z+, z−, with α=1,L=1, γ1=1,γ2=2 61

3.3 Shape of the model for futures for a) α = 1, b) α > 1 and c) α < 1. . . . . . . 67

3.4 Fitting spot rates for a) Gold, b) Soybeans and c) Corn. The smooth curve is

the model’s best fit to data. (Jan 1 2011- Oct 18 2011) . . . . . . . . . . . . . 69

3.5 Model and market correlators for crude oil, with R2 = 0.93. (Sep 20 2014- June

11 2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Spot and futures prices correlation G(t, ξ; 0, 0), plotted against t, ξ, of corn

futures prices with the spot price. a) The empirical propagator. b) The model

propagator. (Jan 1 2011- Oct 18 2011) . . . . . . . . . . . . . . . . . . . . . . 70

3.7 G(t, ξ; 0, 0) of Crude oil futures data. (Jan 1 2011- Oct 18 2011) . . . . . . . . 71

3.8 G(t, ξ; 0, 0) of Rice futures data. (Jan 1 2011- Oct 18 2011) . . . . . . . . . . . 71

3.9 G(t, ξ; 0, 0) of Gold futures data. (Jan 1 2011- Oct 18 2011) . . . . . . . . . . 71

LIST OF FIGURES xvii

3.10 Binning of 10 years oil futures data of D(k)E (a, b, c). a) With α=19.98 and for

14 sample points. b) With α=1 and for 40 sample points. (Nov 20 2015- Sep

8 2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.11 a) Empirical GE(z+; z−) and b) Model GE(z+; z−) for market Oil futures prices.

(Nov 20 2015- Sep 8 2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 ν2(τ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2√ν2/τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3 ξ(τ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 ζ(τ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.5 Shape of parameters with a = 5; b = 8; c = 100. τ is remaining time. . . . . . 92

4.6 The Forex martingale process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7 The t and z values when η < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.8 ν2(z) for Group 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.9 ν2(z) for Group 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.10 ξ(z) for Group 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.11 ξ(z) for Group 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.12 ζ(z) for Group 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.13 ζ(z) for Group 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.14 (a) The model variable x(t, τ) with different calendar times t, t′ but with the

same remaining time τ . (b) Model variable for fixed maturity time T , with

remaining time τ(t) 6= τ(t′). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

LIST OF FIGURES xviii

4.15 (a) Model velocity for fixed remaining time τ . (b) Model velocity for fixed

maturity time T is found by comparing x(t, z(τ) to x(t− δ, z(τ + δ). . . . . . . 99

4.16 Pattern A, 2009-09-23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.17 Pattern B, 2009-02-02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.18 The fourth pattern C, 2013-12-18 . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.19 Irregular data, 2008-08-28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.20 Option price fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.21 ν2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.22 ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.23 ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.24 Option price fitting R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.25 Option fitting rmse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.26 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.27 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.28 λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.29 η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.30 TED and the financial crisis in 2008. . . . . . . . . . . . . . . . . . . . . . . . 108

4.31 (a) R2 of EURUSD and (b) Fx volatility of EURUSD . . . . . . . . . . . . . . 110

4.32 (a) R2 of AUDUSD and (b) Fx volatility of AUDUSD . . . . . . . . . . . . . . 111

4.33 (a) R2 of CHFUSD and (b) Fx volatility of CHFUSD . . . . . . . . . . . . . . 111

LIST OF FIGURES xix

4.34 (a) R2 of GBPUSD and (b) Fx volatility of GBPUSD . . . . . . . . . . . . . . 113

4.35 (a) R2 of JPYUSD and (b) Fx volatility of JPYUSD . . . . . . . . . . . . . . 114

5.1 a) The semi-infinite domain with two boundaries on which f(t, x) and A(t, x)

are defined. b) The zero coupon bond for two different times t0 and T0. . . . . 123

5.2 (a) Volatility of US forward interest rates. (b) Volatility of the spread of the

Singapore -US forward interest rates. Period from 9 May 2011 to 18 January,

2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3 Empirical volatility function σ(θ) =√E[δf 2(t, θ)]c and kurtosis κ(t, θ) = E[δf(t, θ)4]/σ4(t, θ)−

3 of the forward interest rates; θ = x− t. (Reference: [1]). . . . . . . . . . . . 136

5.4 (a) Volatility of the Singapore stand-alone forward interest rates. (b) Compar-

ison of volatility of Singapore stand-alone forward interest rates of the US and

spread of the Singapore -US forward interest rates. Period from 9 May 2011 to

18 January, 2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.5 US forward interest rates. (a) The empirical correlator DE(θ, θ′). (b) The

model correlator D(θ, θ′). Data from 9 May 2011 to 18 January, 2012. . . . . . 139

5.6 Singapore forward interest rates. (a) The empirical correlator CE(θ, θ′). (b)

The model correlator C(θ, θ′). Data from 9 May 2011 to 18 January, 2012. . . 139

5.7 Joint US-Singapore forward curve. (a) The empirical spread correlator CE(θ, θ′).

(b) The model spread correlator C(θ, θ′). Data from 9 May 2011 to 18 January,

2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.8 Inverse of propagator: (a) D−1DE (b) C−1CE (c) The Dirac delta function

δ(θ − θ′). Data from 9 May 2011 to 18 January, 2012. . . . . . . . . . . . . . . 141

LIST OF FIGURES xx

5.9 Correlation of Singapore - US forward interest rates spread with the US forward

interest rates. (a) The cross-correlator TE. (b) ∆E of the US forward interest

rates with the spread with the Singapore forward interest rates. (c) The model

coefficient function ∆. Data from 9 May 2011 to 18 January, 2012. . . . . . . . 142

5.10 (a) The Malaysian forward interest rates volatility v2(θ); half-yearly time steps

in the future time direction. (b) The volatility ζ(θ) of the Malaysian spread

over the US forward interest rates. Data from 9 May 2011 to 18 January, 2012. 144

5.11 (a) The Malaysian stand-alone propagator H(θ, θ′). (b) Propagator for the

spread, given by H(θ, θ′), of the Malaysian above the US forward interest rates.

(c) The model fitting the spread for Malaysian forward interest rates. Data from

9 May 2011 to 18 January, 2012. . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.12 Domain for Gij. (a) For the case of Ti = Tj. (b) For the case of Ti 6= Tj. . . . . 149

5.13 The circles signify payment dates, except at T0;; the first payment is at T1 and

the last payment is at TN ; the interest rate swap becomes operational at time

T0. The shaded area inside the rectangles indicate the set of forward interest

rates that determine the price of a swap. (a) A midcurve forward swap is

entered into at time t0 and exercised at time t∗, before T0. (b) A forward swap

is entered into at time t0 and exercised at time T0. . . . . . . . . . . . . . . . . 150

5.14 The daily price of a US Dollar 1x10 swaption for the period 2013-2015. The

heavy (red) line is data. The blue line is the full model value of the swaption

with C2 The broken line is the value of the swaption withoutthe C2 coefficient.

(b) The value of C2 as a function of time. . . . . . . . . . . . . . . . . . . . . . 151

LIST OF FIGURES xxi

5.15 (a) Swaption of US, Singapore stand-alone and Singapore spread interest rates.

(b) Swaption of US and stand-alone Malaysian interest rates . Data for the

period 12 January 2012 to 20 October 2012. . . . . . . . . . . . . . . . . . . . 151

5.16 The shaded area is the domain of integration Ri. . . . . . . . . . . . . . . . . 157

5.17 The shaded domain of the forward interest rates contribute to Gij. For a typical

point t in the time integration, the figure shows the typical correlation function

M(x, x′; t) connecting two different values of the forward interest rates at future

time x and x′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.18 Forward interest rate and future time lattice. . . . . . . . . . . . . . . . . . . . 174

5.19 The 3x6 block structure, with three elements overlapping between successive

rows, is shown in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

List of Symbols

Symbol Definition

p commodity price

x = ln p logarithmic commodity price

y = x−xσx

x after rescaled

D(p) demand function

S(p) supply function

V(p) potential

S,L action and Lagrangian

CIJ(t, t′) non-equal empirical correlation

GIJ(t, t′) non-equal time propagator

E(...) expectation of variable in the bracket

∆IJ correlation factor

z = λ(t/λ)η market time

C call option price

P put option price

K strike price of option

K(x, x′; v, v′; t) transition amplitude

P(x, x′; v, v′; t) conditional probability

xxii

xxiii

Symbol Definition

r spot rate

f(t, x) forward interest rate, at calendar time t, interest rate for

instantaneous deposit at future time x

for a loan from Tn to Tn + `

` Bond tenor, 90 days

B(t, T ) coupon bond price

R(t) Gaussian white noise

A(t, x) Gaussian quantum field

α(t, x) drift

σ(t, x) volatility

ζ(t, x) drift

D(t;x, x′), C(t;x, x′) forward rate propagator for different products

DE(t;x, x′), CE(t;x, x′) empirical forward rate propagators

Chapter 1

Introduction

§ 1.1 Review of financial market and financial modeling

The financial market is originally formed for easy transactions between buyers and sellers from

all over the world. As economics developing rapidly, bond, equity, commodity, option, futures

and other derivatives were invented to satisfy different requirements of different customers. In

comparison to the traditional economics, the finance nowadays becomes paperless transactions

and a platform for investor to predict the trend of the global economics and make money.

However, anyone interested in financial market soon discovers that the trend of the market is

always hard to forecast by experiences.

Due to the difficulty to predict the market, the quantitative finance is introduced to analyse

the trend of the market and help the country and people avoid the market inflation risk.

Although the mathematicians use quite complicated stochastic theories, partial differential

equations and other theories to analyse the market, some important factors and big waves

like the financial crisis cannot be predicted by these theories. New angle of view and theory

need to be considered to model the market.

As Simon Benninga said in his book [2], I liken the modeling of market to cooking and the

market data to the vegetables; the way one used in the financial modeling is like the sauce

1

§ 1.1. Review of financial market and financial modeling 2

you use. In this dissertation, the path integral and the quantum mechanics are the new sauce.

Quantum mechanics is firstly proposed by Max Planck in 1900. Quantum theory nowadays

has been developed and fully applied to optics, life science, cosmology, and so on, except the

finance. Baaquie opened the quantum finance chapter in 2001. The word “quantum” refers to

the quantum mathematics and theoretical methods of quantum mechanics and quantum field

theory which is powerful for analysing financial market data. Quantum mechanics represents

a random system by elements of a state space, and the time evolution of states is determined

by the Hamiltonian differential operator [3]. The main tool in analysing finance data, path

integral, is used for solving the time-series case in the financial market. The Feynman path

integral [3] calculates the probability amplitude of the element by multiplying together the

contributions of all paths in configuration space. Therefore the Feynman path integral is

efficient and logical for analysing the time dependent behaviour of the market data.

The path integral in finance is applied to the commodity market [4] modeled by an action

functional according to statistical microeconomics. The correlation functions are studied by

using perturbation expansion in Feynman path integral. The calibration fitting shows a very

good result which the index r-square is more than 0.97(1 means perfect fitting) for nine

main commodities. Although the model describes the market data very well, it ignores the

correlations between different commodities. Only in the whole market, the model is complete.

Hence the multiple commodities model and the correlations between the multiple commodities

still remain to be investigated.

Another application of the path integral is to describe and predict financial crisis by

modeling the Forex option pricing. Baaquie and Yang modelled the Forex option price by

the acceleration Lagrangian with the classical solution under strict boundary conditions [5].

Although the signal she found in her model is enough to illustrate the 2008 financial crisis, the

model has a shortage that one typical shape of the data can not be fitted by Baaquie-Yang

model. Some revisions need to be done to find out more reasonable model for the Forex option

§ 1.2. Introduction of financial instrument 3

price.

In addition, future is another important derivative in the market. However, it is very

difficult to model future price because the future price depends on two time parameters - one

is the calendar time; the other one is the future time. In physics, two time parameters are

described by the two-dimensional Lagrangians. Because of this reason, no one has worked out

a well-accepted model for the future price yet.

Finally the risk forward interest rate is a quite complicated concept in financial derivatives.

The model needs to be considered in two parts. Firstly, the risk-free forward interest rate,

US forward interest rate, has its own model. Secondly, the risk forward interest rate, such as

other country’s rate, depends on the risk-free rate and the behaviour of its own development.

So the model for both the risk-free and risk forward rate need to be designed properly. The

biggest challenge for this project is how to keep balance with this two parts.

§ 1.2 Introduction of financial instrument

§ 1.2.1 Commodity price and Futures

In finance, when we talk about commodity, it refers to the basic product instead of the

industrial product. Commodities in financial market are divided into two types. The first

type is called soft commodity, which are mainly agricultural products such as corn, soybean,

and wheat. The other type is called hard commodity, which consists of metal and oil such as

gold, silver, and crude oil. In the old time, people took their products to the market and tried

to sell them at a good price. The trading price is the“spot” price for the current market and

current time. In modern finance, commodity futures are one of the most original investing way

in commodity trading. Hence the spot price that shown on the screen is no longer the “spot”


price as before. It generates from a couple of different future contracts on the corresponding

commodity. Futures contract is a forward contract that allow the buyer or seller to buy or sell

some products at a deal price at an exact time in the future. The deal price that the buyer

and seller agreed is called the forward price. The exact time in the future is the time that the

seller delivers the product and the buyer pays for that.

§ 1.2.2 Option price

Option is another derivative in financial market. Due to its special design, it gives a broad

space for quantitative analysis.

There are two types of option, call option and put option, separately. A call option gives

the right to the holder to buy the underlying asset on an exact price on a determined date.

The put option is just the opposite guarantee for the holder to keep the right to sell the

underlying asset on an exact price on an determined date. The fixed price in option is called

strike price. It is a comparable price with the spot price of the underlying asset and should be

determined after discounting and premium. The exact date in option is called option expiry.

the option expiry is when the option goes to its maturity. When an option has expired, it can

no longer be traded.

There are two popular options that are highly active, which are European option and

American option respectively. The holder of the European option is able to exercise the right

of the option at the expiration date. In contrast, the American option gives the holder the

right to exercise at any time before the option expires.

Similarly as futures, each option has a maturity date in the future. Option is a derivative

that allows the holder to exercise or not. Therefore, assuming that there is an underlying

security S, let K be the strike price, T be the time expiry. For an European option, the pay


off functions are shown in the figure 1.1 .

Figure 1.1: Option payoffs

The figures are Long call (buy call option), Long Put (buy put option), Short Call (write calloption) and Short Put (write put option) and from https://en.wikipedia.org/wiki/Option.

Take the call option as an example for illustration. When time goes to the maturity date

T , the call option payoff function is as below


C(T, S(T )) =

S(T )−K, S(T ) > K,

0, S(T ) < K,

≡ [S(T )−K]+.

(1.1)

§ 1.2.3 Interest rate

The interest rate is originally created to define how much one should pay when he borrow

some amount of money before. Now the concept interest rate mainly used for the special

lender, banks. Assuming that the annual interest rate is r, consider one borrows money M

from a bank. After T years, he should pay M(1 + rT ) back because of the inflation. The

compounded interest occurs if one can reinvest immediately when the money is payed back.

Suppose the money can be reinvested n times per year, the total pay back is M(1 + r/n)nT .

When n goes to infinity, the payback becomes MerT .

The forward interest rate is the interest rate in the future. The forward interest rate for

a specified future time can be determined with the current interest rate and the unbiased

expectation of the future. In comparison with the current interest rate, the forward interest

rate will be shown as after one year, the one year interest rate’ which represents the interest

rate of next year.

§ 1.2.4 Bond

The bond is a fix-income security, under which the issuer owes the holders a debt and is

obliged to pay them the coupon or to repay the principal at a later date, termed the maturity

date. The coupon can be considered as the interest. The current yield shows the annual


return calculated with the annual coupon payments and the current price of the bond. Let

the current market price be MP and the annual coupon paid be C. The current Yield is CMP

.

The current yield simply shows the return for the total year but it doesn’t include the time

variance. The yield to maturity (YTM) is introduced to show the interest rate including the

future cash flow. The expression of the bond price B is as below

B =C1

(1 + y)1+

C2

(1 + y)2+ ...+

Clast(1 + y)n

, (1.2)

where

Cn is the nth coupon payment,

B is the bond price,

y is the yield to maturity,

Clast is equal to the principal P + CN and

N is the total number of payment.

§ 1.2.5 One type of swaption: bond option

Bond option is a typical swaption that gives the holder the right to buy a bond at the maturity

date. Because both option and bond are including the future time, Bond option is invested

highly depending on the personal prospection of the future market and since the Black-Sholes

model assumes constant volatility [6], which cannot describe the bond option, the Black-Sholes

model cannot easily be used in the bond option.

§ 1.3. Introduction of financial models 8

§ 1.3 Introduction of financial models

§ 1.3.1 Lagrangian model based on supply and demand

In microeconomics, the “supply and demand” is the basic model of price determination for

commodities. It describes the market that in a competitive market, the price of any commodity

will keep changing until it reaches to a point that the quantity demanded equals the quantity

supplied.

Figure 1.2: Supply and Demand in economics

Figure copies from http://www.investopedia.com/university/economics/economics3.asp

The point (P ∗, Q∗) in figure 1.2 is the equilibrium point for one commodity. In physics,

the potential is defined based on supply and demand. The equilibrium point is the deepest

point of the potential V as showed in figure 1.3.

The potential V is designed as below


Figure 1.3: Supply and Demand as potential

V [p] = D[p] + S[p]

= m

[N∑i=1

dipaii

+N∑i=1

sipbii

]; di, si > 0 ; a, b > 0, (1.3)

where di, si, ai, bi are independent constants for each commodity. The Lagrangian is combined

with the potential term and kinetic term. The dynamics of the market are described by the

kinetic term of the Lagrangian. With the numerical analyse, the total Lagrangian is optimised

as the following

L(t) =1

2

N∑i,j=1

[Lij

∂2xi∂t2

∂2xj∂t2

+ Lij∂xi∂t

∂xj∂t

]+

N∑i=1

dipaii

+N∑i=1

sipbii (1.4)

The quantities Lij, Lij, di, si, ai, bi are all real and independent parameters. Matrix Lij is

symmetric and positive definite. Because prices (and quantities) are always positive and

hence represented by exponential variables as pi = pi0exi . The Lagrangian is given by


L(t) =1

2

N∑i,j=1

[Lij

∂2xi∂t2

∂2xj∂t2

+ Lij∂xi∂t

∂xj∂t

]+

N∑i=1

dip0i

e−aixi +N∑i=1

sip0iebixi (1.5)

The Lagrangian given in Eq. (1.5) is nonlinear.

§ 1.3.2 Black-Scholes model for option pricing

In 1973, Fisher Black, Myron Scholes, and Robert Merton gives a mathematical model for a

European option price [6], which is well known as the Black-Sholes model. This model is now

widely used in options market.

The assumptions of the Black-Scholes model are shown as the following:

• No arbitrage opportunity in the market (efficient market)

• People can borrow or lend money at a risk-free rate

• Volatility of the underlying is known and constant

• No transaction fees or dividends during the life of the option

• The log security (return of the security) follows a normal distribution

Let t be the calendar time. The underlying asset price S(t) yields

dS(t)

dt= αS(t) + σS(t)R(t), (1.6)

where α is the drift and σ is the volatility of underlying. The white noise R(t) satisfies

E[R(t)] = 0; E[R(t)R(t′)] = δ(t− t′). (1.7)


Baaquie [7] gave the derivation of Black-Scholes model using path integrals. Let x be lnS.

The Lagrangian and action are simply as below

LBS = −1

2R(t)2, (1.8)

SBS = − 1

2σ2

∫ τ

0

(dx

dt+ α

)2

. (1.9)

The boundary conditions chosen are,

x(τ) = x anddx(0)

dt= 0. (1.10)

x(0) is an integration variable and dx(0)/dt = 0 is obeyed by all paths as shown in figure

1.4.

Time t

x (

t)

t0

T

(a) forward in calendar time t

Remaining Time T−t

x (

τ)

t0

τ

(b) backward in remaining time T − t

Figure 1.4: Random paths of the security

Random paths of the security S = ex evolving in calendar time and remaining time. The randompaths are magnified near t = T to make the boundary condition more transparent.

The call price is derived with the conditional probability as the following

C(τ ;K) =e−rτ

ZBS

∫DXeSBS(ex(0) −K)+, (1.11)


where

ZBS =∞∏n=0

N∏i=1

∫ ∞−∞

dxnieSBS . (1.12)

From Eq. (1.11), Black-Scholes formula can be derived with the boundary condition as in

Eq. (1.10) and is shown as below

C = SN(d+)− e−rτKN(d−),

d± =ln(S/K) +

(r ± σ2

2

)τ

σ√τ

,

(1.13)

where K is the strike price of the option, σ is the volatility, r is the spot interest rate, and

σ√τ is the standard deviation of x. N(x) is the cumulative distribution given by

N(x) =1√2π

∫ x

−∞e−

12z2dz. (1.14)

The volatility can be calculated by historical data. Since the Black-Sholes model is recog-

nized by financial market, the market quotes the price of option in implied volatility.

§ 1.3.3 HJM Model for forward interest rate

The most widely used model in bond and interest rate field nowadays is Health-Jarraw-

Morton(HJM) model [8]. The forward interest rate f(t, x), is the prediction of interest rate

fixed by future time x > t at time t. The bond market is determined by f(t, x). Let R(t) be

Gaussian white noise and the expectation and correlation of R(t) is given by

E[R(t)] = 0 ; E[R(t)R(t′)] = δ(t− t′).


The HJM model is a linear model

∂f(t, x)

∂t= α(t, x) + σ(t, x)R(t). (1.15)

where α(t, x) is the drift and σ(t, x) is the volatility of data. A single white noise R(t) is

used to describe the forward interest rates but it cannot fully cover the evolution of forward

rate correlation. It is proper to replace the one-dimension white noise by a two-dimension

quantum field A(t, x).

The derivation of the quantum HJM model [1] is shown as below

∂f

∂t(t, x) = α(t, x) + σ(t, x)A(t, x), (1.16)

f(t∗, x) = f(t0, x) +

∫ t∗

t0

dtα(t, x) +

∫ t∗

t0

dtσ(t, x)A(t, x), (1.17)

where the drift can be calculated by martingale as the following

α∗(t, x) = σ(t, x)

∫ x

t∗

dx′D(x, x′; t)σ(t, x′). (1.18)

Chapter 2

Multiple Commodities in Statistical

Microeconomics: Model and Market

§ 2.1 Introduction

The theory of prices proposed in [9] is based on the concept of the action functional; the

subsequent publication [4] provides strong empirical evidence in support of this formulation for

the case of single commodities. The present paper extends the analysis to multi-commodities

by modifying the single commodity model in a parsimonious manner.

The theory of commodity prices [10] is one of the bedrocks of microeconomics and usually

starts with the concept of the utility function of a typical consumer [11, 12] . A maximization

of the utility function with a budget constraint yields the demand for the commodities as a

function of price. The supply function is obtained by maximizing the profit for the producers

and the market prices of commodities in conventional microeconomics are fixed by equating

supply with the demand[11, 12].

In contrast to conventional microeconomics, in statistical microeconomics [9] the prices

of all commodities are taken to be intrinsically random – and the probability distribution

function of prices is fixed by the exponential of the so called action functional. The action

14

§ 2.1. Introduction 15

functional in turn is the sum of two parts, a ‘kinetic’ term that determines the dynamical

evolution of commodity prices and a microeconomic potential that is the sum of the supply

and demand functions. The action functional contains all the information of the market and

determines the distribution of market prices as well as the change in market prices as the

prices evolve in time [13, 14, 15].

The primary focus in the statistical microeconomic formulation is to describe the unequal

time correlation functions of market prices. The auto- and cross-correlation [16, 17] functions

for multiple commodities is modeled using the action functional and the Feynman path inte-

gral. The action functional is calibrated by matching the prediction of the model’s correlation

functions with the observed market and provides a stringent test of the accuracy of the model.

The microeconomic potential for commodity with price p is given by V [p] and has been

introduced in [9]; the potential has its minimum value at its extrema p, given by

∂V [p]/∂p = 0.

The price p is taken to be the average commodity price.

What happens when the price p is not equal to the average price p, that is, p 6= p? The

microeconomic potential V [p] in this case causes the prices to ‘move’, that is, to change and

tend towards p. Clearly, the more abrupt the change, the more unlikely it is; the change of

price should, for normal market conditions, be gradual and relatively ‘smooth’. To achieve this

smooth movement of the prices in general, a ‘kinetic term’ T [p(t)] is introduced. Although

the concept of the kinetic term is taken from physics, it finds a natural expression in the

evolution of the prices of commodities: the specific form of the kinetic term is determined by

the study of market data [9]. The kinetic term in the action functional is seen to be strongly

supported by market data, and as of now has no clear theoretical explanation. One can only

speculate that demand and supply are determined by consumers and producers, respectively

§ 2.2. The microeconomic action functional 16

and that the kinetic term reflects the process of circulation, distribution and exchange – as

well as the degree of market liquidity – that is necessary for the products to make a transition

from the producer to the final consumer in the market.

One rather unexpected result is that the kinetic term in the action functional has a domi-

nant role in the evolution of commodity prices; due to the high time derivative of prices in the

kinetic term, the short term evolution of commodity prices is completely dominated by the

kinetic term, with the microeconomic potential, containing the supply and demand functions,

come into play for the long term evolution.

§ 2.2 The microeconomic action functional

Consider N commodities, with market prices given by pI ; I = 1, .., N . Prices are always pos-

itive and can be represented by exponential variables as pI = p0exI ; the normalized logarithm

of prices, denoted by xI , is defined as follow

pI = p0IexI ; xI(t) = ln(pI(t)/p0I) ; I = 1, .., N.

The demand function and the supply function are modeled to be [9]

D[p] =N∑i=1

dip0ie−aixi ; S[p] =

N∑i=1

sip0iebixi ; di, si > 0 ; a, b > 0. (2.1)

The coefficients di, si, according to [9], are determined by macroeconomic factors such as

interest rates, unemployment, inflation and so on.

For the purpose of modeling, prices in statistical microeconomics are expressed in terms

of variables that are measured from the average value and normalized by the volatility of the


stock.

yi(t) =xi(t)− xi

σi; i = 1, .., N. (2.2)

xI and σI are the average value of yI . The volatility of xI(t) for the time period being

considered and are given by

xi = E[xi] ; σ2i = E[

(xi − xi

)2].

The normalized variables yi are all of O(1) and hence one can model and compare commodities

with vastly different volatilities and prices. In the statistical microeconomic approach, the

microeconomic potential is the fundamental quantity that combines supply and demand by

considering their sum [9]. The supply and demand yield the microeconomic potential given

by

V =N∑i=1

[dip0ie

aixie−aiσiyi + sip0ie−bixiebiσiyi

]≡

N∑i=1

[die−aiyi + sie

biyi], (2.3)

where

di = dip0ieaixi ; si = sip0ie

−bixi ; ai = aiσi ; bi = biσi.

For the case of multiple commodities, the microeconomic potential for the N -commodities

is further generalized by including a term that depends on the product of the prices of com-

modities – and which cannot be placed either in the demand or in the supply component of


the microeconomic potential. The multiple commodity microeconomic potential is given by

V [p] = D[p] + S[p] + corrrelation term

=N∑i=1

[die−aiyi + sie

biyi]

+1

2

N∑ij;i 6=j

∆ijyiyj. (2.4)

The ∆ij term is introduced to model the cross-correlation of the different commodities.

The motivation for the ∆ij term is the following. The fit for the single commodity using

the microeconomic potential is very accurate [4]. Hence, one would expect that the effect of

multiple commodities should be a perturbation on the single commodities potential. This is

the reason that the simplest modification of the single commodity microeconomic potential is

used for modeling multiple commodities, and for consistency we expect ∆ij to be small.

The dynamics of the prices for N -commodities is determined by the kinetic term T [p(t)]

that, in general, is given by

T [p(t)] =1

2

N∑i,j=1

[Lij

∂2yi∂t2

∂2yj∂t2

+ βij∂yi∂t

∂yj∂t

].

Similar to the reason that led to modeling the cross-correlations by the ∆ij term in the

microeconomic potential V , we continue to model the kinetic term to be solely determined by

the single commodity, with all the correlation coming from the ∆ij term. Hence, the kinetic

term is chosen to be diagonal, with no cross-terms amongst the different commodities and is

given by

T [p(t)] =1

2

N∑i

[Li

(∂2yi∂t2

)2

+ Li

(∂yi∂t

)2]. (2.5)


The Lagrangian is given by the sum of the kinetic and potential factors and yields [9]

L(t) = T [p(t)] + V [p(t)].

The Lagrangian, from Eqs. (2.4) and (2.5), is the following

L(t) =1

2

N∑i

[Li

(∂2yi∂t2

)2

+ Li

(∂yi∂t

)2]

+N∑i=1

[die−aiyi + sie

biyi]− 1

2

N∑ij;i 6=j

∆ijyiyj. (2.6)

The Lagrangian given in Eq. (2.6) is nonlinear.

The action functional determines the dynamics (time evolution) of market prices and is

given by

A[p] =

∫ +∞

−∞dtL(t) =

∫ +∞

−∞dt(T [p(t)] + V [p(t)]

).

All prices of commodities are considered to be stochastic variables and the action functional

is assumed to determine the probability distribution, which is given by

Probability distribution for a specific time evolution ∝ e−A[y]

All correlation functions of the prices are given by the Feynman path integral [9, 3]

D123...n(t1, t2, ...tn) = E[y1(t1)y2(t2) · · · yn(tn)] =1

Z

∫Dye−A[y]y1(t1)y2(t2) · · · yn(tn)

with

Z =

∫Dye−A[y].

§ 2.3. Correlation Function 20

§ 2.3 Correlation Function

We study the leading terms in the Lagrangian by doing a Taylor expansion of the potential

term V about its minima, which will turn out to coincide with an expansion of V in a power

series in yi.

The minima xi is defined by

∂V(x)

∂xi= 0.

Hence from Eq. (2.4)

∂V(x)

∂xi= −aidip0ie

aixi + bisip0iebixi −

∑j,i6=j

∆ij(xj − xjσj

) = 0. (2.7)

In our model, we assume that the equilibrium price of the commodities x is given by its

average value x and yields

xi = xi. (2.8)

Hence, from Eq. (2.7)

− aidieaixi + bisiebixi = 0. (2.9)

Note that Eq. (2.9) is independent of p0i and hence p0i does not enter the calibration of the

model’s parameters. Eq. (2.9) yields

exi =

(aidi

bisi

)(1/(ai+bi))

. (2.10)

Eqs. (2.3), (2.8) and (2.10) yield

aidi = bisi. (2.11)


§ 2.3.1 Expansion of Potential

From the definition of yi given in Eq. (2.2), the minima of the action is about yi = 0. Hence,

expanding the Lagrangian about yi = 0 yields

L =∑i

(1

2Liyi

2 +1

2Liyi

2 +γi2y2i +

αi3!y3i +

βi4!y4i + · · · )− 1

2

∑ij,i6=j

∆ijyiyj.

Define the Lagrangian in terms of the quadratic and nonlinear terms as follow

L = L2 + L3 + L4 +O(y5).

L2(x) are the quadratic terms in the expansion of the Lagrangian given above and L3(x),L4(x)

are the cubic and quartic terms.

The quadratic Lagrangian is given by

L2 = L0 + Lc,

L0 =1

2

∑i

[Liyi

2 + Liyi2 + γiy

2i

]; Lc = −1

2

∑ij;i 6=j

∆ijyiyj,

and the nonlinear terms are

L3 =αi3!y3i ; L4 =

βi4!y4i .

The action is given by the following

A = A0 +Ac +AI =

∫dtL;

A0 =

∫dtL0 ; Ac =

∫dtLc

AI =

∫dt(L3 + L4).


From above we have

γi =1

2(dia

2i + sib

2i ), (2.12)

αi = (−a3i di + b3

i si) = (bi − ai)γi, (2.13)

βi = (a4i di + b4

i si) = (a2i − aibi + b2

i )γi. (2.14)

The linear term in yi is zero due to Eq. (2.11). We will determine the values of α, β, γ, y from

market data; the potential parameter of ai, bi, si, di are then given by the following

ai =±√

4βiγi − 3α2i − αi

2γi; bi = ai +

αiγi

;

si =γi

bi(ai + bi); di =

γiai(ai + bi)

.

The positive branch for ai is used since ai > 0.

§ 2.3.2 Auto-correlation

The correlation function for the A0 is given by the Gaussian propagator

D(0)(t− t′) =1

Z

∫Dye−A0[y]yI(t)yJ(t′).

and the auto-correlation function is given by

D(0)II (t− t′) ≡ D

(0)I (t− t′) =

1

Z

∫Dye−A0[y]yI(t)yI(t

′) +O(∆2).


Using a Fourier transform to evaluate the propagator for the prices, and dropping the

subscript I, yields

D(0)(t− t′) ≡∫ ∞−∞

dk

2π

eik(t−t′)

Lk4 + Lk2 + γ=

e−√a−|t−t′|√a−

− e−√a+|t−t′|√a+

2L(a+ − a−);

a± =L

2L± | L

2L|

√1− 4Lγ

L2.

Case I: Real branch. 4Lγ < L2 and a± is real; let

ω = (γ

L)14 , a± =

√γ

Le±2ϑ, e±2ϑ =

√L2

4Lγ+

√L2

4Lγ− 1.

Hence D(0)(t− t′) is given by

D(0)(t− t′) =ωe−ω|t−t

′| cosh(ϑ)

2γ sinh(2ϑ)sinh[ϑ+ ω|t− t′| sinh(ϑ)].

Case II: Complex branch. 4Lγ > L2 and a± are complex; let

ω = (γ

L)14 , a± =

√γ

Le±i2φ, cos(2φ) =

√L2

4Lγ, sin(2φ) =

√1− L2

4γL. (2.15)

We hence obtain the complex branch propagator

D(0)(t− t′) =ωe−ω|t−t

′| cos(φ)

2γ sin(2φ)sin[φ+ ω|t− t′| sin(φ)].

Define the normalization constant

N =ω

2γ sin 2φ.


and yields the complex branch propagator

D(0)(t− t′) = N e−ω|t−t′| cos(φ) sinφ+ ω|t− t′| sin(φ). (2.16)

The auto-correlation function of commodities will be seen to follow the behavior given by the

complex branch. The real branch cannot describe the data from market.

§ 2.3.3 Cross-correlation

The cross-correlation function is given by I 6= J. The model yields

E[yI(0)yJ(τ)] = DIJ(t) =1

Z

∫Dye−(A0+Ac)yI(0)yJ(τ)

=1

Z

∫Dye−A0[y]yI(0)yJ(τ)

[1 +

1

2

∑ij;i 6=j

∆ij

∫dtyi(t)yj(t) +O(∆2)

].

The first term is zero and hence

DIJ(τ) ' D(0)IJ (t), (2.17)

where

D(0)IJ (t) ≡

∫ ∞−∞

D(0)I (t)D

(0)J (t− τ)dt.

From Appendix Eq. (2.28):

D(0)IJ (t) =

C

LILJ

( 1

αe−|t|α cosφ[

1

R(h1/R) cos φ− (h2/R) sin φ − 1

T(h3/T ) cos φ− (h4/T ) sin φ]

+1

βe−|t|βcosθ[

1

P(h5/P ) cos θ − (h6/P ) sin θ − 1

Q(h7/Q) cos θ − (h8/Q) sin θ]

),

(2.18)


with

C =−1

4

1

α2β2 sin 2φ sin 2θ; φ = φ+ |t|α sinφ; θ = θ + |t|β sin θ.

Coefficients h1− h8, P,Q,R, T are given in Eq. (2.31)

Figures 2.1 and 2.2 below are plots of the cross-correlator for some typical values of the

model’s parameter of the complex branch. The shape of the cross-correlator given by the

model will be seen to be consistent with the result obtained by fitting the model to market

prices.

0 20 40 60 80 100 120 140 160 180 200−4

−3

−2

−1

0

1

2

3

4

5

6x 10

6

α=0.1β=0.15φ=30θ=20

Figure 2.1: α = 0.1, β = 0.15, φ = 30, θ = 20

0 20 40 60 80 100 120 140 160 180 200−1

−0.5

0

0.5

1

1.5

2

2.5x 10

6

α=0.1β=0.15φ=20θ=20

Figure 2.2: α = 0.1, β = 0.15, φ = 20, θ = 20

§ 2.3.4 Nonlinear terms

As discussed in detail in [4], the correlation function to leading order for the nonlinear coupling

yields

E[y2I (t)]c = D

(0)I (0)− βI

2D

(0)I (0)

∫dz(D

(0)I (z))2 +O(∆2), (2.19)

E[y3I (t)]c = −2αI

∫ ∞0

dz(D(0)I (z))2 +O(∆2), (2.20)

E[y4I (t)]c = 3(D

(0)I (0))2 − 2βI

∫ ∞0

dz(D(0)I (z))4 +O(∆2). (2.21)

§ 2.4. Market data and model 26

Some integrations that are useful to solve the potential prameters a, b, s, d are the following

[3] ∫ ∞0

D(0)(τ)dτ = N sin 2φ

ω,

∫ ∞0

(D(0)(τ))2dτ = N 2 secφ− cos 3φ

4ω,

∫ ∞0

(D(0)(τ))3dτ = N 3 2 sin3 φ(11 cosφ+ 2 cos 3φ)

4ω,

∫ ∞0

(D(0)(τ))4dτ = N 4 sinφ3(50 cos 2φ+ 6 cos 4φ+ 47) tanφ

16ω(3 cos 2φ+ 5).

Using four equations 2.11, 2.19, 2.20 and 2.21, potential parameters ai, bi, si, di can be ob-

tained.

§ 2.4 Market data and model

The empirical correlator is denoted by the notation of GIJ(t) and is defined by the expectation

value of the market prices. For a time series data set with time interval of ε, the prices are

given by yI(t) = yI(n), where t = nε and we have τ = kε; for N data points, the correlator is

given by the moving average

GIJ(τ) = GIJ(k) = E[yI(0)yJ(k)]c

∣∣∣market

=1

N

N−k∑n=0

yI(n)yJ(n+ k).

The numerical evaluation of the correlators is obtained by taking the moving average over the

data set. The model always yields a correlator that is a symmetric function of IJ , which is

not necessarily the case for the market correlator [9]. To equate the market correlator with

the model, it is made symmetrically, namely

GIJ(τ) = GJI(τ).

§ 2.4. Market data and model 27

and in terms of the underlying data we have

GIJ(τ) =1

2[

1

N

N−k∑n=0

yI(n)yJ(n+ k) +1

N

N−k∑n=0

yJ(n)yI(n+ k)].

The parameter of time for the market and model are not the same. The reason being that

time for traders is determined by the liquidity of the market and rate of transactions [7]. To

reflect this feature of the market, define market time z(τ) by

τ → z(τ).

The empirical correlator G(τ) is given by the exact model correlator D(τ) by the relation

GIJ(t) = DIJ(z(t)). (2.22)

For single commodity fit

D(0)II (z(t)) = N e−ωIz(t) cos(φI) sinφI + ωIz(t) sin(φI).

For cross-correlation DIJ , we use Eq. (2.18) to fit the market cross-correlator.

According to Ref. [4], for single commodity fit, using GII(t) ≡ GI(t), we have

GI(t) = D(0)I (z(t))− βI

2D

(0)I (0)

∫dτD

(0)I (z(t)− τ)D

(0)I (τ) +O(∆2). (2.23)

Let∫dτ(D

(0)I (τ))2 = CI . When τ equals to zero,

GI(0) ' D(0)I (0)− βI

2D

(0)I (0)CI = D

(0)I (0)(1− βI

2CI). (2.24)

§ 2.5. Fitting with Market Data 28

For the auto-correlation GII , note that the empirical definition of x and σ implies that

GI(0) = 1.

Hence

1 = GI(0) = D(0)I (0)− βI

2D

(0)I (0)CI ⇒ D

(0)I (0) =

1

1− βI2CI. (2.25)

Eq. (2.25) is consistent with Eq. (2.24). Substituting D(0)I (0) into GI(t), for t > 0 – using

Eq. (2.16) for the numerator and denominator – we obtain1

GI(t) =D

(0)I (z(t))

D(0)I (0)

=1

sinφe−ωz(t) cos(φ) sinφ+ ωz(t) sin(φ). (2.26)

We use D(0)I (z(t))/D

(0)I (0) to fit GI(t) in the single commodity case. Hence, once we have

obtained φI , ωI all the parameters of the complex branch of the model can be determined.

§ 2.5 Fitting with Market Data

As a rule, the correlators evaluated from the data are denoted by GIJ and the result obtained

by fitting the model are denoted by DIJ . The numbering for the indices I, J for the various

commodities is the one given in Table 2.1.

We analyze 18 commodities daily data drawn from four major groups of energy, metal,

food, grain, from 2014/01/01 to 2015/02/03 download from

Investing.com/commodities/real-time-futures.2 Since the correlators are symmetric, there are

1The approximation makes the result consistent with the value of GI(0) = 1.2Real-time streaming quotes for the top commodities futures CFDs. The quotes are available for a variety

of futures such as Gold, Crude Oil, Silver, Copper and many more Metals, Energies and Softs futures. Thelatest price as well as the daily high, low and the change for each future. The ”Base” price is the last close of

http://www.investing.com/commodities/real-time-futures


153 correlators in total, with 18 auto-correlation functions and 153 cross-correlation functions.

Table 2.1: Number and Type of Commodity

Number commodity Type Number commodity Type

1 Crudeoil Energy 10 Cocoa Food

2 Heatingoil 11 Soybeansoil

3 Brentoil 12 Orangejuice

4 Natural gas 13 Livecattle

5 Copper Metal 14 Wheat Grain

6 Gold 15 Corn

7 Silver 16 Soybean

8 Platinum 17 Roughrice

9 Palladium 18 Cotton Misc

All the auto-correlation functions can be fit to a high degree of accuracy and confirms the

results found in [4] for single commodities.

The model can fit the majority of the cross-correlators Gij(i 6= j) which are generically

similar to the shape that the model generates from Eq. (2.18) and shown in Figures 2.1 and

2.2. Of the 153 cross-correlators, 110 are of the shape that the model can fit quite well. The

rest of the cross-correlators Gij have features that the model cannot fit; in particular, if the

each future contract (as of 16:30 ET). The change is calculated from the ”Base” price.


cross-correlator has a maximum value at a time lag that is non-zero, then there are no choice

of parameters for Gij that can fit the cross-correlator.

The fitting is based on Eqs. (2.17) and (2.22)

GIJ(τ) = DIJ(z(τ)) ; I 6= J.

To leading order we approximate DIJ(t) by the Gaussian approximation D(0)IJ (t) given in

Appendix § 4.11.1 and obtain

GIJ(τ) ' D(0)IJ (z(τ)) = ∆IJ

∫ ∞−∞

dtD(0)I (t)D

(0)J (t− z(τ)) ; I 6= J.

Hence, the cross-correlation coupling ∆IJ is given by

GIJ(0) = ∆IJ

∫ ∞−∞

dtD(0)I (t)D

(0)J (t) ; I 6= J. (2.27)

The empirical cross-correlation of 18 commodities has been studied. Anticipating results

derived later on in the paper, the parameters ∆IJ are given in Figure 2.3 for all the cross-

correlators.


5

10

15

0

5

10

15

20

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Figure 2.3: Matrix of ∆ij for 18 commodities. Note that for all pairs, |∆IJ | < 0.08 but onecase.

The fitting of market correlator to the model is done in the following.

• Simultaneously fit all the auto-correlation functions DII(z(t)) = DI(z(t)) with the G(t),

as in Eq. (2.26), and evaluate the parameters ω, φ, η, λ. The value of η, λ does not enter

into the calibration of the other parameters.

• Using the values of ω, φ, as in Eq. (2.26), and the nonlinear terms given Eq.(2.19),

(2.20), (2.21), we evaluate L, L, γ, α and β.

• Use the parameters L, L, γ, α and β as in Eq. (2.11), (2.12), (2.13), (2.14) and find [a,

b, s, d].

• Evaluate the cross-correlation function DIJ ; I 6= J and determine ∆IJ .

• The correlators are fitted for a maximum time of lag τ = 200 days.

Note the remarkable fact, that ignoring one cross-correlator, the value of all the ∆IJ ’s is

such that |∆IJ | < 0.08. The fact |∆IJ | < 0.08 provides strong evidence of the correctness


of our approach of considering the multi-commodities model as a perturbation of the single

commodities, which requires |∆IJ | << 1.

Note in extending the statistical model from single to N commodities, N(N − 1)/2 new

parameters ∆IJ are introduced, which in turn are fixed by only a single value of the cross-

correlators GIJ(0), as given in Eq. (2.27).

For our model, the entire dependence of GIJ(τ) for time lag τ > 0 is determined by the

auto-correlation functions D(0)I (z(τ)) and its convolution with itself – as given in Eq. (2.17).

The fact that the model can accurately describe all the auto- and cross-correlators, up to

N = 4 commodities, provides evidence for the correctness of the model. We will discuss

later how the model can be extended to accurately describe a collection of commodities with

arbitrary N .

Calibration for cocoa

1. Use cross-correlation Eq. (2.18) and auto-correlation Eq. (2.16) to fit each data to find ω

φ, λ, and η.

2. Substitute ω, φ, λ, and η into Eq. (2.19), (2.20), (2.21) and (2.15) to get γ, L, L, αI , βI

3. Using the value of all parameters obtained above as in Eq. (2.11), (2.12), (2.13), (2.14),

we can determine the indices [a, b, s, d] [4] of supply and demand function. The result obtained

in Table § 2.6.2 for cocoa from the fitting of the Crude oil, Platinum and Cocoa group is used

§ 2.6. Fits for GII , GIJ 33

below to illustrate the fitting procedure.

ω = 0.1535, φ = 1.2229, λ = 0.700, η = 0.54

⇒ γ = 0.113, L = 203.37, L = −7.35, α = 0.0199, β = 0.0747

⇒∫ ∞

0

dτG(τ) = 4.4413;

∫ ∞0

dτG2(τ) = 6.9973;∫ ∞0

dτG3(τ) = 4.5490;

∫ ∞0

dτG4(τ) = 4.2909

⇒ a = 0.72, b = 0.90, s = 0.094, d = 0.075

§ 2.6 Fits for GII , GIJ

All the fits of the model with data are carried out using the equations for the correlation

functions given in Eqs. (2.16) and (2.18). All auto-correlators are normalized such that

GII(0) = DII(0) = 1. However, for displaying the results clearly, the entire auto-correlator

is rescaled – such that sometimes we scale to GII(0) > 1 or GII(0) < 1 – so that the results

for the different commodities do not overlap and can be viewed clearly. Similarly, the cross-

correlators are also scaled and for four or more commodities are also shifted to avoid an overlap

of the graphs.

§ 2.6.1 Two commodities

Any two commodities, from different types as given in Table 2.1, can be fit to a high degree

of accuracy. The fit is even better if the two commodities belong to the same type. Figure

2.4 shows the fit for Gold and Silver.


0 20 40 60 80 100 120 140 160 180 200−3

−2

−1

0

1

2

3

4

5

R2

11=0.86918

R2

22=0.91679

R2

12=0.84929

η=0.87151

λ=2.0011

Gij(fit)

Gij(data)

Silver

Gold

G(ii)

Gjj

Figure 2.4: Silver and Gold with η = 0.7; λ = 0.1004

The parameters from the auto-correlater are the following

Table 2.2: Gold-Silver. η = 0.7; λ = 0.1004

Two commodities fit R2 γ L L α β

Gold(G11) 0.837 0.136 20.81 -1.52 0.0102 0.257

Silver(G22) 0.863 0.126 37.13 -2.32 -0.0320 0.230

G12 ∆12 = 0.034

The three correlators for gold and silver are fitted well, with R2 given by the following

R2 =

R2G11

R2G12

* R2G22

=

0.837 0.923

* 0.863


§ 2.6.2 Three commodities fit

We study commodities from the same group and from different groups as well.

Three commodities in same group

The figures for three commodities in one group are given below.

Crude oil-Heating oil-Brent oil:

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

6

8

10

Crudoil(G11)

Heatoil(G22)

Brentoil(G33)

D11

D22

D33

0 20 40 60 80 100 120 140 160 180 200−10

0

10

20

30

40

50

D13(fit)

G13(data)

D23(fit)

G23(data)

D12(fit)

G12(data)

(a) (b)

Figure 2.5: Crude oil-Heating oil-Brent oil (a)Autocorrelation and (b)Crosscorrelation withη = 0.7; λ = 0.775

Table 2.3: Crude oil-Heating oil-Brent oil. η = 0.7; λ =

0.775

Three commodities fit R2 γ L L α β

Crude oil(G11) 0.804 0.0539 171.4 -1.434 0.0613 0.0813

Heating oil(G22) 0.797 0.0537 158.6 -1.182 0.0677 0.0688


Brent oil(G33) 0.798 0.0536 167.4 -1.329 0.0638 0.0738

GIJ ∆12 = 0.032 ∆13 = 0.031 ∆23 = 0.032

R2 =

0.804 0.918 0.921

* 0.797 0.923

* * 0.798

Orange juice-Cattle-Soybean:

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

6

8

10

Orangejuice(G11)

Cattle(G22)

Soybean(G33)

D11

D22

D33

0 20 40 60 80 100 120 140 160 180 200−30

−20

−10

0

10

20

30

40

D13(fit)

G13(data)

D23(fit)

G23(data)

D12(fit)

G12(data)

(a) (b)

Figure 2.6: Orange juice-Cattle-Soybean (a)Autocorrelation and (b)Crosscorrelation with η =0.7;λ = 1.132

Table 2.4: Orangejuice-Cattle-Soybean. η = 0.7;λ =

1.132



Orange juice(G11) 0.609 0.0512 116.06 0.0032 -0.0049 0.0057

Cattle(G22) 0.733 0.0477 236.95 -1.48 0.0290 0.0776

Soybean(G33) 0.685 0.0425 225.14 -0.308 -0.0576 0.0595

GIJ ∆12 = −0.030 ∆13 = 0.021 ∆23 = −0.019

R2 =

0.609 0.875 0.727

* 0.733 0.781

* * 0.685


Gold-Silver-Platinum:

0 20 40 60 80 100 120 140 160 180 200−5

0

5

10

Gold(G11)

Silver(G22)

Plati(G33)

D11

D22

D33

0 20 40 60 80 100 120 140 160 180 200−10

0

10

20

30

40

50

D13(fit)

G13(data)

D23(fit)

G23(data)

D12(fit)

G12(data)

(a) (b)

Figure 2.7: Gold-Silver-Platinum (a)Autocorrelation and (b)Crosscorrelation with η = 0.7;λ = 0.344

Table 2.5: Gold-Silver-Platinum. η = 0.7; λ = 0.344


Gold(G11) 0.827 0.0908 58.6 -1.863 0.0071 0.179

Silver(G22) 0.803 0.0752 88.1 -4.17 -0.0217 0.159

Platinum(G33) 0.799 0.0726 112.1 -2.26 -0.0320 0.180

GIJ ∆12 = 0.033 ∆13 = 0.031 ∆23 = 0.025


R2 =

0.827 0.931 0.910

* 0.803 0.895

* * 0.799

Three commodities from different groups

Crude oil-Platinum-Cocoa:

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

6

8

10

Crudoil(G11)

Plati(G22)

Cocoa(G33)

D11

D22

D33

0 20 40 60 80 100 120 140 160 180 200−15

−10

−5

0

5

10

15

20

25

30

35

G13(fit)

G13(data)

G23(fit)

G23(data)

G12(fit)

G12(data)

(a) (b)

Figure 2.8: Crude oil-Platinum-Cocoa (a)Autocorrelation and (b)Crosscorrelation with η =0.70; λ = 0.54.

Table 2.6: Crude oil-Platinum-Cocoa. η = 0.70; λ = 0.54


Crude oil(G11) 0.871 0.0560 286.3 -3.54 0.0566 0.0742

Platinum(G22) 0.835 0.0585 305.2 -4.17 -0.0241 0.1346


Cocoa(G33) 0.920 0.113 203.4 -7.35 0.0199 0.0747

GIJ ∆12 = 0.021 ∆13 = 0.045 ∆23 = 0.023

R2 =

0.871 0.612 0.920

* 0.835 0.943

* * 0.920

The R2 of three commodities fit are normally between 0.8-1. Although they are not quite

high, the values are high enough to be convincing.

§ 2.6.3 Four commodities

When we consider taking more commodities into the fit, such as 4 and 6 commodities, the fit

is not so good.


0 20 40 60 80 100 120 140 160 180 200−30

−25

−20

−15

−10

−5

0

Silver(G11)

Gold(G22)

Crudeoil(G33)

Gas(G44)

D11

D22

D(33)

D(44)

0 20 40 60 80 100 120 140 160 180 200−20

0

20

40

60

80

100

D14(fit)

G14(data)

D24(fit)

G24(data)

D34(fit)

G34(data)

D13(fit)

G13(data)

D23(fit)

G23(data)

D12(fit)

G12(data)

(a) (b)

Figure 2.9: Gold-Silver-Crude oil-Natural gas (a)Autocorrelation and (b)Crosscorrelation withη = 0.70; λ = 0.260.

Four commodities fit R2 γ L L α β

Gold(G11) 0.834 0.0973 54.9 -2.051 0.0074 0.186

Silver(G22) 0.827 0.0835 105.2 -2.940 -0.0220 0.159

Crude oil(G33) 0.749 0.0769 35.3 -0.041 0.0962 0.128

Natural gas(G44) 0.377 0.0667 81.8 -0.921 0.0850 0.015

Table 2.7: Gold-Silver-Crude oil-Natural gas. η = 0.70;

λ = 0.260.


R2 =

0.834 0.885 0.507 0.074

* 0.827 0.725 0.435

* * 0.749 0.809

* * * 0.377

As can can be seen from Table 2.7, the R2 of four commodities fit goes down to 0.7-0.9 and

for some is in the range 0.4-0.7. So the fit seems to fail to give a good result when we consider

four commodities.

§ 2.6.4 Six commodities

0 20 40 60 80 100 120 140 160 180 200−30

−20

−10

0

10

20

30

40

Gold(G11)

D(11)

Silver(G22)

D(22)

Crudoil(G33)

D(33)

Gas(G44)

D(44)

Soyboil(G55)

D(55)

Cattle(G66)

D(66)

0 20 40 60 80 100 120 140 160 180 200−40

−20

0

20

40

60

80

D12fitG12dataD13fitG13dataD14fitG14dataD15fitG15dataD16fitG16dataD23fitG23dataD24fitG24dataD25fitG25dataD26fitG26dataD34fitG34dataD35fitG35dataD36fitG36data’D45fitG45data’D46fitG46data’D56fitG56data

(a) (b)

Figure 2.10: Gold-Silver-Crude oil-Natural gas-Soybean oil-Cattle (a)Autocorrelation and(b)Crosscorrelation with η = 0.70; λ = 0.699.

Table 2.8: Gold-Silver-Crude oil-Natural gas-Soybean

oil-Cattle. η = 0.70; λ = 0.699.


Six commodities fit R2 γ L L α β

Gold(G11) 0.825 0.0794 92.02 -2.259 0.0062 0.155

Silver(G22) 0.566 0.0637 15.36 1.948 -0.0269 0.203

Crude oil(G44) 0.512 0.0863 6.09 1.448 0.1351 0.184

Natural gas(G44) 0.443 0.0520 185.1 -1.394 0.0654 0.0118

Soybean oil(G55) 0.447 0.116 2.51 1.078 -0.1718 0.268

Cattle(G66) 0.447 0.198 2.270 1.050 0.0987 0.2717

R2 =

0.825 0.935 0.281 0.011 0.265 0.727

* 0.566 0.573 0.483 0.291 0.701

* * 0.512 0.789 0 0.033

* * * 0.443 0 0.064

* * * * 0.447 0

* * * * * 0.447

The R2 of six commodities fit can not be considered to be a good fit.

§ 2.7. Comparison of single and multiple commodities fit 44

§ 2.7 Comparison of single and multiple commodities fit

There are two different procedures for doing the fit for a given commodity: by using the

single commodity data and the other by doing a fit for multiple commodities. The purpose

of this exercise is to ascertain how much do the prices of other commodities affect a given

commodity’s prices. We also need to verify that the multi-commodity prices are in fact a

perturbation to the single commodity fit of the prices of a given commodity.

• A given commodity is calibrated using only the data for the given commodity. This is

denoted by S-commodity in Table 2.9.

• The same commodity is calibrated by doing a fit for the commodities for all the groups

and maximizing the goodness of fit for this collection of commodities, including choosing

the best value for the cross-correlation parameters ∆ij’s. This fit is denoted by M-

commodity in Table 2.9.

• In Table 2.9, Group A is a simultaneous fit for three commodities, with two commodities

from metal and the other from energy. In Group B of Table 2.9, three commodities

from different groups, namely one from energy, one from metal and one from food are

calibrated using our model.

• The single commodity fit has an R2 above 0.94 whereas when the multiple commodity

fit is done, R2 is still good but now about 0.80.

Most of the parameters between single commodity fit and multiple commodities fit are

similar. The potential constants a, b, s, d given in Eq.(2.1), as well as the market time param-

eters η, λ changes by about 10% in going from the single to the three commodities fit. This

verifies our intuition that the price of a single commodity is mostly, up to 90%, determined

by its own dynamics, with the other commodities being a perturbation on its prices. Hence,

§ 2.7. Comparison of single and multiple commodities fit 45

while it is true that single commodity fit is more accurate than the multiple fit, the multiple

commodity fit contains the influence from other commodities and hence reflects the market

more accurately.

Table 2.9: Comparison of Single-Commodity fit(S-) with

Multiple-Commodities fit(M-). Group 1 is Gold-Silver-

Crude oil (GSC) and Group 2 is Crude Oil-Platinum-

Cocoa (CPC)

Group1

GSC R2 γ L L α β η λ a b s d

M-Gold 0.78 0.076 111.7 -2.16 0.056 0.073 0.7 0.41 0.38 1.1 0.046 0.14

S-Gold 0.95 0.056 22.6 2.24 0.041 0.059 0.7 0.83 0.44 1.2 0.030 0.08

M-Silver 0.83 0.063 59.7 -2.30 0.006 0.146 0.7 0.41 1.5 1.6 0.013 0.014

S-Silver 0.94 0.056 168.1 -1.68 -0.021 0.127 0.7 0.1 1.7 1.3 0.015 0.012

M-Crudeoil 0.80 0.060 350.1 -4.97 -0.018 0.131 0.7 0.41 1.5 1.5 0.014 0.013

S-Crudeoil 0.99 0.058 144.6 -1.45 0.020 0.107 0.72 0.1 1.2 1.5 0.015 0.019

Group2

CPC R2 γ L L α β η λ a b s d

M-Crudeoil 0.75 0.056 286.3 -3.54 0.057 0.074 0.7 0.54 1.1 1.2 0.020 0.022

§ 2.8. Conclusion 46

M-Platinum 0.90 0.059 305.2 -4.17 -0.024 0.135 0.7 0.54 1.5 1.5 0.013 0.013

S-Platinum 0.94 0.056 167.7 -1.663 -0.043 0.098 0.7 0.1 1.7 1.3 0.016 0.012

M-Cocoa 0.83 0.11 203.4 -7.35 0.020 0.075 0.7 0.54 0.72 0.90 0.094 0.075

S-Cocoa 0.94 0.061 193.4 -2.801 0.016 0.062 0.76 0.51 0.85 1.11 0.028 0.037

In conclusion, the result encoded in Table 2.9 supports our basic premise that the multiple-

commodity behaviour of the market should be considered to be a perturbation on the prices

of single commodities.

§ 2.8 Conclusion

The theory of commodity prices needs to explain the behavior of all commodities, including

their cross-correlations, and the action functional based statistical microeconomic modeling

must provide such a description. The study of multiple commodities provides empirical evi-

dence supporting the approach of statistical microeconomics. The fits have R2 ≈ 0.8 for up

to three commodities, which is reasonable but not excellent.

One of the main empirical result of this chapter is that the market of single commodities

can be viewed as being partially complete, with other commodities affecting the price of any

given commodity only perturbatively, with correlation terms contributing less than 10% to

the price of a single commodity. This result provides an explanation for the excellent results

obtained for the single commodities considered in isolation, as was obtained in [4].

Note a minimal extension of the single commodity action was made, motivated by the need

to preserve the accurate results for the single commodities. One can improve the accuracy of

§ 2.9. Appendix 47

the model by including cubic, quartic and higher order terms of prices and involving different

commodities. This would make the calibration more difficult, but would have the advantage

of being able to simultaneously fit a large number of commodities.

The results of this chapter place the statistical microeconomic theory of commodity prices

on a firm footing. The significance of the various terms in the action functional in terms of

the functioning of the underlying economy need further study.

Future research can study after aspects of market prices. One is for some extreme cases in

the history of the market. For example, the prices of all precious metals increased together in

1976-1980. This created a high cross-correlation between them. To see such a peak value of

cross-correlation, one must use a long time interval, which the model in this chapter cannot

achieve. The other interesting aspect is to investigate the time delay between the behaviour

of two different commodities. It may happen that commodity A triggers a delayed change in

the price of commodity B.

§ 2.9 Appendix

The cross-correlation function is evaluated analytically and a few consistency checks are made

by reducing it to special cases obtained earlier for the single commodity auto-correlator.

§ 2.9.1 Derivation of D(0)IJ

The Gaussian propagate is given by

D(0)IJ (t) =

∫ ∞−∞

dτD(0)I (τ)D

(0)J (t− τ)

=

∫ ∞−∞

dk′

2π

∫ ∞−∞

dk

2π

∫ ∞−∞

dτeikτ

LIk4 + LIk2 + γI

eik′(t−τ)

LJk′4 + LJk′4 + γJ(2.28)

§ 2.9. Appendix 48

Performing two integration yields

D(0)IJ (t) =

1

LILJ

∫ ∞−∞

dk

2π

eikt

(k2 + λ2)(k2 + λ2∗)(k

2 + ω2)(k2 + ω2∗)

with

λ2 =LI2LI

(1 +

√1− 4γILI

LI2

), λ2∗ =

LI2LI

(1−

√1− 4γILI

LI2

),

ω2 =LJ2LJ

(1 +

√1− 4γJLJ

LJ2

), ω2∗ =

LJ2LJ

(1−

√1− 4γJLJ

LJ2

)Define

D(0)IJ (t) =

1

LILJζ(t)

Then

ζ(λ, ω, t) =

∫ ∞−∞

dk

2π

eikt

(k2 + λ2)(k2 + λ2∗)(k

2 + ω2)(k2 + ω2∗)

(2.29)

λ2 = α2e2iφ ; ω2 = β2e2iθ ; λ2∗ = α2e−2iφ ; ω2

∗ = β2e−2iθ

Hence

ζ(λ, ω, t) =1

(λ2 − λ∗)(ω2 − ω2∗)

∫ ∞−∞

dk

2πeikt(

1

k2 + λ2∗− 1

k2 + λ2)(

1

k2 + ω2∗− 1

k2 + ω2)

Define the normalization constant

C =1

(λ2 − λ2∗)(ω

2 − ω2∗)

=−1

4

1

α2β2 sin 2φ sin 2θ

Note the identity

I(z) =

∫ ∞−∞

dk

2π

eikt

k2 + z2=

1

2ze−|t|z

§ 2.9. Appendix 49

Thus

C−1ζ(λ, ω, t) = I(λ)(1

λ2 − ω2∗− 1

λ2 − ω2) + I(λ∗)(

1

λ2∗ − ω2

− 1

λ2∗ − ω2

∗)

+I(ω)(1

ω2 − λ2∗− 1

ω2 − λ2) + I(ω∗)(

1

ω2∗ − λ2

− 1

ω2∗ − λ2

∗)

(2.30)

We make the following definition

C−1ζ(λ, ω, t) = ζ1(λ, ω, t) + ζ1(ω, λ, t)

where

ζ1(λ, ω, t) = I(λ)(1

λ2 − ω2∗− 1

λ2 − ω2) + C.C.

and ζ1(ω, λ, t) = I(ω)(1

ω2 − λ2∗− 1

ω2 − λ2) + C.C.

Define

h1 = α2 cos 2φ− β2 cos 2θ;h2 = α2 sin 2φ+ β2 sin 2θ;

h3 = α2 cos 2φ− β2 cos 2θ;h4 = α2 sin 2φ− β2 sin 2θ;

h5 = −α2 cos 2φ+ β2 cos 2θ;h6 = α2 sin 2φ+ β2 sin 2θ;

h7 = −α2 cos 2φ+ β2 cos 2θ;h8 = −α2 sin 2φ+ β2 sin 2θ;

and

R = h21 + h2

2;T = h23 + h2

4;

P = h25 + h2

6;Q = h27 + h2

8

Let

φ = φ+ |t|α sinφ ; θ = θ + |t|β sin θ;

§ 2.9. Appendix 50

We obtain

ζ1(λ, ω, t) =1

αe−|t|α cosφ

1

R[(h1/R) cos φ− (h2/R) sin φ]− 1

T[(h3/T ) cos φ− (h4/T ) sin φ]

ζ1(ω, λ, t) =1

βe−|t|βcosθ

1

P[(h5/P ) cos θ − (h6/P ) sin θ]− 1

Q[(h7/Q) cos θ − (h8/Q) sin θ]

We obtain the final result that is used for the cross-correlator

ζ(λ(α, φ), ω(β, θ), t) = C(ζ1(λ, ω, t) + ζ1(ω, λ, t))

D(0)IJ (t) =

C

LILJ(ζ1(λ, ω, t) + ζ1(ω, λ, t))

§ 2.9.2 Consistency check for D(0)IJ

We take the limit of t = 0 and λ→ ω as well as the limit of β →∞.

λ→ ω; t = 0

Recall from Eq. (2.30)

ζ(λ, ω, t) =

∫ ∞−∞

dk

2π

eikt

(k2 + λ2)(k2 + λ2∗)(k

2 + ω2)(k2 + ω2∗)

C−1ζ(λ, ω) =1

λ2 − ω2(I(ω)− I(λ)) + C.C.+ (

1

λ2 − ω2∗I(λ)− 1

λ2∗ − ω2

I(ω)) + C.C.

We take the limit of t = 0;λ → ω taking care to cancel the divergent terms that appear

in the expansion. This yields

C−1ζ(λ, ω, 0) =1

2

1

λ2 − ω2(

1

ω− 1

λ) + C.C.+

1

2(

1

λ2 − ω2∗

1

λ− 1

λ2∗ − ω2

1

ω) + C.C.

§ 2.9. Appendix 51

=1

2

1

(λ+ ω)λω+ C.C.+

1

2λ(

1

λ2 − ω2∗− 1

λ2∗ − ω2

) + C.C.

=1

4

e3iφ

α3+ C.C.+

1

2

1

α3

e−iφ

e2iφ − e−2iφ+ C.C.

= − 1

2α2(secφ− cos 3φ)

We hence obtain

ζ(λ, λ, 0) =1

8α7 sin 2φ2 (secφ− cos 3φ)

and we have recovered the result given in [4].

β →∞

From Eq. (2.30) we obtain the following

ζ(λ, ω, t) =

∫ ∞−∞

dk

2π

eikt

(k2 + λ2)(k2 + λ2∗)(k

2 + ω2)(k2 + ω2∗)

Taking the limit

β →∞ : ω2 = β2e2iθ →∞

yields the single commodity auto-correlator

ζ(λ, ω, t) =1

β4

∫ ∞−∞

dk

2π

eikt

(k2 + λ2)(k2 + λ2∗)

(2.31)

In this limit, the coefficients are given by

h1 = h3 = −h5 = −h7 = −β2 cos 2θ;

h2 = −h4 = h6 = h8 = β2 sin 2θ;

§ 2.9. Appendix 52

and

R = T = P = Q = β2

Hence, after some simplifications

ζ(λ, ω, t) =−1

4α2β2 sin 2φ sin 2θ

1

αe−α|t| cosφ 1

β2

(cos(|t|α sinφ+φ−2θ)− cos(|t|α sinφ+φ+2θ)

)

=1

β4

e−|t|α cosφ

2α3 sin 2φsin(|t|α sinφ+ φ)

The final result agrees with the result obtained in [4].

Chapter 3

Statistical Field Theory of Futures

Commodity Prices

§ 3.1 Futures commodity prices

The study of futures prices is fundamental to microeconomics and to the theory of commodity

pricing. Studies of forecasting commodity futures from the point of view of economics and

finance have been carried out in [20, 21]; the dynamics of futures commodity prices have been

carried out by [22, 23] and the effect of financialization on the commodities markets have been

studied by [24].

In this chapter, a statistical field theory for the modeling of futures commodity prices is

proposed. Futures commodity prices are modeled by generalizing earlier work in studying the

spot commodity prices [9, 4, 25]. The model is calibrated and tested using daily data for

commodity futures prices.

The futures commodity price is a futures contract for the price of a commodity, locked-in

at present calendar time t, for buying the said commodity at a future time, denoted by ξ > t

– at which date in the future the contract matures. Exchange traded futures contracts require

53

§ 3.1. Futures commodity prices 54

margin calls. At future time ξ, the buyer of the contract makes the payment and takes delivery

of the commodity. The traders can either buy (long) or sell (short) the contract before the

contract matures at time ξ.

The futures commodity price depends on the price of the underlying commodity, and hence

a futures contract is a derivative product. The futures commodity contracts are fungible and

can be traded; nearly 80% of all futures contracts do not result in the actual delivery of

the said commodity, since long and short positions can be offset. Futures contracts are used

largely for speculating and for hedging.

Market price pi(t) of a commodity, labeled by i, called the spot price, for calendar time t,

is given by

pi(t) = p0exi(t).

The normalized spot commodities prices are given by

yi(t) =xi(t)− xi

σi. (3.1)

The multiple spot Lagrangian is defined in terms of the normalized multiple commodities

prices [25], and is the following

L(t) =1

2

N∑i

Li

[(∂2yi∂t2

)2

+LiLi

(∂yi∂t

)2]

+N∑i=1

[die−aiyi + sie

biyi]− 1

2

N∑ij;i 6=j

∆ijyiyj. (3.2)

Market futures prices pi(t, ξ) of a commodity, labeled by i – for calendar time t and future

time ξ – are given by

pi(t, ξ) = p0exi(t,ξ).

The spot and future prices are shown in Figure 3.1. Similar to Eq. (3.1), the normalized


variables for the futures prices are defined by

yi(t, ξ) =xi(t, ξ)− xi(t, t)

σ(xi)⇒ E[yi(t, ξ)] = 0 ; E[y2

i (t, ξ)] = 1. (3.3)

The normalized futures price of a commodity is given by

ξ ≥ t : yi(t, ξ) : futures price.

The normalized spot price of a commodity is given by

yi(t) = yi(t, t) : spot price.

(t’,x’)

(t,t)

t

x(0,0)

Figure 3.1: Points on the boundary are calendar time (t, t); (t′, t′) and points (t, ξ); (t′, ξ′) arein future time.

The spot Lagrangian, given in Eq. (3.2), needs to generalized to a Lagrangian that is valid

for the futures prices of commodities. One needs to add new terms to the spot Lagrangian

that attenuates the fluctuations in the ξ direction. A symmetric generalization of the multiple


spot Lagrangian [25] is given by

L(t, ξ) =1

2

N∑i

Li

[(∂2yi∂t2

+ α2i

∂2yi∂ξ2

)2

+LiLi

((∂yi∂t

)2+ α2

i

(∂yi∂ξ

)2)]

+N∑i=1

[die−aiyi + sie

biyi]− 1

2

N∑ij;i 6=j

∆ijyiyj. (3.4)

The futures Lagrangian has new terms ∂2yi/∂ξ2, ∂yi/∂ξ that are purely ‘kinetic’ in the sense

that they attenuate and dampen out fluctuations in the future time ξ direction. The microe-

conomic potential – containing the supply and demand terms as well as the correlation term

∆ij – is extended into the future direction.

Higher derivative actions, similar to Eq. (3.4) but in four spacetime dimensions have been

applied to the study of cosmology, quantum and conformal gravity by [26] and [27]. The

Euclidean model was studied by [28] for its role in quantum gravity and D-brane dynamics;

the Euclidean path integral was used by [29] for analyzing the ghost states for Minkowski

time.

Present time is taken to be t = 0; the value of the futures price at present time yi(0, ξ) is

fixed by the market. On the semi-infinite plane with a boundary at ξ = t, as shown Figure

3.1, the action is

S =

∫ ∞0

dt

∫ ∞t

dξL(t, ξ).

The spot commodity price y(t, t) on the boundary is a random variable since the value of

the future spot price is not known at the present time. Hence, the futures commodity price

must obey the Neumann boundary condition and yields

∂yi(t, ξ)

∂ξ

∣∣∣ξ=t

= 0. (3.5)

§ 3.2. Single commodity; Gaussian approximation 57

The range of time is extended to −∞ as the boundary at t = 0 will play no important

role in subsequent analysis. The action is given by

S =

∫ ∞−∞

dt

∫ ∞t

dξL(t, ξ). (3.6)

The partition function for futures commodity prices is given by the Feynman path integral

Z =

∫Dye−S. (3.7)

We see from Eq. (3.7) that the futures prices are modeled by yi(t, ξ), which are nonlin-

ear two dimensional statistical fields. The Lagrangian given in Eq. (3.4) is mathematically

equivalent to a two dimensional nonlinear Euclidean quantum field theory. Due to the higher

derivatives in the Lagrangian the quantum field theory is completely finite, with no divergent

Feynman diagrams.

§ 3.2 Single commodity; Gaussian approximation

The first step in studying a nonlinear quantum field is usually to linearize the Lagrangian by

keeping only the quadratic terms, and study it as a Gaussian (free) quantum field. To simplify

the discussion, consider a single commodity and an expansion about the small fluctuations of

y(t, ξ) by expanding the Lagrangian given in Eq. (3.4) for small values of y(t, ξ).

As in the case of single commodities, for having the minimum of the microeconomic po-

tential at y(t, ξ) = 0, the following condition is chosen [9]

ad = bs.

§ 3.2. Single commodity; Gaussian approximation 58

The Gaussian quantum field, after integration by parts using the Neumann boundary condi-

tion, has the following Lagrangian – given in terms of parameters L, γ1, γ2.

L(t, ξ) =L

2y( ∂4

∂t4+ α4 ∂

4

∂ξ4+ 2α2 ∂4

∂ξ2∂t2− (γ2

1 + γ22)(α2 ∂

2

∂ξ2+∂2

∂t2) + γ2

1γ22

)y. (3.8)

Eq. (3.4) and (3.8) yield the following

γ21 + γ2

2 =L

L; γ2

1γ22 = a2d+ b2s.

The Lagrangian has three branches: real, complex and critical [3]. For convergence of the

path integral, all branches require L > 0 and a2d+ b2s > 0.

The modeling of single and multiple commodities are described by the complex branch,

which requires complex γ1, γ2 [4, 25]; a2d + b2s > 0 imposes the condition that γ1, γ2 are a

complex conjugate pair.

Hence, for the complex branch

γ1 = reiφ ; γ2 = re−iφ ⇒ 2r2 cos(2φ) =L

L.

L can be negative or positive for the complex branch and yields

−1 ≤ cos(2φ) ≤ +1 ⇒ − π

2≤ φ ≤ +

π

2.

In the empirical studies of futures and spot prices, it is found that π/4 ≤ φ ≤ +π/2.

The Gaussian approximation yields a quadratic action S given in Eq. (3.8). The supply

and demand, via the microeconomic potential, yield only a single term in the action, which is

Lγ21γ

22y

2/2; all the remaining terms arise from the kinetic terms dampening rapid fluctuations

§ 3.3. Propagator 59

in the direction of calendar time t and future time ξ.

However, the term Lγ21γ

22y

2/2 is crucial for stabilizing the path integral; without this

term, the path integral given by Eq. (3.7) is divergent. This divergence can be seen clearly

in the calculation of the Gaussian propagator, given in Section § 3.4.3, which is divergent for

γ21 = 0 = γ2

2 .

The semi-infinite domain over which y(t, ξ) is defined, together with the Neumann bound-

ary condition given in Eq. (3.5), yields the following Fourier expansion

y(t, ξ) =

∫ ∞−∞

dω

2π

∫ ∞0

dk

π/2eiωt cos(k(ξ − t))y(ω, k) ; y∗(ω, k) = y(−ω, k).

The action is given by

S =1

2π2

∫ ∞−∞

dω

∫ ∞0

dk y(−ω, k)∆−1(ω, k)(ω, k)y(ω, k), (3.9)

and

∆−1(ω, k) = L(ω2 + α2k2 + γ21)(ω2 + α2k2 + γ2

2). (3.10)

§ 3.3 Propagator

Using Gaussian path integration, the propagator is given by

G(t, ξ; t′ξ′) = E[y(t, ξ)y(t′, ξ′)] =1

Z

∫Dye−Sy(t, ξ)y(t′, ξ′)

=1

π2

∫ ∞−∞

dω

∫ ∞0

dkeiω(t−t′) cos(k(ξ − t)) cos(k(ξ′ − t′))∆(ω, k). (3.11)


Note, as expected, the propagator obeys the Neumann boundary conditions

∂G

∂ξ

∣∣∣ξ=t

= 0 =∂G

∂ξ′

∣∣∣ξ′=t′

. (3.12)

Let

θ = ξ − t ; θ′ = ξ′ − t′.

Simplifying Eq. (3.11) yields

G(t, ξ; t′, ξ′) =1

4π2

∫ ∞−∞

dω

∫ ∞0

dkeiω(t−t′)(eikθ + e−ikθ)(eikθ′+ e−ikθ

′)∆(ω, k)

=1

4π2

∫ ∞−∞

dω

∫ ∞−∞

dkeiω(t−t′)(eik(θ+θ′) + eik(θ−θ′))∆(ω, k)

= D(t− t′; θ − θ′) +D(t− t′; θ + θ′) (3.13)

Using identity

∆ =1

L

1

γ21 − γ2

2

( 1

ω2 + α2k2 + γ22

− 1

ω2 + α2k2 + γ21

)

yields

D(τ, θ) =1

L(

1

γ22 − γ2

1

)

∫ ∞−∞

dωdk

4π2eiωτ+ikθ(

1

ω2 + α2k2 + γ21

− 1

ω2 + α2k2 + γ22

) (3.14)

From Appendix Eq. (3.41)

I(τ, θ) =

∫ ∞−∞

dωdk

4π2eiωτ+ikθ(

1

ω2 + α2k2 + γ2) =

1

2παK0(γ

√τ 2 + (

θ

α)2)

where K0 is the modified Bessel function of second kind. Hence

D(τ, θ) =1

2Lπα

1

γ22 − γ2

1

[K0(γ1

√τ 2 + (

θ

α)2)−K0(γ2

√τ 2 + (

θ

α)2)]

(3.15)


From Eqs. (3.13) and (3.15), for τ = t− t′, the propagator is given by

G(t, ξ, ; t′ξ′)

=1

2Lπα

1

γ22 − γ2

1

[K0(γ1

√τ 2 + (

θ + θ′

α)2)−K0(γ2

√τ 2 + (

θ + θ′

α)2)

+K0(γ1

√τ 2 + (

θ − θ′α

)2)−K0(γ2

√τ 2 + (

θ − θ′α

)2)]

Define

z2+ = τ 2 + (

θ + θ′

α)2 ; z2

− = τ 2 + (θ − θ′

α)2 (3.16)

τ = t− t′ ; θ = ξ − t ; θ′ = ξ′ − t′

0

2

4

6

8

10

0

2

4

6

8

10

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 3.2: Theoretical plot of G(z+; z−) as a function of z+, z−, with α=1,L=1, γ1=1,γ2=2

Then

G(t, ξ, ; t′ξ′) =1

2Lπα

1

γ22 − γ2

1

[K0(γ1z+)−K0(γ2z+) +K0(γ1z−)−K0(γ2z−)

].

≡ G(z+; z−) (3.17)


A plot of G(z+; z−) as a function of z+, z− – for some typical values of the parameters – is

given in Figure 3.2.

Using the normalization condition G(0, 0) = 1 derived later in Eq. (3.24) yields the final

result1

G(z+; z−) = ln(γ1

γ2

)[K0(γ1z+)−K0(γ2z+) +K0(γ1z−)−K0(γ2z−)

]. (3.18)

Note the important fact that the propagator, which can in general depend on four variables

t, t′; ξ, ξ′, is required by the model to depend on only two variables, given by z+, z−. The

empirical study will take this into account and leads to some very stringent conditions of

market data if their model is to work. This point is addressed at length in the empirical

study.

For the complex branch with γ1 = reiφ = γ∗2 , the propagator is given by2

G(z+; z−) = −4φ=[K0(γ1z+) +K0(γ1z−)

]: Real (3.19)

Eq. (3.19) is the key equation in the empirical study of this model

The concept of market time [7, 1] is fundamental for mapping the model’s propagators to

empirically determined correlation functions. Traders operate in the market based on their

subjective perception of time, and this fact is captured by the idea of market time. Both

calendar and future time are modified by the concept of market time: in all propagators and

correlation functions in general, the following replacement is made.

τ → λ(τ

λ)η ; θ → λ(

θ

λ)η ; θ′ → λ(

θ′

λ)η ⇒ z± → m(z±). (3.20)

1The limit of γ1 → γ2 is non-singular and discussed in Section § 3.4.2.2=(f) is the imaginary part of a complex valued function f .

§ 3.4. Propagator for spot prices 63

where the function m(x) can be read off from the definition of market time given in Eq. (3.20).

The connection of the empirical propagator GE(z+; z−) to model’s propagator is given by

GE(z+; z−) = G(m(z+);m(z−)). (3.21)

In fitting the model’s correlators to the empirical correlators, the substitution given in Eq.

(3.21) is always made. To keep the notation simple, the variables t, ξ continue to be used

everywhere – keeping in mind that all fits are made with market time replacing all calendar

and future time variables that appear in the model’s propagators.

§ 3.4 Propagator for spot prices

The spot prices y(t, t) provide a model for commodity prices that is distinct from the model

proposed in [9]. Calibration shows that both the models provide a fairly accurate represen-

tation of commodity spot prices correlation functions. For spot prices, from Eq. (3.16), we

have

ξ = t ; ξ′ = t′ ⇒ z+ = z− = τ.

Hence, from Eqs. (3.11) and (3.17), the spot prices propagator is given by

G(t, t; t′, t′) = E[y(t, t)y(t′, t′) =1

Lπα

1

γ22 − γ2

1

[K0(γ1τ)−K0(γ2τ)] ; τ = t− t′. (3.22)

For taking the limit of t = t′, note that

limx→0

K0(x)→ − ln(x

2),


Eq 3.22 yields the following limit

E[y2(t, t)] = limt′→t

G(t, t; t′, t′) =1

Lπα

1

γ22 − γ2

1

limτ→0

[K0(γ1τ)−K0(γ1τ)]

=1

Lπα

1

γ22 − γ2

1

[− ln(γ1τ

2) + ln(

γ2τ

2)] =

1

Lπα

1

γ22 − γ2

1

ln(γ2

γ1

). (3.23)

The normalization given in Eq. (3.3) yields, from Eq. (3.23)

E[y2(t, t)] =1

Lπα

1

γ22 − γ2

1

ln(γ2

γ1

) = 1. (3.24)

Eq. (3.24) fixes one of the parameters of the model. For the complex branch of the propagator

recall that γ1 = γ∗2 = reiφ. The normalization condition in Eq. (3.24) yields

1 =1

Lπαr2| φ

sin(2φ)| ⇒ L =

1

παr2|φ.sin(2φ)| > 0 (3.25)

Note the nontrivial result that L > 0 for both the complex and real branches, as indeed it

must be for the path integral to be convergent. The model has five independent parameters

and the constraint Eq. (3.25) reduces this to four. Incorporating the constraint Eqs. (3.24)

and (3.25) yields the following spot rate propagator

G(t, t; t′, t′) = E[y(t, t)y(t′, t′) = ln(γ1

γ2

)[K0(γ1τ)−K0(γ2τ)] ; τ = t− t′. (3.26)

§ 3.4.1 Boundary condition

It is verified that the explicit expression for the propagator obeys the Neumann conditions.

From Eq. (3.17), the propagator is given by

G(t, ξ; t′, ξ′) =1

2Lπα

1

γ22 − γ2

1

[K0(γ1z+)−K0(γ2z+) +K0(γ1z−)−K0(γ2z−)

].


Note that

∂z±∂ξ

∣∣∣ξ=t

= ± 1

α2wθ′ ; w2 = τ 2 +

(θ′)2

α2; lim

ξ→tz± → w

Hence, since dK0(x)/dx = −K1(x), we obtain

∂G(t, ξ; t′, ξ′)

∂ξ

∣∣∣ξ=t

=1

2Lπα

1

wα2(γ22 − γ2

1)×[

− γ1θ′K1(γ1w) + γ2θ

′K1(γ2w) + γ1θ′K1(γ1w)− γ2θ

′K1(γ2w)]

= 0

Hence the result satisfies the Neumann boundary condition.

§ 3.4.2 Special case γ = γ2 = γ1

The special case constitutes a critical point for the acceleration Lagrangian of the spot rates

[3]. For the futures prices, from Eq. (3.13), the propagator is given by

G(t, ξ; t′, ξ′) = D(t− t′; θ − θ′) +D(t− t′; θ + θ′)

with

D(τ, θ) =1

L

∫ ∞−∞

dωdk

4π2

eiωτ+ikθ

(ω2 + α2k2 + γ2)2= − 1

2γL

∂

∂γI(τ, θ)

where, from Appendix Eq. (3.41)

I(τ, θ) =1

2παK0(γ

√τ 2 + (

θ

α)2)

Using dK0/dx = −K1 yields the critical propagator

G(z+, z−) =1

4πLαγz+K1(γz+) + z−K1(γz−) (3.27)


The result above is the analog for futures prices of the result obtained for the critical point for

the spot commodity prices given in [3]. The normalization, from Eq. (3.25), for limφ→0 γ =

reiφ has the limit

L = limφ→0

1

παr2| φ

sin(2φ)| = 1

2παr2=

1

2παγ2(3.28)

§ 3.4.3 Limit of γ → 0

For x→ 0 the Bessel function of the second kind has the following asymptotic expansion

K1(x) ≈ 1

x

From Eq. (3.27), the propagator has the limit

limγ→0

G(z+, z−) =1

4πLαγz+

1

γz+

+ z−1

γz+

=1

2πLαγ2

This is a noteworthy result as it shows that all three quantities, namely L, α, γ2 have to be

non-zero for the path integral to converge. These are the coefficients of the three distinct

types of terms in the Lagrangian, and all three coefficients are required to yields a finite path

integral.

Consider imposing the constraint G(0, 0) = 1 on the path integral; then Eq. (3.28) and

above yields

limγ→0

G(0, 0) = 1

In other words, once one imposes the normalization G(0, 0) = 1, the dependent parameter

ensures that the path integral will always converge and in doing so maintains the constraint

§ 3.5. Contour map of G(t, ξ; 0, 0) and α 67

to be G(0, 0) = 1. In particular, note that

G(0, 0) = 1 ⇒ 2πLαγ2 = 1 ⇒ L, α, γ2 > 0

§ 3.5 Contour map of G(t, ξ; 0, 0) and α

G(0

,0;t

,x)

calendar time t(day)

0future time(day)

G(0

,0;t

,x)


0future time(day)

G(0

,0;t

,x)

future time(day)


0

Figure 3.3: Shape of the model for futures for a) α = 1, b) α > 1 and c) α < 1.

The value of G(t, ξ; 0, 0) for three different values of α is shown in Figure 3.3. Recall from Eq.

(3.16) that the definition of z± is given by

z2+ = τ 2 + (

θ + θ′

α)2 ; z2

− = τ 2 + (θ − θ′

α)2;

τ = t− t′ ; θ = ξ − t ; θ′ = ξ′ − t′.

α is a dimensionless pure number, and the numerical value of α determines how important is

future time in the behavior of the correlation of futures price. This is because it is the ratio

(θ ± θ′)/α that determines the contribution of future time to the propagator.

• α = 1: Figure 3.3 (a) shows that the value of G(t, ξ; 0, 0) is constant for equal distance

in future time and calendar time, which implies futures prices affect the spot price for

an equal length of time a given future calendar time.

• α > 1: Figure 3.3 (b) shows that the value of G(t, ξ; 0, 0) for future time greater than

§ 3.6. Spot rate G(t, t; t′, t′): empirical and model 68

calendar have the same values, from which one can conclude futures prices affect the

spot prices for a given future calendar time of shorter duration.

• α < 1 : Figure 3.3 (c) shows that future time smaller than calendar time affects the

spot prices for a given future calendar time.

We see that a small value of α increases the contribution of near future time to the

correlation at future calendar time and a large value of α requires a large future time to affect

future calendar time.

Consider the following example, discussed later in Section § 3.10. Data of daily futures

price of crude oil are given till 10 years into the future yields 10 data points; these are in turn

splined into 50 points. The analysis in Section § 3.10 shows that α = 20 is the best value for

this case, and which implies that future time becomes important for spot prices after four

years.

From the study of the contour map of G(t, ξ; 0, 0) and empirical results for α, one can

conclude that it is primarily the spot rates – reflected in the τ term in z± – that determine

the behavior of futures prices in the near future. It is only far into the future, from about

three years onwards, that the dynamics of futures prices come into play.

§ 3.6 Spot rate G(t, t; t′, t′): empirical and model

The single commodity spot market prices are used for fitting the model’s spot propagator

G(t, t; t′, t′), given in Eq. (3.22), to the empirical propagator. The fitting is at the boundary

of futures price, as in Figure 3.1; both γ1, γ2 are complex. For all cases, the model predicts

the shape of the correlation function for a few hundred points based on four parameters.

Some typical fits are shown in Figure 3.4: the correlation from data have an irregular

§ 3.6. Spot rate G(t, t; t′, t′): empirical and model 69

shape, whereas the smooth lines are the model’s fit and prediction.

When the shape has a deep valley like the case of corn data as shown in Figure 3.4(c), the

model cannot fit such a deep case and the best fit can only give R2 = 0.608.

0 20 40 60 80 100 120 140 160 180 200−0.2

0

0.2

0.4

0.6

0.8

1

1.2

τ(time lag/day)

G(τ)

R2= 0.9714

0 20 40 60 80 100 120 140 160 180 200−0.2

0

0.2

0.4

0.6

0.8

1

1.2

τ(time lag/day)

G(τ)

R2= 0.97601

0 20 40 60 80 100 120 140 160 180 200−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

τ(time lag/day)

G(τ)

R2=0.60818

Figure 3.4: Fitting spot rates for a) Gold, b) Soybeans and c) Corn. The smooth curve is themodel’s best fit to data. (Jan 1 2011- Oct 18 2011)

From Figures 3.4(a) and (b) it is seen that the pattern at the boundary for commodities

like gold and soybeans can be fit almost perfectly, with R2 = 0.971, 0.976, respectively. More

recent data for crude oil, given in Figure 3.5, shows that for different regimes of the time, the

model fits market spot prices quite well (R2 = 0.93).

0 20 40 60 80 100 120 140 160 180 200−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

τ(time lag/day)

G(τ)

Figure 3.5: Model and market correlators for crude oil, with R2 = 0.93. (Sep 20 2014- June11 2015)

§ 3.7. Spot-futures G(t, ξ; 0, 0): empirical and model 70

§ 3.7 Spot-futures G(t, ξ; 0, 0): empirical and model

The propagator G(t, ξ; 0, 0) measures the correlation of the spot and futures prices of com-

modities. This is the first measure of how well the futures Lagrangian can model the behavior

of the futures prices.

The fit of the correlation – determined from market data – of the spot and futures prices

for the corn is very accurate and yields R2=0.96. Figure 3.6(a) shows the market correlation

of the spot price of corn with its forward price G(t, ξ; 0, 0) with the model’s propagator given

in Figure 3.6(b). Similarly, Figures 3.7 and 3.8 show excellent fits of crude oil and rice for the

correlation between spot and futures prices, with R2 = 0.93, 0.98 respectively.

Figure 3.6: Spot and futures prices correlation G(t, ξ; 0, 0), plotted against t, ξ, of corn futuresprices with the spot price. a) The empirical propagator. b) The model propagator. (Jan 12011- Oct 18 2011)

§ 3.7. Spot-futures G(t, ξ; 0, 0): empirical and model 71

Figure 3.7: G(t, ξ; 0, 0) of Crude oil futures data. (Jan 1 2011- Oct 18 2011)

Figure 3.8: G(t, ξ; 0, 0) of Rice futures data. (Jan 1 2011- Oct 18 2011)

Figure 3.9: G(t, ξ; 0, 0) of Gold futures data. (Jan 1 2011- Oct 18 2011)

§ 3.8. Algorithm for empirical GE(z+, z−) 72

Table 3.1: Calibration: Spot-Futures Correlations

Commodity α L γ1 = γ∗2 R2 λ η

Crude oil 6.22 256.97 0.0553-0.0866i 0.93 0.784 0.7043

Corn 10.00 9.93× 103 0.0072-0.0080i 0.96 71.76 1.809

Rice 6.6922 156.85 0.00066-0.1248i 0.98 1.3748 0.7150

Gold 9.418 4.15× 103 0.0161-0.04006i 0.41 47.32 0.1000

The case of gold is included to show that the model cannot fit the prices of all commodities.

Due to the highly irregular shape of the correlation between gold spot and futures, the fit fails

with R2 = 0.41.

The results of the fits for spot and futures prices are shown in Table 3.1.

§ 3.8 Algorithm for empirical GE(z+, z−)

The propagator is empirically evaluated from market data and then matched with the model.

This procedure both calibrates the model’s parameters as well as tests the accuracy of the

model. The main features of the algorithm are summarized below.

• The propagator

G(t, ξ; t′, ξ′) (3.29)

in general depends on four variables, given by t, ξ; t′, ξ′.

• The symmetry of the model yields the result

G(t, ξ; t′, ξ′) = D(t− t′, ξ − t, ξ′ − t′). (3.30)

This shows that the model’s propagator depends on only three variables, given by t −


t′, x−t, x′−t′. This feature of the model is used for evaluating the empirical propagator.

• The model further requires the following

D(t− t′, ξ − t, ξ′ − t′) = G(z+, z−). (3.31)

In other words the model’s propagator depends on only two variables.

The evaluation and calibration of the empirical propagator will be based on algorithms that

reduce the dependence of the empirical propagator from four variables to two variables.

The market data of the futures commodity prices y(t, ξ) are interpreted as the outcome

of sampling stochastic futures prices. A basic assumption of statistical theory of prices is

that the expectation value of stochastic prices is equal to the historical average of the futures

prices. To perform the average over historical data, note from Eq. (3.30), that the model

propagator G(t, ξ; t′, ξ′) depends only on three variables since for constant z

G(t, ξ; t′, ξ′) = G(t+ z, ξ + z; t′ + z, ξ′ + z) : Invariant under time translations (3.32)

Invariance under time translation is a reflection of the assumption that there is no special

instant of time for the model. Of the four variables t, ξ; t′, ξ′, one is auxiliary; in Eq. (3.32)

let the auxiliary variable be z = −t and yields

D(a, b, c) = D(t− t′, ξ − t, ξ′ − t′) ≡ G(0, t− t′, ξ − t, ξ′ − t′). (3.33)

In Eq. (3.33), the following choice variables (for future convenience) is made

ξ − t = b ; t′ − t = −a ; ξ′ − t = c− a

⇒ ξ = b+ t ; t′ = t− a ; ξ′ = t+ c− a (3.34)


The three variables, labeled a, b, c, yield D(a, b, c), which is given by the following

D(a, b, c) = G(t, b+ t, t− a, c− a+ t) = G(t, ξ; t′, ξ′). (3.35)

Eq. (3.35) is the theoretical basis of connecting the model with data. Since D(a, b, c) does

not depend on t, the variable t is taken to be an index labeling the random samples of the

propagator D(a, b, c).

The empirical propagator, denoted by DE(a, b, c), is defined by summing over the random

samples in the following manner. Discretize t → tn = nε, and let ε = 0.2 years. Then the

expectation value required for evaluating the empirical propagator is given by

D(a, b, c) = E[y(t, b+ t)y(t− a, c− a+ t)]

⇒ DE(a, b, c) =1

N

N∑n=1

y(tn, b+ tn)y(tn − a, c− a+ tn). (3.36)

The empirical analysis is carried out using a finite sample size, and N = 200 days is chosen

for commodity data.

From Eq. (3.31), the propagator has a further reduction in the number of independent

variables from

a, b, c → z+, z−.

According to the definition of z+, z− given in Eq. (3.16), and from Eq. (3.34)

z2+ = τ 2 +

(θ + θ′

)2

α2; z2

− = τ 2 +

(θ − θ′

)2

α2

τ = t− t′ = a ; θ = ξ − t = b ; θ′ = ξ′ − t′ = c

⇒ z+ =

√a2 +

(b+ c)2

α2; z− =

√a2 +

(b− c)2

α2(3.37)


A major test for the viability of the model is whether data supports the reduction in the

dependence of the underlying coordinates from a, b, c → z+, z−.

A key link in the mapping from DE(a, b, c) to GE(z+, z−) is that the propagator DE(a, b, c)

is binned into a matrix with coordinates (z+, z−) for all points (a, b, c) with the same value of

z+, z−. The mapping of points (a, b, c) to z+, z− is a mapping of many points to one.

In the model for futures prices, for all points (a, b, c) that are mapped into (z+, z−), the

propagator DE(a, b, c) should approximately have the same value, and yield a sample value

of GE(z+, z−).

No model has so far been assumed for the empirical propagator DE(a, b, c). Consider the

three dimensional parameter space E3 = (a, b, c). The parametric equation for a surface

z+ = A ; z− = B ; A,B > 0 (3.38)

defines a two dimensional surface S(A,B) inside E3. One expects that the appropriate choice

for quantity α should lead to DE(a, b, c) = GE(z+, z−): a constant on the surface S(A,B);

changing A,B changes the surface, and in turn DE(a, b, c) has another numerical value. The

behavior of DE(a, b, c) being constant on surfaces S(A,B), for the appropriate choice of α,

would indicate that market data has a pattern that is required and predicted by the model.

Each point a, b, c is mapped to a unique bin, labeled by [z+, z−]. The samples of DE(a, b, c),

obtained by varying a, b, c, are denoted by

D(k)E (a, b, c) ≡ G

(k)E (z+, z−) : k = 1, 2 · · · , K ; K = K(z+, z−). (3.39)

The number of samples depend on the bin and yields K = K(z+, z−). The average value is

taken of all the sample values D(k)E (a, b, c) that are collected in a given bin. The average is

§ 3.9. Binning of empirical D(k)E (a, b, c) 76

taken to be equal to empirical GE(z+, z−). More precisely

GE(z+, z−) =1

K

K∑k=1

D(k)E (a, b, c)

∣∣∣(a,b,c)→(z+,z−)

=1

K

K∑k=1

G(k)E (z+, z−) (3.40)

The optimum value of α is recursively chosen, together with γ1, γ2, for which the sample

values of G(k)E (z+, z−) yield the best fit of the empirical propagator with the model’s propa-

gator. In other words, from Eq. (3.21)

GE(z+, z−) ≈ G(m(z+),m(z−)) : optimum value for α, γ1, γ2, λ, η

The model’s propagator, from Eq. (3.18), is given by

G(z+; z−) = ln(γ1

γ2

)[K0(γ1z+)−K0(γ2z+) +K0(γ1z−)−K0(γ2z−)

].

To find the best fit, the program has to run through many values of

gamma1, γ2, λ, η. For each choice of γ1, γ2, λ, η, the entire binning process has to be repeated

to find the best α. This is the reason that the algorithm is very slow: because the binning

of the propagator D(k)E (a, b, c) requires the mapping of a three dimensional parameter space,

specified by a, b, c, to coordinates z±, and which is computationally very intensive.

§ 3.9 Binning of empirical D(k)E (a, b, c)

Figure 3.10(a) shows the binning of D(k)E (a, b, c) for a fixed value of z+, z−; the value of α=19.98

gives the best of empirical GE(z+; z−) with the model’s G(z+; z−); recall from Eq. (3.39) that

k stands for the different sample values of D(k)E (a, b, c) in a bin, which is labeled by z+, z−.

The values of D(k)E (a, b, c) are randomly distributed about its mean value in the bin: this is

what one expects since ideally all the D(k)E (a, b, c) differ from the expected value D(a, b, c) due

§ 3.9. Binning of empirical D(k)E (a, b, c) 77

to random errors and due to out lying data points. The identification of D(k)E (a, b, c) with

G(k)E (z+; z−) made in Eq. (3.39) is valid only if all the D

(k)E (a, b, c)’s are approximately equal.

Figure 3.10(b) is the binning of the same data, but with α = 1. In contrast , to the case

of α=19.98, the binning of D(k)E (a, b, c) for α=1 does not yield values that lie randomly above

and below the expected mean value. Instead, the values of D(k)E (a, b, c) tend to cluster and

bunch up – either above the mean value, near 1.0, or below the mean value near 0.1. The

clustering of data points for α=1 reflects the fact that D(k)E (a, b, c) is not equal to sample values

of empirical G(k)E (z+; z−), and which in turn implies that an average value of D

(k)E (a, b, c) in

the same bin would not be an accurate representation of GE(z+; z−). It is for this reason that

our proposed model, for an arbitrary value of α such as α = 1, cannot fit the result obtained

by the binning D(k)E (a, b, c).

0 2 4 6 8 10 12 140.2

0.4

0.6

0.8

1

1.2

1.4

1.6

mean

Sample

DE

k(a,b,c)

0 5 10 15 20 25 30 35 400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

mean

DE

k(a,b,c)

Sample

Figure 3.10: Binning of 10 years oil futures data of D(k)E (a, b, c). a) With α=19.98 and for 14

sample points. b) With α=1 and for 40 sample points. (Nov 20 2015- Sep 8 2016)

It is a rather unexpected empirical result – and one that is necessary for successfully

describing future prices by the proposed model– that for a unique choice of α = 19.98 such

that the mapping of (a, b, c) to the same value of z+, z−, yields sample values D(k)E (a, b, c), for

k = 1, 2 · · · , K that are approximately equal.

The negative result obtained by binning D(k)E (a, b, c) for α = 1 provides a model indepen-

§ 3.10. Empirical results for GE(z+; z−) 78

dent test of the validity of the specific Lagrangian chosen in this chapter. By choosing an

arbitrary α and evaluating the value of the propagators for fixed values of (z+, z−), Figure

3.10(b) shows that for an arbitrary α the values of D(k)E (a, b, c) do not show the reduction of

the number of variables from a, b, c to (z+, z−): the values of D(k)E (a, b, c) are not approximately

equal.

One can do a model independent variation of α to determine the optimum α for which

the binning from a, b, c to (z+, z−) effectively results in a D(k)E (a, b, c) that is approximately a

constant; in this case, an optimum α would be a completely model independent result. It is

of course the expectation from the model that the model independent result for optimum α

would be close to the one obtained from the model.

In summary, Figure 3.10(a) provides reasonable evidence that there is a unique α for which

the binning yields an approximately constant value for DE(a, b, c); hence, one can conclude

that there exists a parameter α that encodes an inherent and model independent property of

the market’s behavior. The choice of α = 19.98 is guided by the requirement of obtaining the

best fit of the empirical propagator with the model’s propagator.

§ 3.10 Empirical results for GE(z+; z−)

The value of GE(z+; z−) for different values of z+, z− is shown in Figure 3.11(a), with the best

fit for the model’s G(z+; z−) shown in Figure 3.11(b). Given the complexity of the fit, a value

of R2 = 0.695 is reasonable. A more computationally intensive study needs to be carried out

for determining GE(z+; z−) more accurately.

The results of the calibration and fitting of the model are summarized in Table 3.2. The

calibration of the model has three distinct sources, namely the correlation function for (a)

spot-spot prices, (b) spot-futures prices and (c) futures-futures prices.

§ 3.10. Empirical results for GE(z+; z−) 79

0 2 4 6 8 10 12 14 160

10

20

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16

0

5

10

15

20

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 3.11: a) Empirical GE(z+; z−) and b) Model GE(z+; z−) for market Oil futures prices.(Nov 20 2015- Sep 8 2016)

• Calibration using the spot-spot correlation function yields values consistent with the

results given in Table 3.1. The value of α is approximately 1 since it does not appear in

the spot-spot propagator G(t, t; t′, t′).

• The spot-future results for oil are also similar to the one obtained in Table 3.1. The

value of α is approximately 10.

• The value of L for the results given in Table 3.1 and Table 3.2 are very different, by

a factor of almost 1000 for spot-spot and spot-futures results. It is not clear why the

value of L should differ so much. Tables 3.1 and 3.2 refer to two different periods of

time, Jan 1 2011- Oct 18 2011 and Nov 20 2015- Sep 8 2016, receptively. It is possible

that the value of L has changed so dramatically due to the different periods used for

the calibration.

• The value of α = 19.98 is highest for the futures-futures case, as can be seen from Table

3.2. The reason, from the discussion in Section § 3.5, is that the highest contributions

to the propagator is only futures prices which are far in the future.


Table 3.2: Crude oil Calibration

Crude oil α L γ1 = γ∗2 R2 λ η

Spot 1.005 9.75× 105 0.0159-0.0278i 0.931 0.9 0.99

Spot-Future 14.22 3.29× 107 0.0019-0.0057i 0.993 1.76 1.99

Future 19.98 1× 10−4 50.7-998.3i 0.695 36.6 0.95

§ 3.11 Conclusion

The empirical study of futures price provides evidence for the suitability of modeling futures

prices based on a two dimensional Euclidean quantum field theory.

The fit of the model of the correlation function of spot prices, for all cases except corn,

is over 95% accuracy (going by the R2 score). The fit of the correlation of the spot to the

futures prices, except for gold, is also accurate to over 93%. The correlation of futures prices

for two different future times is however good to only 70%. This lower accuracy does not

necessarily mean that the model cannot deliver an accuracy of over 90% for this case. The

computing power required for studying the correlation of futures-futures prices if far higher

than is available in a laptop. To fully test the model, one needs to study the futures-futures

prices correlation function using a supercomputer: only then can the full accuracy of the

model be gauged.

A two dimensional Euclidean quantum field has been applied in [7, 1] to the modeling

of forward interest rates f(t, ξ). The quantum field theory of forward interest rates – by a

change of variables – leads to decoupling of the infinite degrees of freedom, and results in a

system with a finite number of degrees of freedom [7].

For example, the decoupling can be done by going from forward interest rates quantum


field f(t, ξ) to a two dimensional quantum field A(t, ξ) by the following change of variables

∂f(t, ξ)

∂t= β(t, ξ) + σ(t, ξ)A(t, ξ)

where β(t, ξ), σ(t, ξ) are deterministic functions [1]. The quantum field A(t, ξ) is decoupled

in the time direction t and the action functional is effectively many copies (labeled by t) of a

quantum mechanical system, with role of ‘time’ being played by ξ.

If one moves from the path integral to the Hamiltonian, the degree of freedom is f(t, ξ) and

the infinitely many coupled degrees of freedom cannot be decoupled. But if one is interested

in only the correlation functions of f(t, ξ), then these are evaluated using the path integral

and the system is effectively quantum mechanical.

In contrast, the model for futures prices is an irreducible two dimensional nonlinear quan-

tum field. Due to the high order derivatives in the action functional, this theory is completely

finite. There is no need for the procedure of renormalization, which is necessary for obtaining

finite results from nonlinear quantum fields without the higher derivative term; these quantum

fields occur in high energy physics and in the theory of phase transitions.

Many questions remain, including studying the nonlinear terms of the model that are

required for determining the potential of the commodities that yield the futures prices. An-

other open question is the coupling of the futures prices of different commodities – and the

calibration and empirical tests for these models.

§ 3.12. Appendix I(τ, θ) 82

§ 3.12 Appendix I(τ, θ)

The integrations appearing in Eq. (3.14) can be obtained from the following integral

I(τ, θ) =1

α

∫ ∞−∞

dωdk

4π2eiωτeikθ/α(

1

ω2 + k2 + γ2)

=1

α

∫ ∞0

dh

∫ ∞−∞

dωdk

4π2e−(ω2+k2+γ2)heiωτeikθ/α

Gaussian integration yields

∫ ∞−∞

dxe−ax2

=

√π

a⇒ I(τ, θ) =

1

4πα

∫ ∞0

dh

he−γ

2he−τ2

4h− ( θα )2

4h

and from the definition of K0, the associated Bessel function of the second kind

I(τ, θ) =1

2παK0

(γ

√τ 2 + (

θ

α)2)

(3.41)

The final result is given by

I(τ, θ) = I(z) =1

2παK0(γz) ; z2 = τ 2 + (

θ

α)2

§ 3.12.1 Appendix: Algorithm for binning the propagator

timespan = 100; points = 100; M(:, :) = 0; zpoints = 100;

for a = 1 : points

for b = 1 : points

for c = 1 : points


n = 0;

D(a, b, c) = 0;

for t = 1 : timespan

t′ = t+ a;

x = t+ b;

x′ = c+ a+ t;

D(a, b, c) = D(a, b, c) +G(t, x; t′, x′);

[Note: transforming from (t, x; t′x′) to (t− t′, x− t, x′ − t′)]

n = n+ 1;

end

D(a, b, c) = D(a, b, c)/n;

z+ = round

(√a2 +

(b+ c)2

α2

)

z− = round

(√a2 +

(b− c)2

α2

)M(z+, z−) = M(z+, z−) + 1

GE(z+, z−,M(z+, z−)) = D(a, b, c);

end

end

end


for z+ = 1 : zpoints

for z− = 1 : zpoints

G(z+, z−) = 0;

for k = 1 : M(z+, z−)

G(z+, z−) = G(z+, z−) +GE(z+, z−, k);

end

G(z+, z−) = G(z+, z−)/M(z+, z−);

end

end

Chapter 4

Option Price and Market Instability

§ 4.1 Introduction

Baaquie introduced the action functional for pricing the Black-Scholes (BS) model in [7] and

the Black-Scholes price was given a path integral derivation starting from the action functional.

In 2014, Baaquie and Yang postulated another action functional for the option pricing; the

Baaquie and Yang (BY) option price depends on the stock price and its velocity – and is

essentially Gaussian since the action functional is quadratic in the logarithm of the security

(equity) or FX rate [5]. The BY model following the Gaussian model provides analytical

solution and explanation in comparison to Non-Gaussian model [30, 31].

The FX option for the exchange rate of various currencies against the US Dollar are studied

for the period from 2011 to 2015; the following behavior is observed.

• For all currencies, there are ‘normal’ periods where the BY model fits the data fairly

well.

• There are sudden intermittent periods that punctuate the normal period. And for these

periods, the model fails dramatically due to market instability.

• The volatility of the underlying FX rate is not a suitable as an indicator of market

85


instability.

Our hypothesis is that the failure of the BY model to fit market data is due to the effects

of instability that are not captured by the BY model since it is essentially Gaussian. This

simplicity of the BY model is used to our advantage by postulating that market instability

introduces nonlinear effects causing the model to fail. This very failure of the model in turn

is used as a barometer and as a gauge for concluding that the FX market has entered an

unstable and potentially a crisis phase.

The behavior of FX options is an accurate gauge of the state of the international financial

system. The FX markets are international and operate 24 hours a day – and are expected to

quickly respond to the changing tides of the major economic powers[32]. Furthermore, there

is a high volume of daily FX transaction: trading in foreign exchange markets averaged $5.3

trillion per day in April 20131. High liquidity and the key role of currencies in the major

economies, in our opinion, makes FX options a sensitive gauge of the international financial

system.

The industry bench mark for pricing European call option is based on the Black-Scholes

model. Let the price of security be S = ex. For concreteness, consider a vanilla call option

with a payoff function [ex − K]+ and maturing at some future time T . The option price in

general has the following time parameters

t : present calendar time ; T : maturity time ; τ : remaining time.

The BS option price is given by[33, 34, 35]

CBS(S(t), K, r, σ, (T − t)) = SN(d+)− e−r(T−t)KN(d−),

1https://en.wikipedia.org/wiki/Foreign exchange market.

§ 4.2. Quantum finance formulation 87

where

d± =ln(S/K) + r(T − t)± σ2

0(T − t)/2σ0

√(T − t)

; N(x) =1√2π

∫ x

−∞e−

12z2dz.

§ 4.2 Quantum finance formulation

Let x = dx/dt be the velocity of the logarithm of S. In general, the option price C can depend

on the price and velocity of the security. A payoff function H(x, v;K) can depend on both

the final stock value and velocity; for remaining time τ = T − t

C(x(t), x(t), τ,K) =

∫dxdx′P (x, x;x′, x′; τ)H(x′, x′;K),

where expression

P (x, x;x′, x′; τ).

is the conditional probability that the future value is x′, x′ at time T , given the value of x, x

at present time t. Furthermore, let

P (x, x;x′; τ) =

∫dx′P (x, x;x′, x′; τ),

where P (x, x;x′; τ) is the marginal conditional probability.

In the quantum finance formulation of option prices, the conditional probability P (x, x;x′, x′; τ)

is given by what is called the transition amplitude

K(x, x;x′, x′; τ).

The conditional probability P(x, x;x′, x′; τ) is given by appropriately normalizing the tran-

§ 4.2. Quantum finance formulation 88

sition amplitude and yields

P (x, x;x′, x′; τ) =K(x, x;x′, x′; τ)∫

dx′dx′K(x, x;x′, x′; τ),

P (x, x;x′; τ) =

∫dx′P (x, x;x′, x′; τ). (4.1)

The description of the stochastic evolution of a security is defined by the Hamiltonian H

operator for the security [7]. The transition amplitude, in general, is given by the matrix

element of the Hamiltonian operator [3]

K(x, x;x′, x′; τ) = 〈x, x|e−τH |x′, x′〉. (4.2)

The transition amplitude K(x, x;x′, x′;T, t) has another representation defined by the (Eu-

clidean) Feynman path integral over all possible paths x(t) of the security from its initial value

x′, x′ at time τ = 0 to its final value of x, x at time τ [3]. More precisely,

K =

∫DxeS . (4.3)

Up to a normalization, the path integral measure is given by

∫Dx =

τ∏t=0

∫ ∞−∞

dx(t).

The boundary conditions for all the allowed paths in the Feynman path integral given in Eq.

(4.3) is the following [36]

x(0) = x′, x(0) = x′; x(τ) = x, x(τ) = x. (4.4)

§ 4.3. Transition amplitude K 89

§ 4.3 Transition amplitude K

The model Hamiltonian H for the option price is given by [3]

H = − 1

2a

∂2

∂x2− x ∂

∂x+ bx2 +

1

2cx2. (4.5)

The Hamiltonian given in Eq. (4.5) yields the following ‘acceleration’ Lagrangian, derived in

[3] and given by

L = −1

2

(ax2 + 2b(x+ j)2 + cx2

); S =

∫ τ

0

dtL. (4.6)

Since the Lagrangian given in Eq. (4.6) is quadractic, the path integral can be solved

exactly using the classical solutions. The stochastic variable x is separated into two parts:

the classical solution xc and stochastic part ξ.

x = xc + ξ (4.7)

with the classical solution xc given by

δS[xc]

δx(t)= 0.

The classical solution has boundary conditions as Eq. (4.4) and we hence obtain

xc(0) = x′, xc(0) = x′; xc(τ) = x, xc(τ) = x. (4.8)

Hence, Eqs. (4.7) and (4.8) yield the boundary condition as below

ξ(0) = 0; ξ(τ) = 0; ξ(0) = 0; ξ(τ) = 0. (4.9)

§ 4.4. BY Model option price 90

The acceleration action S separates into two parts [5] and is given by

S = S[xc] + S[ξ]. (4.10)

Note S[ξ] is independent of x, x, x′, x′ and depends only on τ . The transition amplitude is

given by

K =

∫DxeS =

∫DξeSξ+Sc = N eSc . (4.11)

The functional integration is defined by

∫Dx =

τ∏t=0

∫ ∞−∞

dx(t) ;

∫Dξ =

τ∏t=0

∫ ∞−∞

dξ(t),

and the normalization is given by

N (τ) =

∫DξeSξ .

We obtain the final result that

K(x, x;x′, x′; τ) = N (τ) expSc(x, x;x′, x′; τ). (4.12)

§ 4.4 BY Model option price

The BY (Baaquie-Yang) model for option pricing – proposed in [37] – is based on the La-

grangian and action given in Eq. (4.6), and is unlike the Black-Scholes case. The BY price

for the European call option, at time t, is given by

CBY (x(t), x(t), τ,K) =

∫dx′P (x, x;x′; τ)[ex

′ −K]+.


Since the Lagrangian and action S[xc] given by Eq. (4.6) yields [3]

S[xc] = −1

2

∫ τ

0

dt(ax2c + 2b(xc + j)2 + cx2

c). (4.13)

The classical solution S[xc] with the boundary condition Eq. (4.4) is solved in the Appendix

§ 4.11.1. The conditional possibility is given by Eq. (4.59)

P (x, x;x′; τ) =

∫dv′P (x, x;x′, x′; τ) =

√1

2πν2exp− 1

2ν2(−x′ + ζx+ ξx+ j)

2.

The solution for the classical action Sc yields the following

ν2 =2Ωrω[ω sinh(2rτ)− r sin(2ωτ)]

a (r2 + ω2),

ζ = 4Ωrω[(r2 − ω2

)sinh(rτ) sin(ωτ) + 2rω cosh(rτ) cos(ωτ)],

ξ = −4Ωrω[ω sinh(rτ) cos(ωτ) + r cosh(rτ) sin(ωτ)], (4.14)

where

Ω =1

(r2 + ω2)2 − r2 (r2 − 3ω2) cos(2ωτ)− ω2 (ω2 − 3r2) cosh(2rτ). (4.15)

r and ω are defined in the Appendix Eq. (4.42) and given below

r ≡ Re

√b+ i√ac− b2

a

; ω ≡ Im

√b+ i√ac− b2

a

.

Some typical shapes of ν2, ξ, ζ, as a function of τ , are shown in Figures 4.1, 4.2, 4.3 and

4.4.


0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

τ(year)

ν2

Figure 4.1: ν2(τ)

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

τ(year)

Figure 4.2:√ν2/τ

0 0.5 1 1.5 2 2.5 3−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

τ(year)

Figure 4.3: ξ(τ)

0 0.5 1 1.5 2 2.5 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

τ(year)

Figure 4.4: ζ(τ)

Figure 4.5: Shape of parameters with a = 5; b = 8; c = 100. τ is remaining time.

§ 4.4.1 Martingale condition

Consider the case of FX options. Let the domestic currency be $ and the value of a foreign

currency is given by $ex. and let rf and rd be the risk free foreign and the domestic interest

rates, respectively. As discussed in [5], the Forex option martingale condition[38] is given by

e−rf τex = e−rdτ∫ +∞

−∞dx′P(x, x;x′; τ)ex

′. (4.16)

The Forex exchange martingale process is shown by Fig. 4.6


Figure 4.6: The Forex martingale process.

Recall that ex is the conversion from foreign currency to domestic currency, with rf and

rd being the foreign and domestic risk free interest rates, respectively.

From Eq. (4.17), the Forex option martingale condition is given by

e−rf τex = e−rdτ∫ +∞

−∞dx′P (x, x′; x; τ)ex

′= exp

(−rdτ +

ν2

2+ ζx+ ξv + j

). (4.17)

Hence the drift is given by

j = (rd − rf )τ −ν2

2. (4.18)

The martingale condition further requires that

ζ = 1; ξ = 0. (4.19)

When we fit the model to market data, as given in Figures 4.22, 4.23, we will see that

the martingale condition is violated. In particular, for large τ ζ deviates from 1 whereas ξ

converges to zero. The violation of the martingale condition shows that the market is not free

from arbitrage; one possible explanation of the imperfection of the market is that the market

has long memory that mitigates against the martingale condition. In fact, in our model the

deviation of the parameters ζ, ξ from 1 and 0, respectively, provides a quantitative measure

of the degree to which the market is imperfect.


§ 4.4.2 BY Option: market time

The FX call option BY price CBY (x, x;K, τ) is given by

CBY (x, x; τ,K) = e−rdτ∫ +∞

−∞dx′P (x, x′; x; τ)

[ex′ −K

]+

= e−rf τeζx+ξxN(d+)− e−rdτKN(d−),

where d± =ζx+ ξx− ln(K) + (rd − rf )τ ± ν2

2

ν.

We introduce two new parameter λ and η such that

t→ z = λ(t

λ)η. (4.20)

The function z(t) [7] is called market time to differentiate it from calendar time t. Market

time is the subjetive estimate of time in the minds of the traders whereas calendar time is

physical time. The parameters λ, η is a measure of market time, which is greater than calendar

time for t < λ and less for t > λ. The difference between t and z, when η < 1, is shown in

Figure 4.7.

0

y

t

y=t

y=

Figure 4.7: The t and z values when η < 1


The transition amplitude is now given by

K(x, x;x′, x′; τ) = 〈x, x|e−z0(τ)H |x′, x′〉; z0 = λ(τ

λ)η. (4.21)

where recall τ is the maturity of the option and the action is given by

S =

∫ z0

0

dzL(z). (4.22)

The parameters λ and η allows us to rescale and dilate calendar time τ and in doing so allows

us to fit effective volatilities that have a maximum at a future time that greater than 1.5

years. The recalibration is done in the following manner

ν2(τ)⇒ ν2(λ(τ

λ)η), (4.23)

ξ(τ)⇒ ξ(λ(τ

λ)η), (4.24)

ζ(τ)⇒ ζ(λ(τ

λ)η). (4.25)

The changes of the fitting parameters by different λ and η are shown by the Figures in Group1

and Group2.

Table 4.1: Fitting Parameters for Group 1 and 2

Group a b c λ η

Group1 1 2.48 19.54 1 1

Group2 1 2.48 19.54 0.3 0.4

Group 1: Group2:


0 500 1000 1500 2000 2500 30000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Figure 4.8: ν2(z) for Group 1

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

ν2

τ(year)

Figure 4.9: ν2(z) for Group 2

0 500 1000 1500 2000 2500 3000−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Figure 4.10: ξ(z) for Group 1

0 0.5 1 1.5 2 2.5 3−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

τ(year)

Figure 4.11: ξ(z) for Group 2

0 500 1000 1500 2000 2500 3000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Figure 4.12: ζ(z) for Group 1

0 0.5 1 1.5 2 2.5 30.7

0.75

0.8

0.85

0.9

0.95

1

τ(year)

Figure 4.13: ζ(z) for Group 2

It is obvious that when λ and η change from [1, 1] to [0.3, 0.4], the shape of the parameters

§ 4.5. Mapping BY Model to data 97

[µ, ξ, ζ] also varies significantly. For example, Figures. 4.16 and 4.17 can be fit very well with

λ = 1, but it is graphs of the type shown in Figure 4.18 that can only be fitted by the model

with λ 6= 1.

§ 4.5 Mapping BY Model to data

Every option has a remaining time τ = T − t; the option price is studied for a fixed remaining

time τ . Options can be studied for which remaining time τ remains constant, namely a

collection of options issued at different calendar time and maturing at a constant time in the

future. Another class of options is having a maturity time at some fixed calendar τ = T − t,

for which T is held fixed. Both these options are shown in Figure 4.14. In this chapter, we

find the options for which τ is held fixed, which is even complicated than the LIBOR market

[39].

For each instant t, the value of the security is represented by the model variable x(z(τ)).

To make the mapping of the model variables to market data more clear, let t be calendar time

and x = τ + t be the future time, as shown in Figure 4.14(a). We adopt the notation that

x(z) = x(t, z(τ)) : z(τ) = λ(τ

λ)η ; τ = x− t : constant τ

For the case of options with remaining time τ that is fixed, as shown in Figure 4.14(a), the

notation x(t, z(τ)) means the value of the stock price at time t for an option maturing at

future time t + τ . For options that mature at some fixed time T in the future, as shown in

Figure 4.14(b), we have the notation

x(z) = x(t, z(T − t)) : z(τ) = λ(T − tλ

)η ; τ = T − t : constant T


future time0

x(t’,τ)

x(t,τ)

t’

t

τ

t(calendar time)

future time0

t’

t

T

T

(a) (b)

x(t’,τ)

x(t,τ)

x(t’,τ(t’))

x(t,τ(t))

t(calendar time)

Figure 4.14: (a) The model variable x(t, τ) with different calendar times t, t′ but with thesame remaining time τ . (b) Model variable for fixed maturity time T , with remaining timeτ(t) 6= τ(t′).

Let xD(t) be the value of the data at time t. The calibration of the model is based on the

following mapping

xD(t) = x(t, z(τ)) : constant τ (4.26)

xD(t) = x(t, z(T − t)) : constant T (4.27)

The mapping from t to z, and the connection of data xD(t) to its representation in the model

x(z(t)), is shown in Figure 4.15. Eqs. (4.26) and (4.27) may look a bit strange since xD(t)

does not depend on τ ; the role played by τ in mapping the model to data will become more

clear when we compute the velocity of the model.

The velocity in the model is defined by

dx

dz=dt

dz

dx

dt.


From the definition of market time we have

dz

dt=

d

dt

(λ(x− tλ

)η)

= −η(τ

λ)η−1.

To relate the model velocity to the velocity given by data, we compare the value of data at

two nearby instants given by t and t− δ – as shown in Figure 4.15. For fixed remaining time

τ , as shown in Figure 4.15(a), we have the following

dx(t, z(τ))

dz=

(−η(

τ

λ)η−1

)−1 dx(z)

dt

=(−η(

τ

λ)η−1

)−1 x(t, z(τ))− x(t− δ, z(τ))

δ.

From the mapping given in Eq. (4.26) we obtain

dx

dz= −1

η

(τλ

)1−η· xD(t)− xD(t− δ)

δ. (4.28)

In the analysis of this chapter δ is chosen equal to 1 day=1/360 years.

. . . . . .

xD(t-δ))=x(t-δ,τ)

t

. . . . . .

x(T,Z(τ))

x(T,Z(T-t+δ))ε

(a) (b)

xD(t-δ))=x(t-δ,τ)

xD(t)=x(t,τ)xD(t)=x(t,τ)t

t-δ t-δx(t-δ,τ)

0 0T1 T1

t(calendar time) t(calendar time)

future time future time

Figure 4.15: (a) Model velocity for fixed remaining time τ . (b) Model velocity for fixedmaturity time T is found by comparing x(t, z(τ) to x(t− δ, z(τ + δ).


The mapping for the option with fixed maturity time T is given in Eq. (4.27) and yields

dx

dz=

x(t, z(T − t))− x(t− δ, z(T − t)− ε)ε

=xD(t)− xD(t− δ)

ε(4.29)

ε = λ(τ

λ)η − λ(

τ + δ

λ)η = −η(

τ

λ)η−1δ +O(δ2) : τ = T − t

where ε is shown in Figure 4.15(b). From Eq. (4.29), note that a change of δ in calendar time

is equal to a change of ε in market time z(T − t).

Note that the result given in Eq. (4.29) for an option with a fixed maturity T is the same

as the result for constant remaining time τ given in Eq. (4.28).

The mapping of the model velocity to data is given in Eq. (4.28). Note that dx/dz is the

velocity for an option maturing after remaining time τ , as shown in Figure 4.15. The options

for different remaining times have the empirical velocity dxD/dt scaled by the remaining time

τ for obtaining the effective model velocity dx/dz. The scaling factor is the same for options

with constant remaining time and with a fixed maturity time, and is the following

1

η

(τλ

)1−η.

Since η < 1 we see that the model’s velocity is enhanced for large remaining time τ , with the

effect of market time become more significant for τ >> λ.

Note for η → 1, z(τ) → τ , and hence market time becomes equal to remaining calendar

time, and the scaling factor becomes 1 as expected.

§ 4.6. Calibration of the BY Model 101

§ 4.6 Calibration of the BY Model

The fitting of the data is with the price of call option given by Eq. (4.20) 2

Cdata(τ) = CBY (rf , rd, z(τ), x, x). (4.30)

The market price Cdata is obtained from the Black-Sholes formula

Cdata(τ) = CBS(S,K, rf , rd, σATM , T − t0). (4.31)

Where σATM is provided by the market. The testing and calibration is given by using

Eqs. (4.30) and (4.31).

The at-the-money (ATM) options are often used to calculate the implied volatility because

they are the most traded contracts; implied volatility differs with strike prices and time to

expiration as well as depending on calendar time. We use the data from Bloomberg to obtain

implied volatility σATM in the form of a dimensionless number τG2data(τ), with at-the-money

implied volatility given by

σATM =√τG2

data(τ).

The FX volatility data is downloaded from Bloomberg and the following moving remaining

times are chosen as

τn|n = 1, 2, ...8 = [0.0833, 0.1667, 0.25, 0.5, 1, 1.5, 2, 3] years.

2The calibration of the model in [5] is for market ATM volatility Gdata and uses the formula

ν2(τη) = G2data(τ)τ

The shortcoming of this calibration is that the parameters λ is not used. This leads to the maximum of theeffective volatility in the paper being fixed to be at around 1.5 years in the future, and hence is unable to fitmany cases.

§ 4.7. Fitting Results 102

R-square and root mean square (RMSE) error are chosen to measure the goodness of fit. For

each calendar date t, there is a fit of volatility Cdata, so the R-square and RMSE error are

functions of calendar time t. R2 is defined as

R2(t) = 1−

8∑n=1

[Cdata(t, τn)− Cfit(t, τn)]2

8∑n=1

[Cdata(t, τn)− Cdata(t, τn)]2

. (4.32)

where τn is the remaining time and Cdata(t, τn) is the mean of Cdata(t) at calendar time t.

Higher R2 means better fit, and the exact fit has an R2 value equal to 1.

RMS error is defined by

RMSE(t) =

√√√√ 1

N

N∑n=1

[Cdata(t, τn)− Cfit(t, τn)

Cdata(t, τn)

]2

. (4.33)

§ 4.7 Fitting Results

In general, the data results in three typical shapes that can be fitted by the model and irregular

shapes that have no fit. The pattern C is in fact the same as pattern A; with the difference

that for pattern C the maximum has been pushed out far into the future. Pattern C can be

fitted by choosing a suitable λ.


0 0.5 1 1.5 2 2.5 310

10.5

11

11.5

12

Expiration time / year

vola

tili

ty σ

/ p

erce

nt

per

yea

r

σ data

σ fit

Figure 4.16: Pattern A, 2009-09-23

0 0.5 1 1.5 2 2.5 315

16

17

18

19

20

21


vola

tili

ty σ

/ p

erce

nt

per

yea

r

σ data

σ fit

Figure 4.17: Pattern B, 2009-02-02

0 0.5 1 1.5 2 2.5 36.5

7

7.5

8

8.5

9

9.5


vo

lati

lity

σ /

per

cen

t p

er y

ear

σ data

σ fit

Figure 4.18: The fourth pattern C, 2013-12-18

The graph below shows a set of data that is irregular and can not be fitted.


0 0.5 1 1.5 2 2.5 310.3

10.32

10.34

10.36

10.38

10.4

10.42

10.44

10.46

10.48


vo

lati

lity

σ /

per

cen

t p

er y

ear

σ data

σ fit

Figure 4.19: Irregular data, 2008-08-28

The model has five free parameters, as shown in Table 1, and options with 8 different

remaining time are being fitted. The fits are usually very good, except for exceptional periods.

To test our hypothesis, we mark those periods for which the fit fails, namely for which R2 <

0.99.

We analyze the exchange rate of the major international currencies against the US Dollar,

namely the Euro, the Pound (GBP), the Japanese Yen, the Swiss Franc (CHF), the Australian

Dollar (AUD), the Canadian Dollar and the New Zealand Dollar.

Of these we analyze the option of the exchange rate of five major currencies against the US

Dollar, which are the Euro, GBP, CHF, Yen, AUD. The Canadian Dollar is highly correlated

with the US Dollar and the New Zealand Dollar to the AUD, and hence their analysis does

not give any new insights.

The scheme of our analysis is the following.

• We find the periods for which the model fails.

• We seek to explain the reason of their failure as being due to market turbulence.

• If the failure of model for a time period occurs only for a particular a country, we look


for reasons such as policy changes or international developments that specifically impact

on that country.

• If for a time period the model fails for all the major currencies, we seek an explanation

that originates in the international financial system.

The graphs below shows one example of the EURUSD fitting at 2, January, 2008.

Table 4.2: Parameters for EURUSD Fitting

R2 a b c λ η

0.9992 23.96 0.801 2.48 0.843 0.325


0 0.5 1 1.5 2 2.5 30.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

data

fit

(year)Remaining time

Figure 4.20: Option price fitting

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(year)

Remaining time

Figure 4.21: ν2

0 0.5 1 1.5 2 2.5 3−1.5

−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

(year)

Remaining time

Figure 4.22: ξ

0 0.5 1 1.5 2 2.5 30.994

0.995

0.996

0.997

0.998

0.999

1

1.001

(year)Remaining time

Figure 4.23: ζ


The graphs below shows the parameters fitting for EURUSD from 2009/01/01 to 2009/04/21.

0 10 20 30 40 50 60 700.9997

0.9997

0.9998

0.9998

0.9999

0.9999

1

time lag(day)

Figure 4.24: Option price fitting R2

0 10 20 30 40 50 60 702

3

4

5

x 10−4

time lag(day)

Figure 4.25: Option fitting rmse

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time lag(day)

Figure 4.26: r

0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time lag(day)

Figure 4.27: ω

0 10 20 30 40 50 60 700.5

1

1.5

2

2.5

3

time lag(day)

Figure 4.28: λ

0 10 20 30 40 50 60 700.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

(time lag(day)

Figure 4.29: η

§ 4.8. Global crisis 108

§ 4.8 Global crisis

Our focus is from 2008 to the present (2016). Before embarking on a study of specific cur-

rencies, we expect that events that have had a major impact on the international financial

system should be enumerated and we should be able to find signals of these globally. This

would provide a check on our analysis as well as show the consistency of using FX option

price as a gauge of the international financial system.

Two international events with a potentially significant impact on almost all countries of

the world are discussed below.

• The period of international significance is the 2007-2008 financial crisis – that is indicated

by dashed lines for all the currencies.3

Figure 4.30: TED and the financial crisis in 2008.4

The TED spread is an acronym for the difference between the interest rates on interbank

loans and on short-term U.S. government debt. The TED spread is calculated in the

3https://en.wikipedia.org/wiki/Financial crisis of 2007%E2%80%9308.

§ 4.8. Global crisis 109

market by measuring the difference between the three-month LIBOR and the three-

month US Treasury-bills interest rate. A large TED spread indicates an increase in

the risk of short term commercial loans and a lack of liquidity. The sudden spike in

TED reaching almost 5%, as shown in Figure 4.30 (where USGG3M is the US 3-month

treasury bill), is a reflection of the 2008 financial meltdown.

The model in [5] showed that the Euro FX option price data could also be used to gauge

the market. The model could correctly reflect the occurrence of the global financial crisis

on the EURUSD exchange rate. This result is re-produced in Figure 4.31 and indicated

by I.

• We examine the impact of the 2007-2008 financial separately for the five major curren-

cies.

• United States debt-ceiling crisis of 2013 lasted January-October, 2013. 5 Our study

shows that the crisis had a strong impact on Europe and Japan but little impact on

Australia and Switzerland; this is because both Australia and Switzerland are not that

closely tied to the US financial policy as are Europe and Japan.

• According to our model based on the pricing of FX options for 2014-2015, the Euro,

British and Swiss Franc were impacted by the drastic fall in oil-prices and the deterio-

ration of the relation of the US and Europe with Russian.

• The black swan event for the Swiss Franc was caused internally by removing the pegging

of the Swiss Franc to the Euro.

Our study also shows many other instabilities, some of which are country specific.

5https://en.wikipedia.org/wiki/United States debt-ceiling crisis of 2013.

§ 4.9. Result for five major FX options 110

§ 4.9 Result for five major FX options

From the graphs we can see that all of Europe except the Swiss Franc were heavily influenced

by the 2008 financial crisis. Each graph for the FX options is alongside another graph showing

the volatility of the FX rate. We compare these graphs to see if the FX volatility is also an

accurate gauge of market instability. We find that the volatility of the FX rate is not a very

precise criterion of financial instability. For example, as shown in Fig. 4.31 (b), although

FX rate had a high volatility during the 2008 financial crisis, there are other periods of large

volatility with apparently no market instability.

§ 4.9.1 Euro

2008 2009 2010 2011 2012 2013 2014 2015 Date(year)0

50

100

150

200

250

300

350

400

450

500

2

0.975

0.98

0.985

0.99

0.995

1

1.005

2008 2009 2010 2011 2012 2013 2014 2015

Op

tio

n p

rice

R2

Date(year)

(a) (b)

(I) (II)

Ex

ch

an

ge

Vo

lati

lity

Figure 4.31: (a) R2 of EURUSD and (b) Fx volatility of EURUSD

The price of EURUSD option was disordered for a short period after the 2008 financial crisis.

The 2008 financial crisis, however, triggered sovereign debt crisis in Europe in 2013, and this

is marked as II in Figure 4.31. The crisis denoted by II could also have had contributions

from the US financial crisis of 2013. Hence, financial instability was correctly gauged by the

failure of the model.


§ 4.9.2 Australia Dollar

2008 2009 2010 2011 2012 2013 2014 2015 Date(year)0

80

160

240

320

400

480

560

640

720

800

2

Date(year)0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2008 2009 2010 2011 2012 2013 2014 2015

Op

tio

n p

rice

R2

(a) (b)E

xc

ha

ng

e V

ola

tili

ty

Figure 4.32: (a) R2 of AUDUSD and (b) Fx volatility of AUDUSD

The graph shows that for the period of the 2008 crisis, the model correctly reflects the financial

meltdown.

§ 4.9.3 Swiss Franc

2008 2009 2010 2011 2012 2013 2014 2015 Date(year)0

50

100

150

200

250

300

350

400

450

500

2

0.75

0.8

0.85

0.9

0.95

1

2008 2009 2010 2011 2012 2013 2014 2015 Date(year)

Op

tio

n p

rice

R2

(a) (b)

(I)

(II)

Ex

ch

an

ge

Vo

lati

lity

Figure 4.33: (a) R2 of CHFUSD and (b) Fx volatility of CHFUSD


The behavior of the model for the case of the Swiss Franc is one of the most interesting.

To start with, the 2007-2008 financial crisis left the Swiss Franc untouched since it did not

take part in the leveraging and high risk instruments that primarily the US and UK banks,

and to a lesser extent the European banks were engaged in. The fittings above confirms

our expectation that the Swiss Franc was not affected much in 2008; in constrast, the Euro,

including the British Pound, were highly impacted.

The Swiss dollar has a very big FX volatility from 2010 to 2012 as shown in Fig 4.33 (b).

This is because of a policy of the Swiss government. After the crisis broke out in 2008, there

was a flight to safety in Europe, with large flows of money to Swiss Bank. This raised the

value of the Swiss Franc and led to the Swiss policy, announced in September 2011, that set

an upper limit to the valuation of the CHF to EURO to be capped at 1.2. So the fluctuations

marked II in Figure 4.33(b) describes this high volatility period.

In 2015,January, 15th, the Swiss suddenly canceled the upper limit of CHF against the

Euro, which is a rare and unpredictable event and can be called a ”Black Swan” event. This

rare event is correctly captured by the failure of the model, and is marked by I in Figure

4.33(a).


§ 4.9.4 British Pound

2008 2009 2010 2011 2012 2013 2014 2015 Date(year)0

50

100

150

200

250

300

350

400

450

500

22008 2009 2010 2011 2012 2013 2014 2015 Date(year)

0.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

Op

tio

n p

ric

e R

2

(a) (b)

Ex

ch

an

ge

Vo

lati

lity

Figure 4.34: (a) R2 of GBPUSD and (b) Fx volatility of GBPUSD

The British Pound, as can be seen from Figure 4.34, was highly unstable in 2007 leading to the

crash of 2008. This is because it is closely tied to the behavior of the US Dollar. The British

Pound was fairly stable after 2008, becoming unstable in 2013-2014. It is relatively stable

now, compared to other major currencies. The volatility of the FX rate does not provide

any sign of instability, in contrast to the instability shown by the failure of the model for the

period of 2013 and end of 2014.


§ 4.9.5 Japanese Yen

2008 2009 2010 2011 2012 2013 2014 2015 Date(year)0

50

100

150

200

250

300

350

400

450

500

Ex

ch

an

ge

Vo

lati

lity

2

Date(year)0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

2008 2009 2010 2011 2012 2013 2014 2015

Op

tio

n p

rice

R2

(a) (b)

Figure 4.35: (a) R2 of JPYUSD and (b) Fx volatility of JPYUSD

According to Figures 4.31(a) and 4.34(a), from 2008 to 2009, the EUR and GBP has imme-

diately disordered when the crisis of 2007-2008 occurred. The Japanese Yuan option, as seen

in Figures 4.35, was immediately affected between 2009 and 2010. This fact seems to show

that although Japan has the same financial policy as the Western countries, the crisis had a

delayed effect on Japan.

§ 4.10 Conclusion

The model for FX options is based on a Gaussian Lagrangian. The hypothesis that the option

price cannot fit the market’s behavior when the market is undergoing large scale fluctuations

and changes seems to be borne out by data. The accuracy of the fit of the model to the market

value of the FX option price is seen to be quite sensitive to the markets characteristics, with

the model’s accuracy falling suddenly with the onset of instability.

The BY option model is sensitive enough to represent the differences in the various markets.

§ 4.11. Appendix 115

For example, the BY model for the FX options for CHF behaves quite differently from those

for the AUD. The fittings double demonstrates that the model could accurately describe the

trend of each country.

The volatility of the FX rates do not seem to be an accurate gauge of the market’s insta-

bility, with the FX volatility providing no clear signal of the onset of market instability.

§ 4.11 Appendix

§ 4.11.1 Classical Solution

From Eq. (4.6), the lagrangian is given by

L = −1

2

(ax2 + 2b(x+ j)2 + cx2

); S =

∫ τ

0

dtL (4.34)

The transition amplitude in Feynman path integral is given by Eq. (4.12)

K = N eSC (4.35)

B.C : x(0) = x′, x(0) = v′; x(τ) = x, x(τ) = v (4.36)

The classical solution satisfies the equation below

∂S

∂x(t)|x(t)=xc(t) = 0 (4.37)

From the Euler-Lagrangian equation, the classical solution xc(t) satisfies equation as below

a....x c(t)− 2bxc(t) + cxc(t) = 0. (4.38)


According to the market data, the solution should in the complex root and hence

b2 − ac < 0 (4.39)

Difine yc as the four conjugate roots of the equation

ay4c − 2by2

c + c = 0 (4.40)

The four complex solution are as below

y = ±r ± iω (4.41)

Where,

r ≡ Re

√b+ i√ac− b2

a

(4.42)

ω ≡ Im

√b+ i√ac− b2

a

(4.43)

Then the relationship from [a, b, c] to [r, ω] is as below

b = −a(r2 − ω2)

c = a(r2 + ω2)2

Using the notation of r and ω, the general solution of xc(t) is given by

xc(t) = ert(a1 sinωt+ a2 cosωt)e−rt(a3 sinωt+ a4 cosωt) (4.44)

where a1,2,3,4 are constants solved by the boundary conditions 4.36.


The action S yields

S = S[xc + ε]

= −1

2

∫ τ

0

dt(a(xc + ε)2 + 2b(xc + ε+ j)2 + c(xc + ε)2

)= S[xc] + S[ε] +R (4.45)

where Sc is the classical action

S[xc] = −1

2

∫ τ

0

dt(ax2

c + 2b(xc + j)2 + cx2c

)(4.46)

S[ε] = −1

2

∫ τ

0

dt(aε2c + 2bε2 + cε2 + 2b · 2jε

)

The residual term R is

R = −∫ τ

0

dt (axcε+ 2bxcε+ cxcε)

= (−axcε− 2bxcε+...x cε)|τ0 −

∫ τ

0

dt ε(a....x − 2bxc + cxc). (4.47)

From Eqs. (4.38) and (4.9),

R = 0. (4.48)

Integrating the classical action Sc in Eq. (4.46) by part, and applying the equations of


motion, the action can be expressed only in terms of the boundary conditions

Sc = −1

2

∫ τ

0

dt d(−a...x cxc + axcxc + 2bxcxc + 4bjxc + bj2) + xc(a

....x c − 2bxc + cxc)

= − 1

2

(−a...

x cxc + axcxc + 2bxcxc + 4bjxc + bj2)∣∣∣∣τ

0

= −1

2

4∑I,J=1

xIMIJxJ − 2bjx1 + 2bjx4 − bj2τ. (4.49)

To find out coefficient MIJ , assuming j=0,

Sc = −1

2

4∑I,J=1

xIMIJxJ (4.50)

For the notation, we rewrite the xi, vi, xf , vf as

xi = x1; vi = x2; (4.51)

xf = x3; vf = x4 (4.52)

From Eq. (4.49), the derivatives of Sc yield MIJ given by

MIJ = − ∂2S

∂xI∂xJ(4.53)

According to the symmetry of the transition amplitude discussed in [3]

M11 = M33;M22 = M44;M12 = −M34; M14 = −M23; (4.54)


The result for MIJ is given as below

M11 =2arω (r2 + ω2) ((−1 + e4rτ )ω + 2e2rτr sin[2τω])

ω2 + e4rτω2 − 2e2rτ (r2 + ω2) + 2e2rτr2 cos[2τω]

M12 = −(−2a (1 + e4rτ ) r2ω2 + b (ω2 + e4rτω2 − 2e2rτ (r2 + ω2)) + 2e2rτr2 (b+ 2aω2) cos[2τω])

(ω2 + e4rτω2 − 2e2rτ (r2 + ω 2) + 2e2rτr2 cos[2τω])

M13 = −4aerτrω (r2 + ω2) ((−1 + e2rτ )ω cos[τω] + (1 + e2rτ ) r sin[τω])


M14 =4aerτ (−1 + e2rτ ) rω (r2 + ω2) sin[τω]


M22 = − 2arω (ω − e4rτω + 2e2rτr sin[2τω])


M23 = − 4aerτ (−1 + e2rτ ) rω (r2 + ω2) sin[τω]


M24 =4aerτrω (− (−1 + e2rτ )ω cos[τω] + (1 + e2rτ ) r sin[τω])


Hence we obtain Sc given by

Sc(xi, vi, xf , vf ) = −1

2M11(x2

i + x2f )−

1

2M22(v2

i + v2f )−M13xixf −M24vivf (4.55)

−M12xivi −M34xfvf −M14xivf −M23xfvi − 2bjxi + 2bjxf − bj2τ (4.56)

Expression of xi, vi, xf , vf are transferred to paper by

x′ = xi, v′ = vi; x = xf , v = vf (4.57)

According to the definition for the kernel K

K(x, v;x′, v′, τ) = N eSc(x,v;x′,v′,τ)

Sc(x, v;x′, v′, τ) = −1

2M11(x′2 + x2)− 1

2M22(v′2 + v2)−M13x

′x−M24v′v

−M12x′v′ −M34xv −M14x

′v −M23xv′ − 2bjx′ + 2bjv − bj2τ


The conditional probability distribution is given by

P (x, v;x′, v′, τ) =eSc(x,v;x′,v′,τ)∫

dx′dv′eSc(x,v;x′,v′,τ)(4.58)

The marginal conditional probability distribution is given by

P (x, v;x′; τ) =

∫dv′P (x, v;x′, v′; τ)

=

√1

2πν2exp− 1

2ν2(−x′ + ζx+ ξv + j)

2 (4.59)

where

ν2 =2Ωrω[ω sinh(2rτ)− r sin(2ωτ)]

a (r2 + ω2)

ζ = 4Ωrω[(r2 − ω2

)sinh(rτ) sin(ωτ) + 2rω cosh(rτ) cos(ωτ)]

ξ = −4Ωrω[ω sinh(rτ) cos(ωτ) + r cosh(rτ) sin(ωτ)] (4.60)

and

Ω =1

(r2 + ω2)2 − r2 (r2 − 3ω2) cos(2ωτ)− ω2 (ω2 − 3r2) cosh(2rτ)(4.61)

Chapter 5

Risky Forward Interest Rates and

Swaptions: Quantum Finance Model

and Empirical Results

Both the US forward interest rates and the term structure for the spread are modeled by a two

dimensional Euclidean quantum field. As a precursor to the evaluation of put option of the

Singapore coupon bond, the quantum finance model for swaptions is tested using empirical

study of swaptions for the US Dollar – showing that the model is quite accurate. A prediction

for the market price of the put option for the Singapore coupon bonds is obtained. The

quantum finance model is generalized to study the Malaysian case and the Malaysian forward

interest rates are shown to have anomalies absent for the US and Singapore case. The model’s

prediction for a Malaysian interest rate swap is obtained.

§ 5.1 Introduction

Coupon and zero coupon bonds form the bedrock of the debt market [40, 41, 42]. Quantum

finance models of risk free coupon bonds match market data very well [1] and so it is natural

121


to extend the analysis to coupon bonds that carry issuer country’s risk of default. The

possibility of default results, as is well known, results in an issuer having to pay coupons on

risky bonds[43, 44] that are higher than the risk free case – the spread being determined by

rating agencies.

In Sections § 5.2, § 5.3 and § 5.4, the quantum finance model for the risk free and risky

forward interest rates are defined; the correlation functions of the model are evaluated and

an explicit expression for the stiff propagator is obtained. The market correlator is defined

in Section § 5.5, and the empirical calibration of all the parameters of the model, for the US,

Singapore and Malaysian forward interest rates are obtained and summarized in Sections § 5.6,

§ 5.8 and § 5.9. The swaption for the forward interest rates is defined in Section § 5.10 and the

US, Singapore and Malaysian markets are empirically analyzed. The results of the paper are

summarized in Section § 5.11.

Sections § 5.12 to § 5.16 are Appedicies to the main text. The derivation of the risky

coupon bond option is carried out in Section § 5.12, and in Section§ 5.13 the equations used

in the empirical analysis are derived from first principles. Swaption data in the market is

given in terms of Black’s swaption formula, and a derivation of this is given in Section § 5.14.

Section § 5.15 is a summary of the algorithm required to obtained the zero coupon bonds from

coupon bonds and Section § 5.16 derives an efficient algorithm to obtain an accurate result fo

the forward interest rates from the zero coupon bonds.

Although we did not use Sections § 5.15 and § 5.16 in our empirical analysis since we

obtained risk free and risky forward interest rates from Bloomberg, for many emerging mar-

kets forward interest rates data is not available and only the sovereign coupon bond data is

available; for these cases one needs the results of Sections § 5.15 and § 5.16 to carry out the

analysis.

§ 5.2. Quantum finance model of forward interest rates 123

§ 5.2 Quantum finance model of forward interest rates

The risk free US zero coupon bonds B(t, T ) are determined by the forward interest rates

f(t, x) as follows

B(t, T ) = exp−∫ T

t

dxf(t, x).

As shown in Figure 5.1(a), let calendar time t be the vertical axis, and let the horizontal

axis x > t be future time. The shaded portion represents the forward interest rates f(t, x):

every point in the shaded domain corresponds to one forward interest rate.

t0

t0Future Time x

Cal

endar

Tim

e t

x = t

f(t0, x)

0T

0t

0T iT

Future Time

Cal

endar

Tim

e

0t

B ( , )0T iT

B ( , )0t iT

Figure 5.1: a) The semi-infinite domain with two boundaries on which f(t, x) and A(t, x) aredefined. b) The zero coupon bond for two different times t0 and T0.

The zero coupon bonds B(t, T ) is shown in Figure 5.1(b) at two different calendar time t0

and T0 . The shaded domain shows the forward interest rates that are bounded by the two

coupon bonds.

In the quantum finance formulation of forward interest rates, the two-dimensional Eu-

clidean quantum field A(t, x) is a random stochastic field driving the forward interest rates;

the forward rates f(t, x) are given by [1]

∂f

∂t(t, x) = α(t, x) + σ(t, x)A(t, x) (5.1)

⇒ f(t, x) = f(t0, x) +

∫ t

t0

dt′α(t′, x) +

∫ t

t0

dt′σ(t′, x)A(t′, x). (5.2)


The deterministic functions α(t, x), σ(t, x) are the drift and volatility, respectively, of the

forward interest rates. A(t, x) is a drift-less Gaussian quantum field, defined on a semi-infinite

plane with two boundaries: one at t = t0 and the other along x = t, as shown in Figure 5.1

(a). The field A(t, x) satisfies the Neumann boundary condition [1]

∂A(t, x)

∂x

∣∣∣x=t

= 0.

The action functional for the field A(t, x), called the stiff action because of the second order

derivative in the future time direction, is defined by [1]

S[A] = −1

2

∫ ∞t0

dt

∫ ∞t

dxA2(t, x) +

1

µ2(∂A(t, x)

∂x)2 +

1

λ4(∂2A(t, x)

∂x2)2

(5.3)

=

∫PL[A]. (5.4)

The partition function is given by Feynman path integral [1]

Z =

∫DA eS[A] ;

∫DA =

∏t

∏x≥t

∫ +∞

−∞dA(t, x).

The risky coupon bond, denoted by R(t, T ), is determined by the risky forward rates g(t, x),

as follows

R(t, T ) = exp−∫ T

t

dxg(t, x).

The time evolution of the risky forward rates g(t, x) is given by

∂g

∂t(t, x) = β(t, x) + σ(t, x)A(t, x) + γ(t, x)ξ(t, x) (5.5)

⇒ g(t, x) = g(t0, x) +

∫ t

t0

dt′β(t′, x) +

∫ t

t0

dt′σ(t′, x)A(t′, x) + γ(t′, x)ξ(t′, x). (5.6)

The deterministic function β(t, x) is the drift of the risky forward interest rates g(t, x). Due


to the trapezoidal structure of the domain of the forward interest rates, one can consistently

choose

α(t, x) = α(θ) ; β(t, x) = β(θ) ; σ(t, x) = σ(θ) ; γ(t, x) = γ(θ)

, where remaining future time θ is defined by

θ = x− t ≥ 0.

The basic assumption of our model is that the risky forward interest rates are driven by

the risk free forward interest rates f(t, x) – with the risky rates having a spread above the

risk free rates. The spread is given by the quantum field for the spread term structure, and is

denoted by ξ(t, x). The volatility of the spread is given by the deterministic function γ(t, x)

and is determined by its correlation function.

The spread also obeys the Neumann boundary condition

∂ξ(t, x)

∂x

∣∣∣x=t

= 0.

The stiff action functional for the risky forward interest rates is obtained by extending the

risk free case given in Eq. (5.3), and yields

S[A; ξ] = −1

2

∫ ∞t0

dt

∫ ∞t

dxA2(t, x) +

1

µ2(∂A(t, x)

∂x)2 +

1

λ4(∂2A(t, x)

∂x2)2

− 1

2

∫ ∞t0

dt

∫ ∞t

dxξ2(t, x) +

1

µ2(∂ξ(t, x)

∂x)2 +

1

λ2(∂2ξ(t, x)

∂x2)2

+

∫ ∞t0

dt

∫ ∞t

dxdx′∆(x− t, x′ − t)A(t, x)ξ(t, x′) (5.7)

=

∫PL[A; ξ].

The cross-term ∆(x− t, x′ − t) connects the fields A(t, z), ξ(t, z′) at the same calendar time,

§ 5.3. Correlation functions 126

but for different future times: this is consistent with the other terms in the Lagrangian, for

which all the terms are defined for the same calendar time.

The partition function is given by Feynman path integral

Z =

∫DADξ eS[A;ξ]. (5.8)

§ 5.3 Correlation functions

The auto- and cross-correlation functions of the risky and risk free interest rates can be

computed from the correlation functions of the quantum fields A(t, x), ξ(t, x). Using the

Neumann condition for the fields A(t, x), ξ(t, x), the action given in Eq. (5.7) yields, after an

integration by parts and in matrix notation, the following

S[A; ξ] = −1

2

∫ ∞t0

dt

∫ ∞t

dxdx′

[A(t, x) ξ(t, x)

]

×

(1− 1

µ2∂2

∂x2+ 1

λ4∂∂x4

)δ(x− x′) −∆(x− t, x′ − t)

−∆T (x− t, x′ − t) (1− 1µ2

∂2

∂x2+ 1

λ4∂∂x4

)δ(x− x′)

A(t, x′)

ξ(t, x′)

≡ −1

2

∫ ∞t0

dt

∫ ∞t

dxdx′

[A(t, x) ξ(t, x)

]M−1(x− t, x′ − t)

A(t, x′

ξ(t, x′)

(5.9)

where

M−1(x− t, x′ − t) =

D−1(x− t, x′ − t) −∆(x− t, x′ − t)

−∆T (x− t, x′ − t) C−1(x− t, x′ − t)

(5.10)


The risk free forward interest rates f(t, x) should be weakly correlated with the risky

forward interest rates g(t, x) since one does not expect, for instance, the Singapore bonds to

strongly influence the US zero coupon bonds. Hence, we expect that

|∆(x− t, x′ − t)| ≤ 0.1 ∀ x− t, x′ − t

An empirical analysis, discussed later in Section § 5.8, will approximately confirm this expec-

tation. All the calculations are done to O(∆).

Let

θ = x− t ; θ′ = x′ − t

To leading order in ∆, we have, in matrix notation

M(θ, θ′) ≡

M11(θ, θ′) M12(θ, θ′)

M21(θ, θ′) M22(θ, θ′)

=

D(θ, θ′) (D∆C)(θ, θ′)

(C∆TD)(θ, θ′) C(θ, θ′)

+O(∆2)(5.11)

The correlation function between the field’s fluctuations at two different future times is

given by the

E[A(t, x)A(t′, x′)] =1

Z

∫DADξ A(t, x)A(t′, x′)eS[A;ξ]

= M11(θ, θ′) = δ(t− t′)D(θ, θ′) +O(∆2). (5.12)

where the δ(t− t′) has been factored out for future convenience. The correlation function of

the spread is determined by

E[ξ(t, x)ξ(t′, x′)] =1

Z

∫DADξ ξ(t, x)ξ(t′, x′)eS[A;ξ]

= M22(θ, θ′) = δ(t− t′)C(θ, θ′) +O(∆2). (5.13)


For the action given in Eq. (5.7), it follows that

1− 1

µ2

∂2

∂x2+

1

λ4

∂4

∂x4

D(θ, θ′)

∣∣∣Neumann

= δ(x− x′) +O(∆2), (5.14)

and

1− 1

µ2

∂2

∂x2+

1

λ4

∂4

∂x4

C(θ, θ′)

∣∣∣Neumann

= δ(x− x′) +O(∆2). (5.15)

D(θ, θ′), C(θ, θ′) are propagators that measure the effect that the fluctuations of the fields

A(t, θ+ t), ξ(t, θ+ t) at point t, x has on the fluctuations of A(t′, θ′+ t), ξ(t′, θ′+ t′) at another

point t′, x′, respectively.

The cross-correlation function of the risk free forward rates f(t, x) with the spread ξ(t, x)

is given by

E[A(t, x)ξ(t′, x′)] =1

Z

∫DADξ A(t, x)ξ(t′, x′)eS[A;ξ]

= M12(θ, θ′) = δ(t− t′)(D∆C)(θ, θ′) +O(∆2). (5.16)

Note that the result above for E[A(t, x)ξ(t′, x′)] is not a symmetric matrix since the two fields

are in-equivalent.

The concept of market time is required for relating the action functional to the empirical

correlation functions. The Lagrangian is written in terms of market future time z given by

x→ z = ω

(x− tω

)η: x ∈ [t,∞] ⇒ z ∈ [0,∞]

§ 5.4. Stiff propagator 129

The action functional describing the market’s behavior is given by [1]

S[A; ξ] = −1

2

∫ ∞t0

dt

∫ ∞0

dzA2(t, z) +

1

µ2(∂A(t, z)

∂z)2 +

1

λ4(∂2A(t, z)

∂z2)2

− 1

2

∫ ∞t0

dt

∫ ∞0

dzξ2(t, z) +

1

µ2(∂ξ(t, z)

∂z)2 +

1

λ4(∂2ξ(t, x)

∂z2)2

+

∫ ∞t0

dt

∫ ∞0

dzdz′∆(z, z′)A(t, z)ξ(t, z′) (5.17)

=

∫PL[A; ξ].

§ 5.4 Stiff propagator

The Neumann boundary condition leads to the following Fourier expansion

A(t, x) =

∫ +∞

0

dk

π/2eiωt cos(kθ)A(t, k) ; θ = x− t (5.18)

The stiff propagator is given by [1]

D(θ; θ′) = λ4

∫ +∞

0

dk

π/2

cos(k(x− t)) cos(k(x′ − t))λ4 + λ4

µ2k2 + k4

= λ4

∫ +∞

−∞

dk

2π· e

ik(x+x′−2t) + eik(x−x′)

λ4 + λ4

µ2k2 + k4

(5.19)

Note that

λ4 + (λ2/µ)2k2 + k4 = (k2 + α+)(k2 + α−)

with α± =λ4

2µ2[1±

√1− 4(

µ

λ)4 ]

§ 5.4. Stiff propagator 130

and yields

1

λ4 + (λ2/µ)2k2 + k4=( 1

α+ − α−)[

1

k2 + α−− 1

k2 + α+

] (5.20)

Define new variables

θ± = θ ± θ′ where θ = x− t ; θ′ = x′ − t (5.21)

Eqs. (5.19) and (5.20) yield

D(θ+; θ−) =( λ4

α+ − α−)[ 1

α−d(θ+; θ−;

√α−)− 1

α+

d(θ+; θ−;√α+)

](5.22)

where

d(θ+; θ−;√α±) =

√α±

2

[e−√α±θ+ + e−

√α±|θ−|

](5.23)

The solution for α± yields three distinct cases, namely, when α± is real, complex or

degenerate, each with unique characteristics [3]. From the previous studies of quantum finance

models of interest rates, the real branch of the stiff propagator is realized in the debt market

[1], for which

µ <√

2λ ⇒ α± : Real

Choose the following parametrization

α± = λ2e±b ; e±b =λ2

2µ2

[1±√

1− 4(µ

λ)4]

; b ≥ 0 (5.24)

§ 5.5. Market correlators 131

Note the system is critical for λ2 = 2µ2 ⇒ b = 0, and yields

α+ = α−

In this parametrization, from Eqs.(5.22) and (5.23)

Gb(θ+; θ−) =λ

2 sinh(2b)

[e−λθ+ cosh(b) sinhb+ λθ+ sinh(b)+

e−λ|θ−| cosh(b) sinhb+ λ|θ−| sinh(b)]

(5.25)

§ 5.5 Market correlators

Both calendar and future time are discretized; for ease of notation, only calendar time is

explicitly discretized and future time is written in the continuum notation. Hence

f(t, x) = f(t, θ + t)→ f(tn, θ + tn) ; g(t, x)→ g(tn, θ + tn) ; tn = εn ; θ = x− tn.

Time derivative is approximated by a finite difference and

∂f(t, x)

∂t→ 1

ε(f(tn, θ + tn)− f(tn−1, θ + tn−1)) ≡ 1

εδf(tn, θ + tn)

similarly

∂g(t, x)

∂t→ 1

εδg(tn, θ + tn)

The evolution equation yields

δf(tn, θ + tn) = εα(tn, θ + tn) + εσA(tn, θ + tn), (5.26)

δg(tn, θ + tn) = εβ(tn, θ + tn) + εσA(tn, θ + tn) + εγξ(tn, θ + tn). (5.27)


All correlation functions depend only on remaining time θ = x − t; this property of the

quantum field theory model is crucial in the empirical analysis.For any correlation function

O(t, x; t′x′), one has for the expectation value

E[O(t, x; t′, x′)] = E[O (t, t+ (x− t); t′, t′ + (x′ − t′))] = δ(t− t′)Φ(θ, θ′).

Setting t = t′, and using the fact that for discretized time, we have

δ(0) =1

ε

yields

E[O(t, x; t, x′)] = E[O (t, θ + t; t, θ′ + t)] =1

εΦ(θ, θ′). (5.28)

Correlation functions are evaluated by assuming averages over historical data are equal to

ensemble averaging. Hence, due to Eq. (5.28), holding θ, θ′ fixed, a sum is taken over the past

historical data to yield

Φ(θ, θ′) = εE[O(t, x; t, x′)] ≡ ε

N

n=−1∑−N

O (tn; tn + θ; tn + θ′) .

Define the connected correlator by

E[AB]c = E[AB]− E[A]E[B].


The connected correlation functions, for t = t′, from Eqs. (5.12) and (5.16) and using δ(0) =

1/ε, are given by

E[δf(t, x)δf(t′, x′)]c = δ(t− t′)ε2σ(θ)DE(θ, θ′)σ(θ′) = εσ(θ)DE(θ, θ′)σ(θ′) (5.29)

E[δ(g − f)(t, x)δ(g − f)(t′, x′)]c = δ(t− t′)ε2γ(θ)CE(θ, θ′)γ(θ′) = εγ(θ)CE(θ, θ′)γ(θ′) (5.30)

For equal time t = t′, note that the definition of volatilities σ, γ in the defining equations Eq.

(5.1) and (5.5) can be changed up to a scaling factor, which is taken to be equal to 1/εD(θ, θ)

and 1/εC(θ, θ), respectively. Using this scale factor, one obtains [7]

εDE(θ, θ) = 1 = εCEθ, θ) (5.31)

To simplify the notation, for equal time one can ignore the time index since the correlator

does not depend on it. We adopt the notation

E[δf(t, t+ θ)δg(t, t+ θ′)] ≡ E[δfδg](θ, θ′) (5.32)

In this notation, the result given in Eqs. (5.29) and (5.30) is written as

E[δfδf ]c(θ, θ′) = σ(θ)DE(θ, θ′)σ(θ′) ; E[δ(g − f)δ(g − f)]c(θ, θ

′) = γ(θ)CE(θ, θ′)γ(θ′)

The normalization given in Eq. (5.31) yields, from Eqs. (5.29) and (5.30), that

E[(δf(t, x)

)2

]c = E[δfδf ]c(θ, θ) = σ2(θ) (5.33)

E[(δ(g − f)(t, x)

)2

]c = E[δ(g − f)δ(g − f)]c(θ, θ) = γ2(θ) (5.34)

The empirical propagators DE(θ, θ′), CE(θ, θ′) and volatilities σ, γ are evaluated from Eqs.


(5.29), (5.30), (5.33) and (5.34). Note the important result that the volatilities σ, γ are not

model dependent but rather, are directly obtained from the market without any fitting. This

greatly increases the accuracy of the model and incorporates important market information

into the results of the model.

The models propagators D(z(θ), z(θ′)), C(z(θ), z(θ′)) provide a fit of the model’s param-

eter using the relation

DE(θ, θ′) = D(z(θ), z(θ′)) = D(z, z′) ; CE(θ, θ′) = C(z(θ), z(θ′)) = C(z, z′)

where recall market time is given by

z(θ) = ω

(θ

ω

)η; θ = x− t

The cross-correlator is fixed uniquely by Eqs. (5.33) and (5.34) and yields

E[δfδ(g − f)]c(θ, θ′) = σ(θ)(DE∆ECE)(θ, θ′)γ(θ′)

Define the empirical cross-correlator

TE(θ, θ′) ≡ E[δfδ(g − f)]c(θ, θ′)

σ(θ)γ(θ′)(5.35)

The empirical cross-correlator, in matrix notation, is given by

∆E(θ, θ′) =(D−1E TEC

−1E

)(θ, θ′) (5.36)

The model cross-correlator ∆(z(θ), z(θ′)) is given by the mapping from the empirical cross-

§ 5.6. Empirical volatility and propagators 135

correlator ∆E(θ, θ′) using the concept of market time z(θ). We hence obtain

∆(z(θ), z(θ′)) = ∆E(θ, θ′)

Defining

z−1(θ) = ω(θ

ω)1/η ⇒ ∆(θ, θ′) = ∆E(z−1(θ), z−1(θ′)) (5.37)

It is the coefficient function ∆(θ, θ′) that appears in Lagrangian, with θ, θ′ being replaced by

independent integration variables z, z′ in the action.

In summary, the calibration of the risky forward rates yields the following results.

• Parameters µ, λ, µ, λ and ω, η.

• Functions σ(θ), γ(θ).

• Function ∆(z, z′).

§ 5.6 Empirical volatility and propagators

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

−6

0 1 2 3 4 5 6 7 8 9 10

Time to maturity(year)

σ2(θ

) (\

ye

ar)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−6

0 1 2 3 4 5 6 7 8 9 10


σ2(θ

) (\

ye

ar)

Figure 5.2: (a) Volatility of US forward interest rates. (b) Volatility of the spread of theSingapore -US forward interest rates. Period from 9 May 2011 to 18 January, 2012.


All the data for the US, Singapore and Malaysian forward interest rates is daily data taken

from Bloomberg, for the period from 9 May 2011 to 18 January, 2012.

The volatility of the US forward interest rates f(t, x) and of the Singapore spread ξ(t, x) =

g(t, x)−f(t, x) is determined from market data using Eqs. (5.33) and (5.34) respectively, and

shown in Figure 5.2. Both the volatilities are of the same order of magnitude. The volatility

of the spread ξ(t, x) follows the US case, as is to be expected since the US debt market drives

the international debt market.

Compared to earlier period of 1999-2001, there has been a dramatic change in the volatility

of the US forward interest rates since – or more likely due to – the 2008 financial meltdown,

and shown in Figure 5.3. The volatility for 2011-2012 is seen, in Figure 5.2, to increase

indefinitely as one goes to future time, as opposed to the volatility for the 1999-2001 period,

which has a maximum for future time of about 1.5 years in the future around 2001-2 and then

leveling off as one goes indefinitely into the future.

Figure 5.3: Empirical volatility function σ(θ) =√E[δf 2(t, θ)]c and kurtosis κ(t, θ) =

E[δf(t, θ)4]/σ4(t, θ)− 3 of the forward interest rates; θ = x− t. (Reference: [1]).

One can interpret the change of behavior of the US forward interest rates from 2000 to

2011 as implying that, unlike during the pre-2008 period, the volatility – and hence the risk

of the debt market – since 2008 has been increasing indefinitely for future time pointing to a

future that is increasingly unstable.


The test of the model is to compare the model’s correlators of the forward rates for the

risk free and risky case, given by D(z(θ), z(θ′)), C(z(θ), z(θ′)), with the empirical correlators

DE(θ, θ′), CE(θ, θ′). The calibration is done via the equation

D(z, z′) = D(z(θ), z(θ′)) = DE(θ, θ′) =E[δfδf ]c(θ, θ

′)

σ(θ)σ(θ′)

and

C(z, z′) = C(z(θ), z(θ′)) = CE(θ, θ′) =E[δ(g − f)δ(g − f)]c(θ, θ

′)

γ(θ)γ(θ′)

One of the major advantages of the model is that the functions σ(θ), γ(θ) and ∆(z, z′) can be

directly obtained from the empirical correlator without the need to do any fits. This feature

greatly increases the accuracy of the model.

§ 5.6.1 Stand-alone Singapore rates

Consider a stand-alone model for the Singapore forward rates g(t, x), similar to the risk free

case as in Eq. (5.1) and given by

∂g

∂t(t, x) = d(t, x) + s(t, x)B(t, x) (5.38)

and the Euclidean quantum field B(t, x) has an action similar to A(t, x). The stand-alone

empirical volatility is given by

s2(t, x) = E[(δg)2]c

Note that

E[(δ(g − f))2]c = E[(δg)2]c + E[(δf))2]c − 2E[δgδf)]c (5.39)

§ 5.7. Calibration of US and Singapore models 138

This yields

γ2 = s2 + σ2 − 2E[δgδf)]c (5.40)

Eq. (5.40) is shown in Figure 5.4, and the volatilities are consistent.

0

0.2

0.4

0.6

0.8

1

1.2x 10

−6

0 1 2 3 4 5 6 7 8 9 10


σ2(θ

) (\

ye

ar)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−6

γ2

s2+σ

2−2E[dgdf]

0 1 2 3 4 5 6 7 8 9 10

σ2(θ

) (\

ye

ar)


Figure 5.4: (a) Volatility of the Singapore stand-alone forward interest rates. (b) Comparisonof volatility of Singapore stand-alone forward interest rates of the US and spread of theSingapore -US forward interest rates. Period from 9 May 2011 to 18 January, 2012.

The model’s propagator is given by

C(z(θ), z(θ′)) = CE(θ, θ′) =E[δgδg]c(θ, θ

′)

s(θ)s(θ′)

A fit of the model’s parameters with the empirical result is carried out to ascertain the accuracy

of the stand-alone simplified model.

§ 5.7 Calibration of US and Singapore models

The models of the forward interest rates for the US and Singapore are calibrated using the

empirical correlators. The empirical value and the best fit of the model is evaluated for the

§ 5.7. Calibration of US and Singapore models 139

0

5

10

15

0

2

4

6

8

10

12

14−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

0.2

0.4

0.6

0.8

1

Figure 5.5: US forward interest rates. (a) The empirical correlator DE(θ, θ′). (b) The modelcorrelator D(θ, θ′). Data from 9 May 2011 to 18 January, 2012.

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0.5

0.6

0.7

0.8

0.9

1

0

2

4

6

8

10

12

0

2

4

6

810

12

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.6: Singapore forward interest rates. (a) The empirical correlator CE(θ, θ′). (b) Themodel correlator C(θ, θ′). Data from 9 May 2011 to 18 January, 2012.

US forward interest rates as follows

DE(θ, θ′) =E[δfδf ]c(θ, θ

′)

σ(θ)σ(θ′)= D(z(θ), z(θ′))

The results are shown in Figure 5.5. For the stand alone Singapore forward interest rates

CE(θ, θ′) =E[δgδg]c(θ, θ

′)

s(θ)s(θ′)= C(z(θ), z(θ′))

and the results are shown in Figure 5.6.

§ 5.8. Determination of ∆(θ, θ′): Coupling of US-Singapore rates 140

The empirical correlator for the spread of the Singapore forward interest rates above the

US forward interest rates is given by

C(z, z′) = CE(θ, θ′) =E[δ(g − f)δ(g − f)]c(θ, θ

′)

γ(θ)γ(θ′)= C(z(θ), z(θ′))

The results of calibrating the propagators is given in Table 5.1.

02

46

810

12

0

2

4

6

8

10

12

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

0.2

0.4

0.6

0.8

1

Figure 5.7: Joint US-Singapore forward curve. (a) The empirical spread correlator CE(θ, θ′).(b) The model spread correlator C(θ, θ′). Data from 9 May 2011 to 18 January, 2012.

Note that the unit of the axis in the 3D graphs from Section 8 to Section 9 are for the

payment period, which is half a year in this chapter.

§ 5.8 Determination of ∆(θ, θ′): Coupling of US-Singapore

rates

The empirical cross-correlator, from Eq. (5.36) and in matrix notation, is given by

∆E(θ, θ′) =(D−1E TC−1

E

)(θ, θ′)


Note that all the matrices on the right hand side of above equation are empirically determined.

The value of ∆E(θ, θ′) evaluated from above equation has large errors. The reason is that on

numerically inverting DE and CE, both D−1E and C−1

E have large errors, of order 107. This is

due to the fact that both are differential operator, and which are approximately given by the

model’s values as in Eqs. (5.14) and (5.15).

0

5

10

1520

0

5

10

15

20

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

5

10

15

20

0

5

10

15

20

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

5

10

15

20

0

5

10

15

20

0

0.2

0.4

0.6

0.8

1

Figure 5.8: Inverse of propagator: (a) D−1DE (b) C−1CE (c) The Dirac delta function δ(θ−θ′).Data from 9 May 2011 to 18 January, 2012.

Hence, instead of inverting the empirical propagators DE and CE, the empirical inverses

are replaced by the model’s value for their inverses D−1, C−1 given by Eqs. (5.14) and (5.15);

more precisely

(D−1DE)(θ, θ′) =

(1− 1

µ2D

∂2

∂θ2+

1

λ4D

∂4

∂θ4

)DE(θ, θ′) = δ(θ − θ′)

and

(C−1CE)(θ, θ′) =

(1− 1

µ2c

∂2

∂θ′2+

1

λ4c

∂4

∂θ′4

)CE(θ, θ′) = δ(θ − θ′)

Figure 5.8(a) and (b) shows that the model’s values for the inverse of DE and CE is fairly

accurate, with the off-diagonal elements all falling to small values. Comparing Figure 5.8 with

the numerical representation of the Dirac δ−function, given in Figure 5.8(c), shows that there

are significant errors. Approximating D−1E and C−1

E by the models values given by D−1, C−1

introduce errors that are far smaller than inverting DE and CE.


The model’s inverse of the propagator yields the empirical cross-correlator, which in matrix

notation is given by

∆E(θ, θ′) =(D−1TEC

−1)

(θ, θ′)

Rrecall the matrix TE(θ, θ′) is empirically evaluated by

TE(θ, θ′) ≡ E[δfδ(g − f)]c(θ, θ′)

σ(θ)γ(θ′)

Hence, one obtains the following final result for ∆E(θ, θ′)

∆E(θ, θ′) =

(1− 1

µ2D

∂2

∂θ2+

1

λ4D

∂4

∂θ4

)(1− 1

µ2c

∂2

∂θ′2+

1

λ4c

∂4

∂θ′4

)TE(θ, θ′) (5.41)

The computation of ∆E(θ, θ′) requires, as can be seen from Eq. (5.41), both the empirical

value of the cross-correlator TE as well as the inverse of model’s propagators. The empirical

result for TE is shown in Figure 5.9(c) and ∆E(θ, θ′) is shown in Figure 5.9(b). The coefficient

function, from Eq. (5.37), is given by

∆(θ, θ′) = ∆E(z−1(θ), z−1(θ′))

and is shown in Figure 5.9(c).

0

5

10

15

20

0

5

10

15

20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

05

1015

20

05

1015

20−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0

2

4

6

8

10

0

2

4

6

8

10−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Figure 5.9: Correlation of Singapore - US forward interest rates spread with the US forwardinterest rates. (a) The cross-correlator TE. (b) ∆E of the US forward interest rates with thespread with the Singapore forward interest rates. (c) The model coefficient function ∆. Datafrom 9 May 2011 to 18 January, 2012.


The diagonal value of ∆(θ, θ) is near the value of 1, with the off-diagonal values of ∆(θ, θ′)

being less than 0.4. We can tentatively conclude that our assumption of the US and Singapore

forward interest rates being weakly coupled is approximately correct. The fact that ∆(θ, θ) ' 1

can be due to the fact that the spread ξ(t, z) is most strongly correlated with A(t, z) for equal

future time.

§ 5.8.1 Malaysian forward interest rates

The Malaysian forward interest rates h(t, x) are irregular, with features that are absent for

both the US and Singapore cases. The model for the Malaysian forward interest rates and its

spread over the US is defined by the following

∂h

∂t(t, x) = m(t, x) + v(t, x)M(t, x) (5.42)

∂(h− f)

∂t(t, x) = n(t, x) + ζ(t, x)(M − A)(t, x) (5.43)

The Euclidean quantum field M(t, x) and the spread M(t, x) − A(t, x) has an action similar

to A(t, x).

The volatility function, defined similar to the US and Singapore case, is given by

E[(δh(t, θ))2]c = v2(θ) ; θ = x− t

The empirical volatility of the Malaysian forward interest rates, v2(θ), is shown in Figure 5.10.

A noteworthy feature is that unlike the US forward interest rates, volatility v2(θ) plateaus

after 9 years.


2

4

6

8

10

12

14x 10

−4

0 1 2 3 4 5 6 7 8 9 10


σ(θ

) (\

ye

ar)

2

4

6

8

10

12

14

16x 10

−4


σ(θ

) (\

ye

ar)

0 1 2 3 4 5 6 7 8 9 10

Figure 5.10: (a) The Malaysian forward interest rates volatility v2(θ); half-yearly time steps inthe future time direction. (b) The volatility ζ(θ) of the Malaysian spread over the US forwardinterest rates. Data from 9 May 2011 to 18 January, 2012.

The propagator is given by the normalized correlation function (θ = x− t; θ′ = x′ − t)

H(θ, θ′) =E[δh(t, θ)δh(t, θ′)]

v(θ)v(θ′); H(θ, θ′) =

E[δ(h− f)(t, θ)δ(h− f)(t, θ′)]

ζ(θ)ζ(θ′)

The propagator H(θ, θ′) has values that are negative for future times such that |θ − θ′| >

1.5 years, as shown in Figure 5.11(a). This implies that the forward interest rates 1.5 years

in the future move in the opposite direction to present day rates. Negative correlations

are almost absent in both the US (only a few points are slightly negative) and Singapore

correlation functions. It needs to be studied if there are any regulations on the Malaysian

debt market that is responsible for this behavior.

0

5

10

15

20

0

5

10

15

20

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

5

10

15

20

0

5

10

15

20

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0

5

10

15

20

0

5

10

15

20

0

0.2

0.4

0.6

0.8

1

Figure 5.11: (a) The Malaysian stand-alone propagator H(θ, θ′). (b) Propagator for thespread, given by H(θ, θ′), of the Malaysian above the US forward interest rates. (c) Themodel fitting the spread for Malaysian forward interest rates. Data from 9 May 2011 to 18January, 2012.

§ 5.9. Summary of Calibration Results 145

If one studies the spread of the Malaysian forward interest rates above the US, given by

h(t, x) − f(t, x), one obtains the result given in Figure 5.11(b); the negative correlation is

alleviated a bit.

The model generates only positive values propagators, as can be seen from the fit of the

US and Singapore forward interest rates. The spread for the Malaysian can be fit by the

model , but with large errors, and the result is shown in Figure 5.11(c). The R2 = 0.58 for

the Malaysian spread is not a very good fit, primarily because of the negative values of the

empirical propagator.

§ 5.9 Summary of Calibration Results

Recall, from Eq. (5.24), the following are the parameters of the action

α± = λ2e±b ; e±b =λ2

2µ2

[1±√

1− 4(µ

λ)4]

; b ≥ 0

The calibration of the US, Singapore and Malaysian forward interest rates yields the results

given in Table 5.1.

The calibration for the forward interest rates has a number of key features. Firstly, the

accuracy of the calibration of the US and Singapore stand-alone model is comparable, with

R2 equal to 0.79 and 0.74 respectively. Secondly, modeling the Singapore forward interest

rates as being driven by the US rates is marginally more accurate, yielding a R2 equal to 0.80.

Lastly, the value of market time, given by η, ω, for the two cases for the Singapore forward

interest rates is quite different from the US.

Different model’s propagators R2 λ b η ω µ

§ 5.9. Summary of Calibration Results 146

Only US D 0.79 2.08 0.63 0.98 0.01 1.34

Only Singapore C 0.74 0.55 1.08 0.41 7.30 0.31

Joint US-Singapore C 0.80 5.78 0.92 0.94 0.02 3.89

Joint US-Malaysian H 0.58 3.79 0.02 0.74 0.13 2.68

Table 5.1: Model’s parameters

The model has a fairly good fit with data and

|DE(θ, θ′)−D(z(θ), z(θ′))| ≤ 0.19 ; |CE(θ, θ′)− C(z(θ), z(θ′))| ≤ 0.20 ; ∀ θ, θ′

A possible conclusion from the calibration is that if one is interested in studying the

response of the Singapore forward interest rates to changes of the US forward rates, then

studying the spread is going to yield more accurate results. On the other hand, if one is

studying questions related only to the home market for Singapore sovereign bonds, then the

stand-alone model may be more appropriate.

The Malaysian-US spread fit is not good, for reasons discussed earlier. The values of the

parameters are quite different than the US or Singapore case. A rather unusual result is that

b ≈ 0, showing that the Malaysian-US spread is near the critical value of the system. The

financial implications of the forward interest rates being near criticality needs to be studied

further.

§ 5.10. Interest rate swaptions 147

§ 5.10 Interest rate swaptions

Interest rate swaps for US$ are widely traded instruments. The empirical study of swaptions

– options on swaps – can be used for testing and calibrating the quantum finance models for

coupon bond options.

There are swaptions traded in the market in which the floating rate is paid at ` = 90 days

intervals, and the fixed rate payments are paid at 2` = 180 days intervals. For a swaption

with fixed rate payments at 90 days intervals – at times T0 +n`, with n = 1, 2.., N – there are

N payments. For payments made at 180 days intervals, there are only N/2 payments 1 made

at times T0 + 2n` , n = 1, 2, ..., N/2, and of amount 2RS.

Define the following positive valued function

[A]+ =

A : A ≥ 0

0 : A < 0

The payoff function for a swaption is2

CL(t, T0;RS) = V[B(t, T0)−B(T0, T0 +N`)− 2`RS

N/2∑n=1

B(t, T0 + 2n`)]

+

= V[B(t, T0)−

N/2∑n=1

cnB(t, T0 + 2n`)]

+. (5.44)

The equivalent coupon bond put option, maturing at time t∗ and with payoff function is

1Suppose the swaption has a duration such that N is even. Note that for N = 4 the underlying swap hasa duration of one year.

2The price of CR for the case of 90 days floating and 180 days fixed interest payments is given from CL byusing the put-call relation similar to the one given in [1].


given by

(K −

N/2∑n=1

cnB(t∗, T0 + 2n`))

+. (5.45)

and has the coefficients ci and strike price given by

cn = 2lRs; n = 1, 2....(N − 1)/2 : semi− annual payments at T0 + 2n`

cn/2 = 1 + 2lRs; : annual payments at T0 + n`

K = B(t, T0)

For the US Dollar swaption, let

Ji = ciFi ; F = exp−∫ Ti

t∗dxf(t0, x) ; F =

∑i

Ji

where f(t, x) is the risk free US Dollar forward yield curve; f(t0, x) is the yield curve at time

t0 and is taken from the market. The value of the risk free swaption – receiving floating Libor

and paying fixed interest rate – in the quantum finance model is given by [1]

C(t0, t∗, K) = B(t0, t∗)

√C2

2π− 1

2B(t0, t∗)(K − F ) +O(X2) (5.46)

The put swaption – for receiving fixed interest rate and paying the floating Libor rate – is

given by put-call parity and yields [1]

P (t0, t∗, K) = B(t0, t∗)

√C2

2π+

1

2B(t0, t∗)(K − F ) +O(X2) (5.47)


From Eq. (5.81)

C2 =N∑ij=1

JiJj[eGij − 1] '

N∑ij=1

JiJjGij +O(σ2γ2, σγ3, σ3γ) (5.48)

Ti--t

0

Ti-t

0t*-t

0

t*-t

0

‘Tj-t

0

Ti-t

0t*-t

0

t*-t

0

‘

0 0

(a) (b)

Figure 5.12: Domain for Gij. (a) For the case of Ti = Tj. (b) For the case of Ti 6= Tj.

The integrations for Gij are written in terms of future time θ = x− t, θ′ = x′ − t as this

is required for the empirical analysis. Hence

Gij =

∫ t∗

t0

dt

∫ Ti

t∗

dx

∫ Tj

t∗

dx′σ(x− t)D(x− t, x′ − t)σ(x′ − t)

=

∫ t∗

t0

dt

∫ Ti−t

t∗−tdθ

∫ Tj−t

t∗−tdθ′E[δfδf ]c(θ, θ

′) (5.49)

The quantum finance model for the forward interest rates yield the expression for the option

price C(t0, t∗, K) in terms of the correlators E[δfδf ]c(θ, θ′). The correlator in turn is taken

directly from the market data.

Note that the integrand of Gij in Eq. (5.49) is over future calendar time – from t0 to t∗.

However, using the fact that the correlators depend only on θ, θ′ one can re-write the integral

entirely in terms of the correlators evaluated from historical data that precede time t0. The

various domains for different values of Gij are shown in Figure 5.12.


This symmetry of the correlator – depending on only future time θ, θ′ and not explicitly

on calendar time t – is crucial in pricing the coupon bond option (or swaption); empirical

studies show that this symmetry is valid for periods of up to to 2-3 years or longer, depending

on the regime of the market [1].

§ 5.10.1 US swaptions

A US Dollar 1x10 swaption is an option on an interest swap [45] that matures in one years

time and with 10 coupons in the future and is shown in Figure 5.13.

1 - -

Figure 5.13: The circles signify payment dates, except at T0;; the first payment is at T1 andthe last payment is at TN ; the interest rate swap becomes operational at time T0. The shadedarea inside the rectangles indicate the set of forward interest rates that determine the priceof a swap. (a) A midcurve forward swap is entered into at time t0 and exercised at time t∗,before T0. (b) A forward swap is entered into at time t0 and exercised at time T0.

The US swaption is analyzed for the period 2013-2015, with the results shown in Figure

5.14. The maturity for the option is taken to be t∗ = 1/2 year. Figure 5.14(a) shows the

daily market value of the swaption together with the model’s prediction. The coefficient C2

is the main prediction of the model and is shown in Figure 5.14(b). The R2 for the error of

swaption’s market price compared to the model price is 0.6 for 500 days and 0.8 for 200 days.

Recall that the expression for the US put $ swaption, from Eq. (5.47), is given by

P (t0, t∗, K) = B(t0, t∗

√C2

2π+

1

2B(t0, t∗)(K − F ) +O(X2)


0 50 100 150 200 250 300 350 400 450 5000.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Red(heavy): data

Blue(light): model

Black(break): model without c2

0 50 100 150 200 250 300 350 400 450 5000.015

0.02

0.025

0.03

0.035

0.04

0.045

Figure 5.14: The daily price of a US Dollar 1x10 swaption for the period 2013-2015. Theheavy (red) line is data. The blue line is the full model value of the swaption with C2 Thebroken line is the value of the swaption withoutthe C2 coefficient. (b) The value of C2 as afunction of time.

consists of two parts: a contribution due to C2 and another contribution that is model inde-

pendent and given by (1/2)B(t0, t∗)(K−F ). Both terms are of the same magnitude and both

are essential. The stochastic behavior of the swaption is captured by (1/2)B(t0, t∗)(K − F );

the C2 coefficient changes the overall drift of the swaption and corrects the behavior of the

model’s price, as shown in Figure 5.14(b) so as to provide an accurate price of the swaption,

as shown in Figure 5.14(a).

0 20 40 60 80 100 120 140 160 180 2000.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

US Option

Singapore Option depending on US

Singapore Option

0 20 40 60 80 100 120 140 160 180 2000.05

0.1

0.15

0.2

0.25

0.3

US

malaysia1

malaysia2

Figure 5.15: (a) Swaption of US, Singapore stand-alone and Singapore spread interest rates.(b) Swaption of US and stand-alone Malaysian interest rates . Data for the period 12 January2012 to 20 October 2012.


§ 5.10.2 Singapore swaptions

Let g(t, x) be the Singapore forward yield curve. The quantum finance model for g(t, x) is

based on the assumption that it is driven by the US Dollar yield curve, plus a spread yield

curve given by ξ(t, x), as given in Eq. (5.5). The Singapore bond is given by

Ri = exp−∫ Ti

t∗

dxg(t0, x) ; R =N∑i=1

ciRi

The price of the risky coupon bond put option, from Eq. (5.47), is given

C(t0, t∗, K) = R(t0, t∗)

√C2

2π+

1

2R(t0, t∗)(K −R) +O(X2) (5.50)

with

C2 '∑ij

cicjRiRjGij

The C2 coefficients given in Eq. (5.82) for the risky bond are directly computed from

market correlators and yield the following

Gij =

∫ t∗

t0

dt

∫ Ti−t

t∗−tdθ

∫ Tj−t

t∗−tdθ′

×σ(θ)D(θ, θ′)σ(θ′) + γ(θ)C(θ, θ′)γ(θ′) + 2σ(θ)(D∆C)(θ, θ′)γ(θ′)

(5.51)

Expressing the result given above in Eq. (5.51) directly in terms of the empirically evaluated

correlators yields the following

Gij =

∫ t∗

t0

dt

∫ Ti−t

t∗−tdθ

∫ Tj−t

t∗−tdθ′

×E[δfδf ]c(θ, θ

′) + E[δ(g − f)δ(g − f)]c(θ, θ′) + 2E[δfδ(g − f)]c(θ, θ

′)

(5.52)


The expression of Gij in terms of the correlators is a result of the general framework

of the quantum finance model for forward interest rates; however, the empirical correlators

themselves are model independent, and encode the full information of the market.

Eq. (5.52) above is evaluated using empirical results. Clearly using the market correlators

will yield more accurate results since a specific choice of an action is calibrated against the

market’s correlators.

Singapore and Malaysian currency based interest rate swaptions are not traded instru-

ments. The price of a swaption given in Eq. (5.47) can be used to generate the model’s

predictions on the value of the Singapore and Malaysian swaptions, and is shown in Figures

5.15(a) and (b).

The Singapore swaptions – for modeling both the spread of the Singapore forward interest

rates above the US forward interest rates as well as for the stand-alone Singapore forward

interest rates – are seen to follow the US swaption values. As can be seen from Figure 5.15(a),

whenever the Singapore swaption value deviates from the US swaption, the value of the

Singapore swaption changes and criss-crosses the US swaption value. It is logical to expect

the US swaption market should drive the value of Singapore swaption, given that the US

swaption is the most liquid instrument and has the greatest range of maturities and coupon

payments.

§ 5.10.3 Malaysian swaptions

The model’s fit for the Malaysian forward interest rates, discussed in Section § 5.8.1, is not

very good with R2 = 0.58; the general framework of quantum finance nevertheless can still be

applied to the Malaysian case. The reason being that the specific form of the action chosen in

Eq. (5.7) is not necessary, and instead one can instead start with a generic propagator for the

§ 5.11. Conclusions 154

action denoted by H−1(θ, θ′) without specifying its dependence on θ, θ′. One can generalize

the action given in Eq. (5.7) to the following

S[A; ξ] = −1

2

∫ ∞t0

dt

∫ ∞t

dxA2(t, x) +

1

µ2(∂A(t, x)

∂x)2 +

1

λ4(∂2A(t, x)

∂x2)2

− 1

2

∫ ∞t0

dt

∫ ∞t

dxM(t, x)− A(t, x)

H−1(θ, θ′)

M(t, x′)− A(t, x′)

+

∫ ∞t0

dt

∫ ∞t

dxdx′∆M(x− t, x′ − t)A(t, x)(M(t, x′)− A(t, x′))

where M(t, x)−A(t, x) yields the spread of the Malaysian over the US forward interest rates.

The earlier analysis for the price of the swaption goes through and one only needs to

empirically evaluate the propagator H(θ, θ′). Gij given in Eq. (5.52) required for pricing a

swaption is evaluated empirically and hence the expression of Gij can be evaluated in terms

of the empirical correlators. Evaluating Gij empirically for the Malaysian swaption, the gen-

eralized quantum finance model’s prediction for the Malaysian swaption, using the Malaysian

empirical propagator, is given in Eq. (5.15)(b).

The Malaysian swaption seems to be fairly insensitive to the movements of the US swap-

tion, which can be attributed to the Malaysian debt market being decorrelated from the

US debt market. The Malaysian swaption’s price is always much higher than the US swap-

tion, which can be explained by the higher risk of Malaysian swaptions compared to the US

sovereign swaptions.

§ 5.11 Conclusions

The study of risky coupon bonds reveals many new features of the forward interest rates of

emerging markets, of which Singapore is one of the more robust examples. The Singapore

forward interest rates were studied both as stand-alone and as being driven by the US forward

§ 5.11. Conclusions 155

interest rates. The empirical study shows that both approaches give reasonable results.

The modeling the Singapore forward interest rates by a spread over the US extends the

quantum finance formulation of forward interest rates to the study of two coupled forward

interest rates as given by the action in Eq. (5.7), with the coupling ∆ being small. This allows

one to study the two rates perturbatively. The extension of the martingale condition for two

coupled forward interest rates leads to new terms for the drift, as in Eq. (5.75), that couple

both the forward interest rates through the spread of the Singapore forward interest rates.

The result of the empirical calibration show that modeling the Singapore forward interest

rates by a spread over the US is more in line with the parameters of the US rates, in contrast

to the value of market time index η from the calibration of the stand-alone US forward interest

rates.

The Malaysian forward interest rates seem to have anomalies not present in the US or

the Singapore case; for this reason, an accurate fit of the model could not be obtained for

the Malaysian case. However, one can extend the model, as discussed in Section § 5.10.3, by

using the quantum finance framework of an action functional and path integrals, by letting

the propagator in the action be determined empirically. Such a generalization of the quantum

finance framework allows one to study the Malaysian forward interest rates and to price its

options – such as the interest rate swaptions and coupon bond call and put options.

In summary, if one is concerned with only the home market, one can model the forward

interest rates as a stand-alone case and ignore its coupling to the international debt market.

However, if an emerging economy wants to issue sovereign debt in the international capital

markets, it is more consistent with the international market to model the emerging markets

debt market as being driven by the US debt markets.

§ 5.12. Appendix 1: Risky coupon bond option 156

§ 5.12 Appendix 1: Risky coupon bond option

The price of the risk free coupon bond option has been evaluated in [1]. The derivation is

generalized to the case of the risky bond.

A coupon bond is a portfolio of zero coupon bonds with fixed coupons an paid out at

pre-determined future times Tn = n`, where ` is the tenor. The risky coupon bond, having a

principal L and maturing at future time TN , is given by

R(t) =N∑i=1

anR(t, Tn) + LR(t, TN) ≡N∑i=1

cnR(t, Tn)

Let C(t) be the price of a call option at time t. A call option for the risky coupon bond,

maturing at future time t∗ and with strike price K, has a payoff function given by [1]

C(t∗) = [R(t∗)−K]+

The price of a call option at time t is defined to be the discounted value of the conditional

expectation value of the payoff function, given the value of the security at time t, which in

our case are the forward interest rates at time t given by g(t, x). There is a wide choice of

instruments that can be used for discounting future cash flows, all of which yield a martingale

evolution of the underlying security. For coupon bond options, the most suitable numeraire

for discounting is the forward bond measure, with the zero coupon bond R(t, t∗) used for

discounting future cash flows. The call option is then given by

C(t0)

R(t0, t∗)= E[

C(t∗)R(t∗, t∗)

] : Martingale condition


and which yields the option price at time t0 given by

C(t0) = R(t0, t∗)[R(t∗)−K]+ (5.53)

The risky zero coupon bond is represented as follows

R(t∗, Ti) = exp−∫ Ti

t∗

dxg(t∗, x) = e−βi−QiRi(t0, t∗, Ti) (5.54)

where we make the following defintions

Ri = R(t0, t∗, Ti) = exp−∫ Ti

t∗

dxg(t0, x) (5.55)

Qi =

∫Di

[σ(t, x)A(t, x) + γ(t, x)ξ(t, x)]

≡∫ t∗

t0

dt

∫ Ti

t∗

dx[σ(t, x)A(t, x) + γ(t, x)ξ(t, x)] (5.56)

*

*

Figure 5.16: The shaded area is the domain of integration Ri.

The domain of integration Di is given in Figure 5.16. The coefficient βi, the integrated

form of the forward measure drift, and is fixed (later) by the martingale condition and is given


by

βi =

∫Di

β(t, x) =

∫ t∗

t0

dt

∫ Ti

t∗

dxβ(t, x) (5.57)

The payoff requires the price of the coupon bond at time t∗, which is re-written as

N∑i=1

ciR(t∗, Ti) =N∑i=1

ciRi +N∑i=1

ci[R(t∗, Ti)−Ri] ≡ R + V

with forward bond price R =∑i

ciRi =∑i

Ji (5.58)

The break-up of the coupon bond into R + V is based on the fact that all deviations of the

coupon bond from it’s forward price R are due to fluctuations in the forward interest rates

that are controlled by it’s volatility σ(t, x). In fact, V has an order of magnitude equal to

O(σ) + O(γ) and hence an expansion in power of V results in the volatility expansion that

one is aiming for [1].

From Eq. (5.54), the potential is given by

V ≡N∑i=1

ci[R(t∗, Ti)−Ri] =N∑i=1

ci[Rie

−βi−Qi −Ri

]=

N∑i=1

Ji[e−βi−Qi − 1

](5.59)

Hence, from Eqs. (5.56) and (5.59)

V =N∑i=1

Ji[e−βie

−∫Di

(σA+γξ) − 1]

(5.60)

and the payoff function is given by

[ N∑i=1

ciR(t∗, Ti)−K]

+=[R +

N∑i=1

Ji(e−βi−Qi − 1)−K

]+

(5.61)

=[R + V −K

]+


The payoff function is re-written using the properties of the Dirac delta function as follows

( N∑i=1

ciR(t∗, Ti)−K)

+=

1

2π

∫ +∞

−∞dpdηeiη(V−p)(F + p−K

)+

The price of the call option, from Eq. (5.53), can be written as

C(t0, t∗, K) = R(t0, t∗)1

2π

∫ +∞

−∞dpdη

(R + p−K

)+e−iηpZ(η) (5.62)

with the partition function for the call option price is given by

Z(η) =1

Z

∫DADξeSeiηV ; Z =

∫DADξeS (5.63)

From the expression for the partition function given above, the effective action for the

pricing of the coupon bond option, from Eqs. (5.60) and (5.63), is given by

SEff ≡ S[A; ξ] + iηV (5.64)

= S[A; ξ] + iηN∑i=1

Ji[e−βi−Qi − 1

]= S[A; ξ] + iη

N∑i=1

Ji[e−βie

−∫Di

(σA+γξ) − 1]

(5.65)

where recall

Ji = ciFi ; Fi = exp−∫ Ti

t∗dxg(t0, x) (5.66)

A cumulant expansion [46] of the partition function in a power series in η yields

Z(η) = eiηC1− 12η2C2−i 1

3!η3C3+ 1

4!η4C4+... (5.67)


The coefficients C2, C3, C4, ... are evaluated using Feynman diagrams.

Expanding the right hand side of Eq. (5.63) in a power series to fourth order in η yields

Z(η) =1

Z

∫DADξeiηV eS[A;ξ]

=1

Z

∫DADξeS[A;ξ]

[1 + iηV +

1

2!(iη)2V 2 +

1

3!(iη)3V 3 +

1

4!(iη)4V 4 + · · ·

](5.68)

Comparing Eqs. (5.67) and (5.68) yields, to fourth order in η, the following

C1 = E[V ] (5.69)

C2 = E[V 2]− C21 (5.70)

C3 = E[V 3]− C31 (5.71)

C4 = E[V 4]− 3C22 − C4

1 (5.72)

As shown in [1], the coefficient C1 must be exactly zero to obey the martingale condition for

the forward measure. Hence, the martingale condition fixes the drift β by the requirement

that

C1 = E[V ] =N∑i=1

Ji[EF(e−βi−

∫Di

(σA+γξ))− 1]

= 0 (5.73)

Recall the volatility functions σ, γ are functions of only x− t. Performing the Feynman path

integration yields, in matrix notation

eβi = EF (e−

∫Di

(σA+γξ))=

∫DADξeSe

−∫Di

(σA+γξ)(5.74)

= exp1

2

∫ t∗

t0

dt

∫ Ti

t∗

dx

∫ Ti

t∗

dx′[σ(x− t) , γ(x− t)]M(x− t;x′ − t)

σ(x′ − t)

ξ(x′ − t)


Hence the drift is given by

β(x− t) =

∫ x

t

dx′[σ(x− t) , γ(x− t)]M(x− t;x′ − t)

σ(x′ − t)

ξ(x′ − t)

(5.75)

and yields

C1 = 0

The drift α(t, x) for the risk free forward interest rates f(t, x) can be obtained by setting γ to

zero and yields the result given in [1]

α(t, x) =

∫ x

t

dx′σ(x− t)D(x− t;x′ − t)σ(x′ − t)

It is shown in [1] that the put-call parity for the coupon bond requires two conditions

Z(0) = 1 ; C1 = 0

The condition Z(0) = 1 is fulfilled by the normalization of the partition function Z(η) and the

second condition follows from the martingale condition. The call option partition function is

given by

Z(η) = e−12η2C2−i 1

3!η3C3+ 1

4!η4C4... (5.76)

The price of the call option, from Eqs. (5.53) and (5.62)

C(t0, t∗, K) = R(t0, t∗)1

2π

∫ +∞

−∞dpdη

(F + p−K

)+e−iηpZ(η) (5.77)


and yields the following for the price of a coupon bond call option [1]

C(t0, t∗, K) = R(t0, t∗)

√C2

2π− 1

2R(t0, t∗)(K − F ) +O(X2) (5.78)

To evaluate C2, using the martingale condition given in Eq. (5.74), we have

C2 = EF [V 2] =N∑ij=1

JiJjEF[(e−βi−

∫Di

(σA+γξ) − 1)(e−βj−

∫Dj

(σA+γξ) − 1)]

⇒ C2 =N∑ij=1

JiJjEF[e−βi−βj−

∫Di

(σA+γξ)−∫Dj

(σA+γξ) − 1]

≡N∑ij=1

JiJj(eGij − 1) '

N∑ij=1

JiJjG2ij : Ji = ciRi (5.79)

Performing the Gaussian integration to evaluate Gij yields two types of terms.

• Terms that depend only on the individual domains Di, Dj. The drifts βi, βj cancel these

term.

• Terms that link domains Di and Dj, and which are generically shown in Figure 5.17

Hence, the C2 coefficient for the risky bond is given by

Gij = G(1)ij +G

(2)ij +G

(3)ij ; θ = x− t ; θ′ = x′ − t (5.80)

G(1)ij =

∫ t∗

t0

dt

∫ Ti

t∗

dx

∫ Tj

t∗

dx′σ(θ)D(θ, θ′)σ(θ′)

G(2)ij =

∫ t∗

t0

dt

∫ Ti

t∗

dx

∫ Tj

t∗

dx′γ(θ)C(θ, θ′)γ(θ′)

G(3)ij = 2

∫ t∗

t0

dt

∫ Ti

t∗

dx

∫ Tj

t∗

dx′σ(θ)(D∆C)(θ, θ′)γ(θ′)

The integration domain for Gij is illustrated in Figure 5.17, and shows it’s dependence on

Ti and Tj. Gij is the forward bond propagator that expresses the correlation in the fluctuations

§ 5.13. Appendix 2: Swaptions 163

t

0t

0t jTiT

Future Time

Cal

endar

Tim

e

jFiF

M(t,x,x’)

x x’

t*

t*

Figure 5.17: The shaded domain of the forward interest rates contribute to Gij. For a typicalpoint t in the time integration, the figure shows the typical correlation function M(x, x′; t)connecting two different values of the forward interest rates at future time x and x′.

of the forward bond prices Fi = F (t0, t∗, Ti) and Fj = F (t0, t∗, Tj). The computation for the

cumulant’s coefficients yields, from Eq. (5.79), the result

C2 'N∑ij=1

JiJjGij +O(σ2γ2, σγ3, σ3γ) (5.81)

The integrations for Gij are written in terms of future time θ = x − t as this is required

for the empirical analysis. Hence, we obtain

Gij =

∫ t∗

t0

dt

∫ Ti−t

t∗−tdθ

∫ Tj−t

t∗−tdθ′

× σ(θ)D(θ, θ′)σ(θ′) + γ(θ)C(θ, θ′)γ(θ′) + 2σ(θ)(D∆C)(θ, θ′)γ(θ′) (5.82)

§ 5.13 Appendix 2: Swaptions

Interest swaptions are studied in detail in [7] and we summarize the results.

To quantify the value of the swap, let the swap start at Libor time T0, with payments

made at fixed times Tn = T0 + n`, with n = 1, 2, ..., N ; the first payment is made at T1 and


the last payment is made at time TN + `TN . Summing upIn summary, at time t0, the values

of the forward swaplets – corresponding to the interest rate payments made at future times

Tn – yields the following forward price for the floating rate receiver swap

swapL(t0, RS) = `VN∑n=0

N − 1B(t0, Tn + `)[L(t0, Tn)−RS

](5.83)

and fixed rate receiver swap

swapR(t0, RS) = `VN∑n=0

N − 1B(t0, Tn + `)[RS − L(t0, Tn)

](5.84)

They obey the identity

swapL(t0, RS) + swapR(t0, RS) = 0 (5.85)

One can simplify the expression for the swaps. The Libor Zero Coupon Yield Curve

represents Libor in terms of Libor zero coupon bonds, and yields [1]

L(t, T ) =1

`

B(t, T )−B(t, T + `)

B(t, T + `)

Hence

`V

N∑n=0

N − 1B(t0, Tn + `)L(t0, Tn) = V

N∑n=0

N − 1[B(t0, Tn)−B(t0, Tn + `)

]= V

[B(t0, T0)−B(t0, TN + `TN)

]Hence, from Eq. (5.83)

swapL(t0, RS) = V[B(t0, T0)−B(t0, TN + `TN)− `RS

N∑n=0

N − 1B(t0, Tn + `)]

(5.86)


with a similar expression for swapR.

An interest rate swaption, denoted by CL and CR, is an option on a floating or a fixed

interest rate receiver swap, swapL and swapR, respectively.

Consider a swap with N payments dates given by Tn = T0 + n` ; n = 1, 2...., N ; the swap

starts at time T0, the first payment is made at time T1 and the last payment is made at time

TN + `TN . A midcurve swaption, similar to a midcurve caplet, is contracted at time t0 and

matures at time t∗ < T0. The payoff function for a midcurve swaption is given in Figure

5.13(a) and is the same as a midcurve forward swap. The swaption is an option on the swap

and hence has the same cash flow as a swap if it is exercised.

The swaption that will studied henceforth is the one that matures at t∗ = T0, when

the swap becomes operational and is shown in Figure 5.13(b). Almost all market data on

swaptions is exclusively given for this case and is, consequently, the most important one for

empirical studies of swaptions.

The swaption, on maturing, will be exercised only if the value of the swap at time T0 is

greater than it’s initial par value of zero. Hence, the payoff function for the swaption for the

floating and fixed receivers swap, from Eqs. (5.83) and (5.84), is given respectively by the

following

CL(T0, T0;RS) =[swapL(t0, RS)

]+

= `V[ N∑n=0

N − 1B(t0, Tn + `)(L(t0, Tn)−RS)]

+(5.87)

CR(t, T0, RS) =[swapR(t0, RS)

]+

= `V[ N∑n=0

N − 1B(t0, Tn + `)(RS − L(t0, Tn)

)]+

(5.88)


In terms of zero coupon bonds, the swaption payoff function, from Eq. (5.86), is given by

CL(t0, T0;RS) = V[B(t0, T0)−B(t0, TN)− `RS

N∑n=1

B(t, T0 + n`)]

+(5.89)

and a similar expression for CR. The value of the swaption at an earlier time t < T0 can be

obtained by discounting the payoff function using the money market numeraire and yields

CL(t0, T0, RS) = V E[e−

∫ T0t0

r(t′)dt′CL(T0;RS)]

= V E[e−

∫ T0t0

r(t′)dt′(B(t0, T0)−B(T0, TN)− `RS

N∑n=1

B(T0, T0 + n`))

+

](5.90)

and similarly for CR(t, T0, RS). One can see that a swap is equivalent to a particular portfolio

of coupon bonds, and all techniques that are used for coupon bonds options can be used for

analyzing swaptions.

Discounting by the forward bond numeraire B(t, T0), similar to the case of coupon bond

options given in Eq. (5.53), makes the swaption price computationally more tractable; the

price of the swaption, from Eq. (5.89), is given by

CL(t, T0, RS)

B(t, T0)= V E

[CL(T0;RS)

]⇒ CL(t, T0, RS) = V B(t, T0)E

[B(t, T0)−B(t, TN)− `RS

N∑n=1

B(t, T0 + n`)]

+(5.91)

A change of numeraire changes the drift for the forward interest rates [1].

§ 5.14. Appendix 3: Black’s Model for Swaption 167

§ 5.14 Appendix 3: Black’s Model for Swaption

Market data for swaptions, including the data provided by Bloomberg, is quoted as an effective

volatility for the swaption based on Black’s model – similar to the implied volatility surface

being provided for the price of vanilla stock options. From the effective volatility, the swaption

price can be re-constructed. Hence, to obtain the empirical value of the swaptions, we briefly

discuss Black formula for a swaption [47].

Consider coupon payments paid at fixed times Ti = `i. In our case ` = 90 days. Tn is the

time for each payment. m is the times of swaption payments per year; we take m = 2 for

semiannual swaptions. The total number of payments is N . The last payment of the coupon

bond, which includes the payment of the principal, is given at time TN . Hence

N = mn ; TN = `mn (5.92)

The payer swaption has the value, at time t when it is issued, given by [1]

Cps (t) = V [swap(t)]+

For payments at equal time intervals Ti − Ti−1 = `, which is the period for each fixed

payment, the swaption is given by

Cps (t) = V

[R(t, T0)−R(t, TN)− `Rk

N∑i=1

R(t, Ti)]]

+

Let Πt = Cps (t) be the value of the payer swaption at present time t; we use `L

∑Ni=1R(t, Ti)

as the numeraire. The discounted value of payoff function of an option maturing at future

§ 5.14. Appendix 3: Black’s Model for Swaption 168

time t∗ is given by

Πt

`L∑N

i=1R(t, Ti)= E[t,t∗]

[ Cps (t∗)

`L∑N

i=1R(t∗, Ti)

]= V E[t,t∗]

[[R(t∗, T0)−R(t∗, TN)]− `Rk

∑Ni=1R(t∗, Ti)+

`L∑N

i=1R(t∗, Ti)

]= E[t,t∗][

R(t∗, T0)−R(t∗, TN)

`∑N

i=1R(t∗, Ti)−Rk+]

= E[t,t∗]

[S(t∗, T0, TN)−Rk

]+

The forward swap rate S is defined as

S(t, T0, TN) =R(t, T0)−R(t, TN)

`∑N

i=1R(t, Ti)(5.93)

The Black-76 model for the value of the swaption is based on the assumtion that forward

swap rate S follows a geometric Brownian Motion with constant volatility. The value of the

swaption is given by

Πt = `V( N∑i=1

R(t, Ti))× E[t,t∗]

[S(t∗, T0, TN)−Rk

]+

(5.94)

Based on the Black-Scholes analysis, the value of swaption is given by the following

Cs(t) = `V( N∑i=1

R(t, Ti))× [S(t, T0, TN)N(d1)−RkN(d2)] (5.95)

where N(x) is the cumulative normal distribution with

d1 =ln(S(t, T0, TN)/Rk) + 1

2σ2T

σ√T

; T = t∗ − t (5.96)

d2 = d1 − σ√T (5.97)

§ 5.15. Appendix 4: Zero coupon bonds from coupon bonds 169

Bloomberg provides the daily value of σ as well as the fixed value of the fixed leg of the

swaption, given by Rk. The daily value of the zero coupon bonds R(t, Ti) are available from

Bloomberg, and one can then compute the daily price of the swaption using Eq. (5.95).

§ 5.14.1 Par value of fixed payments

The forward price, at time t0 for a swaption maturing at time t∗ is given by

Cps (t, t∗;Rk) = L

[F (t, t∗, T0)− F (t, t∗, TN)− `Rk

N∑i=1

F (t, t∗, Ti)]

+

where the forward bond price is given by

F (t, t∗, Ti) = exp−∫ Ti

t∗

dxf(t, x)

The par value RP of the swaption is given by the forward swaption being zero. Hence

Cps (t0, t∗;RP ) = 0

and yields

`RP =F (t0, t∗, T0)− F (t0, t∗, TN)∑N

i=1 F (t0, t∗, Ti)

§ 5.15 Appendix 4: Zero coupon bonds from coupon

bonds

In this Section, we discuss how to use the boot-strapping method to extract the zero coupon

bonds from coupon bonds. The data provider, such as Bloomberg, gives the yield to maturity


yi(t) (YTM), every day t and for the coupon payments at future time, specified by i.3

Consider a coupon bond, denoted by CN(t). The coupons can be annual, semi-annual or

quarterly; a coupon bond have coupons that are paid pay quarterly, semiannual or annual. As

in Eq. (5.92), define ` = 90 days. Payments can be made annually, semiannually or quarterly,

and yield the following

N = mn ; TN = `mn : m = 4 annual ; m = 2 semiannual ; m = 1 quarterly

Note that the maturity of the bond is given by n = TN/(`m), m` is the tenor of the coupon

bond, N is the total number of payments and TN is the total number of years for the coupon

bond to expire. The coupon payments are made at times

Ti = `mi ; i = 1, 2, · · · , N

The coupon bond pays N fixed coupons at time Ti = `mi, with the final payment made

at time TN . The principal L is returned the end of TN years. The price of the coupon bond

is the sum is the sum of the discounted future cash flows. Fix t = 0 and let the coupon for

the ith payment be fixed at ci; the coupon bond has the following expansion in terms of the

zero coupon bonds

CN(t) =N∑i=1

ciB(t, Ti) + LB(t, TN)

=N∑i=1

ci(1 + 1

myi(t))i

+L

(1 + 1myN(t))N

(5.98)

where the YTM discounting factor yi(t) is given yearly. Hence, in terms of the discounting

3Some data providers give the price of the coupon bond by varying the coupons ci so that the bond hasits the par value. We will not analyze this case.


factors, the zero coupon bond is given by

B(t, Ti) =1

(1 + 1myi(t))i

(5.99)

The price of a coupon bond CN(t) = CN is written in terms of yield to maturity (YTM)

y(t) = y by the following equation

CN(t) ≡ CN =c1

1 + 1my

+c2

(1 + 1my)2

+ · · · cn(1 + 1

my)N

+L

1m

(1 + y)N(5.100)

If the coupon if fixed at cn = c, we have

CN(y) =c

y/m+y/m− c/L

y/m· L

(1 + y/m)N(5.101)

The par value of YTM is y/m = c/L and the price of the bond is then equal to its face value

L. In other words

CN(y) = L : y = cm/L = par YTM

The data is usually given by the yield to maturity y = y(t); hence

CN(y(t)) : price of coupon bond

Each issuer of coupon bonds, be it a sovereign bond or a corporate bond, has its own

complete forward yield curve. The price for the coupon bond of a given issuer is provided by

the data provider by specifying all the coupons cn as well as giving the daily price using the

YTM y(t).

Consider a coupon bond with one payment of coupon c1 and let the YTM be y(t) = y;


then, from Eqs. (5.98) and (5.100)

C1 =c1

(1 + 1my1)

+L

(1 + 1my1)

=c1

(1 + 1my)

+L

(1 + 1my)⇒ y1 = y : y1 is fixed

For a coupon bond with two payments, let the coupons be c1, c2 and let the YTM be y; one

has, from Eqs. (5.100) and (5.98)

c2

(1 + 1my)2

+c1

(1 + 1my)

+L

(1 + 1my)2

= C2 =c2

(1 + 1my2)2

+c1

(1 + 1my1)

+L

(1 + 1my2)2

⇒ c2 + L

(1 + 1my2)2

=c2

(1 + 1my)2

+c1

(1 + 1my)

+L

(1 + 1my)2− c1

(1 + 1my1)

: y2 is fixed

Similar to above relation, once the values of y1, y2, · · · , yn−1 have been evaluated, the value of

yn can be determined recursively.

Hence, of all the discounting factors y1, y2, · · · , yn, · · · , yN can be obtained from the price

of coupon bonds C1, C2, · · · , CN with the different of coupon payments given by c1, c2, · · · , cN .

We can then obtain the zero coupon bond from Eq. (5.99).

In general, for coupons cn and for time t in between Tm−1 and Tm, the coupon bond price

is given in terms of YTM y by the following

CN(t) = (1 +1

my)t−Tm−1

×(

cm(1 + 1

my)1

+cm+1

(1 + 1my)2

+ ...+cN

(1 + 1my)N−m+1

+L

(1 + 1my)N−m+1

)Tm−1 ≤ t ≤ Tm

Suppose that the data is given in a manner in which the maturity of the coupon bond is

always at a fixed θi = t + Ti. The coupon bond then has a price, similar to Eq. (5.98), at

§ 5.16. Appendix 5: Forward interest rates and zero coupon bonds 173

time t given by

CN(t) =N∑i=1

ciB(t, t+ Ti) + LB(t, t+ TN) ; ciL = αi

CN(t)

L=

N∑i=1

αi(1 + 1

myi(θi))i

+1

(1 + 1myN(θN))N

; θi = t+ Ti (5.102)

The yield to maturity yi(θi) can be extracted exactly as the case where the coupon bond

data is given with coupon bonds have a fixed maturity. Eq. (5.99) is then modified to yield

the zero coupon bond as follows

B(t, t+ Ti) =1

(1 + 1myi(t+ Ti))i

= exp−∫ t+Tn

t

dxf(t, x) = exp−∫ Tn

0

dθf(t, θ)(5.103)

§ 5.16 Appendix 5: Forward interest rates and zero coupon

bonds

The forward interest rates are sometimes given directly, as is the case for the US. In other

cases, the price of coupon bonds for different maturities is given, from which the zero coupon

bond price can be obtained as discussed in Section § 5.15 and given in Eq. (5.99).

In this Section, the forward interest rates f(t, x) are obtained from the price of zero coupon

bonds B(t, Tn).

Consider a collection of zero coupon bonds B(t, Tn) maturing at future time Tn in the

future. Let the present time t = 0 to simplify the notation, and define

B(0, Tn) ≡ B(Tn) ; f(0, x) ≡ f(x)


The forward interest rates, from Eq. (5.99) (suppressing the index t = 0) are given by

B(Tn) = exp−∫ Tn

0

dxf(x) =1

(1 + 1myn)n

(5.104)

The future times Tn define a lattice for the zero coupon bond. The forward interest rates

are defined on the future lattice in the following manner

f(x) = fn(x) ; x ∈ [Tn−1, Tn]

The forward interest rates and the corresponding future times are shown as the Figure 5.18.

Hence

B(Tn) = exp−N∑n=1

∫ Tn

Tn−1

dxfn(x)

and

ln(B(Tn−1)

B(Tn)) =

∫ Tn

Tn−1

dxfn(x)

Note that

................0 1 2 N-1 N

f1 f

2 ................ f

N

T1 T

2 ................. T

N-1 T

N

Figure 5.18: Forward interest rate and future time lattice.

The forward interest rates are assumed, piecewise, to be second order polynomials and

given below

fn(x) = an + bnx+ cnx2 (5.105)


Hence

ln(B(Tn−1)

B(Tn)) =

∫ Tn

Tn−1

dxfn(x)

= an(Tn − Tn−1) +1

2bn(T 2

n − T 2n−1) +

1

3cn(T 3

n − T 3n−1) (5.106)

Let

f ′n−1(Tn) =dfn−1(x)

dx

∣∣∣x=Tn

For each period from Tn−1 to Tn, to ensure that the forward interest rates yield a smooth

function for f(x), the following conditions are imposed.

• fn−1(Tn) = fn(Tn) : forward interest rates are continuous. Hence

an−1 + bn−1Tn + cn−1T2n = an + bnTn + cnT

2n (5.107)

• f ′n−1(Tn) = f ′n(Tn) : forward interest rates first derivatives are continuous. Hence

bn−1 + 2cn−1Tn = bn + 2cnTn (5.108)

We count the number of independent equations that the scheme provides.

• There are N conditions are in Eq. (5.106).

• The boundary conditions given in Eqs. (5.107) and (5.108) are for points from 1 to

N − 1, and hence yield 2(N − 1) conditions.

• One more conditions is that the spot interest rate r is obtained from the market. Using

f(t, t) = r yields

f1(0) = r = a1 (5.109)


• The last condition is the Neumann condition at the end point, that is f ′N(TN) = 0; this

follows from the fact that the final value of the interest rate is taken to be random.

Hence

f ′N(TN) = 0 = bN + 2cNTN (5.110)

In summary, from Eqs. (5.106), (5.107), (5.108), (5.109) and (5.110), the number of

equation we have is

N + 2(N − 1) + 2 = 3N

and this is sufficient to fix the 3N unknown parameters an, bn, cn.

There are 3N parameters and 3N linear equations. These linear equation are written as

AX = b, where the matrix A is defined by Eqs. (5.106), (5.107), (5.108), (5.109) and (5.110)

and given below. The structure of matrix A is in overlapping blocks of size 3x6 and organized

as shown in Figure 5.19. More precisely, each block has the following entries

• The first row is the continuity equation given in Eq. (5.107). The first entry A11 = 1 is

due to the boundary condition given in Eq. (5.109). In general, there are 6 entries to

this row.

• The second row enters the data given in Eq. (5.106). In general, there are 3 entries to

this row.

• The third row is the condition given in Eq. (5.108), with 4 entries per row. The last

row of A encodes the boundary condition given in Eq. (5.110).

The matrix structure of A is shown in Figure 5.19.


. . . . . .

. . . . . .

3 conditions

6 parameters

A=

Figure 5.19: The 3x6 block structure, with three elements overlapping between successiverows, is shown in the figure.

The matrix elements are written as follows.

1 2 3 4 5 6 ... 3n− 2 3n− 1 3n 3n+ 1 3n+ 2 3n+ 3 ... 3N − 2 3N − 1 3N

1 0 0 0 0 0... 0 0 0 0 0 0... 0 0 0

T112T

21

13T

31 0 0 0... 0 0 0 0 0 0... 0 0 0

0 1 2T1 0 −1 −2T1... 0 0 0 0 0 0... 0 0 0

... ... ... ... ... ...... ... ... ... ... ... ...... ... ... ...

... ... ... ... ... ...... 1 Tn T 2n −1 −Tn −T 2

n ... 0 0 0

... ... ... ... ... ...... 0 0 0 Tn − Tn−112(T 2

n − T 2n−1) 1

3(T 3n − T 3

n−1) 0 0 0

... ... ... ... ... ...... 0 1 2Tn 0 −1 −2Tn... 0 0 0

... ... ... ... ... ...... ... ... ... ... ... ...... ... ... ...

... ... ... ... ... ...... 0 0 0 0 0 0... 0 1 2TN


×

a1

b1

c1

...

an−1

bn−1

cn−1

an

bn

cn

...

aN

bN

cN

=

r

ln(B(0)/B(T1))

0

...

0

0

ln(B(Tn − 1)/B(Tn))

...

0

The solution to the forward interest rates determination is given by

AX = b ⇒ X = A−1b (5.111)

From N data points, we could obtain a fit with 3N parameters for the forward interest rates, and

hence obtaining a far superior result that using only the N data points would yield. The reason being


we used the continuity and differentiability of the forward interest rates, and in this way encoding

more information into the fit for the forward interest rates. One can of course, use a higher power

fit to get even better results, but in our studies we find the quadratic fit to be adequate [48].

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