Inflation – linked Option Pricing: a Market Model Approach with

Stochastic Volatility

LIANG LIFEI

(B.Sc. (Hons), NUS)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2010

i

Acknowledgements

I have been interested in the area of financial modeling for the whole of my undergraduate and

graduate years, and I am very glad that this thesis has given me the chance to gain more

modeling knowledge of inflation-linked derivatives. I would like to extend my deepest

appreciation to the following people whose support has made my research project an enjoyable

experience.

First and foremost, I am very lucky to be a student of my supervisor Prof. Xia Jianming and co-

supervisor Dr. Oliver Chen. They have provided me with valuable suggestions and

encouragement for the research, and their insights have inspired me and broadened my

knowledge in this field. I am very grateful to have them as my supervisors.

I would like to thank Mr. Pierre Lalanne as well as inflation trading desk of UBS who have

answered my queries with practical knowledge and helped me generously throughout the thesis.

I would also like to thank Mr. Zhang Haibo and Ms. Zhang Chi from Department of Chemical

and Biomolecular Engineering, NUS, whose expertise in optimization has helped me greatly.

Finally, I would like to thank Department of Mathematics and National University of Singapore

for providing necessary resources and financial support; my friends for their cheerful company

and my parents who have always been supportive throughout the years.

ii

Table of Contents

Acknowledgements ........................................................................................................................ i

Summary ....................................................................................................................................... iv

List of Symbols .............................................................................................................................. 1

List of Tables ................................................................................................................................. 3

List of Figures ................................................................................................................................ 5

1. Introduction ........................................................................................................................... 6

2. Two Factor Stochastic Volatility LMM Model ................................................................. 10

2.1. Forward CPI and Forward Risk Neutral Measure .......................................................... 10

2.2. Two Factor Model and Derivation of Pricing Formula.................................................. 11

2.3. Implementation Issues .................................................................................................... 15

3. Hedging of Inflation - Linked Options .............................................................................. 17

4. Convexity Adjustment ......................................................................................................... 20

5. Calibration ........................................................................................................................... 25

5.1. Parameterization ............................................................................................................. 26

5.2. Interpolation – Based Calibration .................................................................................. 28

5.3. Non – Interpolation – Based Calibration........................................................................ 38

6. Conclusions........................................................................................................................... 44

Bibliography ................................................................................................................................ 46

Appendices ................................................................................................................................... 49

iii

Appendix I Derivation of YoY caplet price under one factor stochastic volatility................... 49

Appendix II Riccati equation .................................................................................................... 59

Appendix III Structural deficiency of one factor model ........................................................... 60

Appendix IV Interpolation based on flat volatilities ................................................................. 61

iv

Summary

Inflation-linked derivatives‟ modeling is a relatively new branch in financial modeling.

Originally it was adapted from interest rate models; but attention is currently turning to market

model. In this thesis, we extend stochastic volatility market model to two-factor setting. The

analysis in this thesis shows that two-factor model offers more profound structure and greater

flexibility of fitting volatility surface while retaining the tractability of one-factor model.

We then apply the two-factor model to two related issues. Hedging analysis is conducted from a

new perspective where zero-coupon (ZC) options are used to hedge year-on-year (YoY) options.

This can be of great practical interest as it leverages on a complicated trading book and saves on

transaction cost. Convexity adjustment is also approximated under the model. Furthermore, we

have illustrated in detail how it can be captured via concrete trading activities.

The new two-factor model regime and broader scope which aims to calibrate both ZC and YoY

options with one model, call for new calibration procedures. In this thesis, two approaches have

been proposed.

Firstly, we devise an interpolation scheme that yields a market consistent interpolation. A

calibration against these interpolated prices can reveal mispricing and, thus, arbitrage

opportunities between the two options markets. However, a more thorough analysis is necessary

to determine if a misprice can really constitute an arbitrage opportunity.

Secondly, to mitigate the arbitrary nature of interpolation, we propose a non-interpolation-based

calibration scheme. In this approach, only market-quoted prices are inputs of calibration. ZC and

YoY option prices are weighted differently to reflect their respective market liquidity and bid-

offer spreads.

v

With this thesis, we fulfilled the aim to build a comprehensive framework under which an

inflation-linked option pricing model can be calibrated and applied.

1

List of Symbols

The Consumer Price Index at time

- forward CPI at time t

Swap break-even of a zero – coupon inflation-linked swap

Price at time t of nominal zero coupon bond with maturity

Short rate at time s

Period between and

Forward rate between times and as seen at time t

Volatility of

The j th factor loading of , j = 1 & 2

The j th factor loading of extended by time, i.e. it is equal to

when

and 0 afterwards

The j th common variance process of forward CPIs, j = 1 & 2

Mean reversion of

Long-term variance of

Volatility of variance

Brownian motion associated to of

Brownian motion that drives

Brownian motion that drives

Correlation between the j th factor of and , i.e.

Correlation between and

2

C

Correlation between and the j th factor of , i.e.

YoY caplet price

Price of ZC caplet with maturity at

Weight assigned to errors of ZC caps in non-interpolation-based calibration

Weight assigned to errors of YoY caps in non-interpolation-based calibration

3

List of Tables

Table 4.1 Dynamic hedging when moves by 1.

23

Table 4.2 Dynamic hedging when moves up (above) and down (below)

by 1.

24

Table 5.2.1

EUR HICP ZC Cap prices, with maturities from 1yr to 10 yr and

strikes from1% to 5%. Highlighted are market prices and others

prices are interpolated.

31

Table 5.2.2 EUR HICP ZC Cap implied volatilities, with maturities from 1yr to

10 yr and strikes from1% to 5%.

31

Table 5.2.3 EUR HICP YoY Cap spot vol, with maturities from 1yr to 10 yr and

strikes from1% to 5%.

32

Table 5.2.4 EUR HICP YoY Cap implied correlations, with maturities from 1yr

to 9 yr and strikes from1% to 5%.

32

Table 5.2.5

EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and

strikes from1% to 5%. New Interpolation.

33

Table 5.2.6

EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and

strikes from1% to 5%. Old Interpolation.

33

Table 5.2.7

Model parameters of interpolation-based calibration. Left: volatility

coefficients. Center: volatility factor loadings. Right: forward CPI /

volatility correlations.

36

Table 5.2.8 Relative percentage error of ZC option prices with maturities from

1yr to 10 yr and strikes from 1% to 5%.

37

Table 5.3.1 YoY implied correlation. Above: perturbed. Below: original. 38/39

4

Table 5.3.2 Relative percentage error of YoY cap prices. Bolded are relative error

of extended caps.

40

Table 5.3.3 Relative percentage error of ZC option prices with maturities from

1yr to 10 yr and strikes from 1% to 5%.

42

Table 7.4.1 EUR HICP YoY Cap prices.

61

Table 7.4.2 EUR HICP YoY Cap flat vol. Bolded are quoted and others are

linearly interpolated.

61

Table 7.4.3 EUR HICP YoY Cap prices. Bolded are quoted and others are from

interpolated flat vols.

62

5

List of Figures

Figure 4.1 Structure of forward starting ZC swap

22

Figure 5.2.1 EUR HICP YoY Caplet spot volatility, with maturities from 1yr

to 10 yr and strikes from1% to 5%. Left: new interpolation in this

thesis; right: old interpolation (refer to figure 7.4.2 of

Appendices).

34

Figure 5.2.2 Market and calibrated YoY implied volatility with x – axis

representing the strikes (%).

35

Figure 5.2.3 Model parameters of interpolation-based calibration. Left:

. Center left:

. Center right:

. Right:

.

36

Figure 5.3.1 YoY implied correlation. Left: perturbed. Right: original.

38

Figure 5.3.2 EUR HICP YoY Caplet spot volatility. Left: perturbed; right:

original.

39

Figure 5.3.3 Model parameters of non-interpolation-based calibration. Left:

. Right:

.

41

Figure 5.3.4 Model parameters of non-interpolation-based calibration. Left:

. Center left:

. Center right:

. Right:

.

41

Figure 5.3.5 Calibrated YoY spot volatility from non-interpolation-based

calibration.

41

Figure 7.4.1 EUR HICP YoY Cap flat vol surface

62

Figure 7.4.2 EUR HICP YoY Cap resulted spot vol 63

6

1. Introduction

Inflation-linked derivatives market was born around 2002 out of hedging needs of market

makers. Currently, inflation – linked swap is the most liquid product whose volume has

increased from almost zero in 2001 to $110 billion in 2007. Trading of inflation-linked options is

also picking up gradually.

Most inflation models so far have been derived from interest rate models. Currently, the pricing

of inflation-linked options is addressed by resorting to a foreign currency analogy. In Jarrow and

Yildrim (2003), the dynamics of nominal and real rates are modeled by one-factor Gaussian

process in the framework proposed in Heath, Jarrow and Morton (1992), or the HJM framework.

Inflation is then interpreted as the exchange rate between the nominal and the real economies.

However, this model suffers two major drawbacks. Firstly, it is based on the market non-

observable of real interest rate. Secondly, it generates volatility skew at the expense of over –

parameterization as remarked in Ungari (2008).

As such, alternative approaches are gaining popularity. For example, Kazziha (1999), Belgrade

et al (2004) and Mercurio (2005) considered a market model in which the underlying variables

are forward CPIs evolving as driftless geometric Brownian motions. Mercurio and Moreni

(2005) took one step further to incorporate a mean – reverting stochastic volatility process to the

forward CPIs while Mercurio and Moreni (2010) built a more complete model with SABR

stochastic volatility process. Details of SABR model can be found in Hagan (2002).

In Liang (2010), a variation of Mercurio and Moreni (2005) model is built. All forward CPIs are

assumed to share one common volatility process - differentiated only by respective factor

loadings. The model is then implemented and calibrated with a boot-strapping algorithm. As

7

explained in Liang (2010) and presented again in Appendix III of this thesis, the effect of the

factor loading of stochastic volatility on volatility surface is relatively limited, the model has

encountered structural difficulty in generating both skew and smile configurations in one

volatility surface. Moreover, we note that inflation - linked zero – coupon option resembles an

equity vanilla option while YoY option is nothing but a series of forward starting options. In

Bergomi (2004) and Fonseca, Grasselli and Tebaldi (2008), it is well explained that there is a

structural limitation which prevents one – factor stochastic volatility models to price consistently

forward starting options with vanilla options.

This thesis corrects the original model deficiency encountered in Liang (2010) and documented

in other literatures detailed above and addresses related issues such as hedging, convexity

adjustment and calibration. As a whole, we strive to build a comprehensive framework under

which an inflation-linked pricing model can be calibrated and applied.

Works on multi – factor stochastic volatility model, such as Bates (2000) and Christoffersen

(2007) have inspired us to extend the model in Liang (2010) to two-factor setting. Our

calibration shows that one factor can have relatively fast mean-reversion to determine short-run

variance while the other can have relatively slow mean-reversion to determine long-run variance.

Despite the seemingly straight-forward extension from one to two-factor setting, it has

significant implication to the calibration scheme. The boot-strapping style schemes proposed in

Liang (2010) will no longer work. Calibration taking into consideration of the global

configuration of the volatility surface is employed in this thesis. The goal of consistently pricing

YoY and ZC options further complicates the calibration. Belgrade et al. (2004) attempted to

address this consistency, though their model setting was too simplistic and there was no

8

calibration against actual data to measure the accuracy. We have, thus, designed a number of

schemes, which can be broadly categorized as interpolation- and non-interpolation-based

schemes.

Variable reduction is important in global optimization as we seek to avoid over-parameterization

as well as to increase the efficiency of optimization. To this end, we have adopted various

parameterization functions for different parameter sets. For example, the parameterization of

correlations is based on Mercurio and Moreni (2010) while that of the factor loadings of

volatility is as proposed in Zhu (2007).

The interpolation-based scheme conforms more to the current practice in the interest rate market.

The advantage of this scheme is that there is more control and more information extracted on the

individual caplets and floorlets. However, the parsimony of the model and the regularity of the

parameters are sacrificed, rendering the model parameters sometimes arbitrary.

The non-interpolation-based scheme does not carry out any form of interpolation and depends

solely on market-quoted prices. It improves on the deficiencies of interpolated scheme. However,

the model is sensitive to the choice of the parameterization functional.

The contributions of the thesis are the followings:

1. We extend and explore the inflation-linked stochastic volatility market model under a

multi - factor setting.

2. The hedging analysis is conducted from a new perspective that ZC options are used to

hedge YoY options – a strategy of great practical interest.

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3. While most articles treat the subject of calibration as if it is no different from that of

interest rate models, it is actually more delicate in an inflation-linked context. This thesis

fills the gap by proposing and testing two original calibration schemes.

The thesis is structured as follows. In chapter 2, the extended two-factor stochastic volatility

model and the derived pricing formula for inflation-linked options are presented. Following that,

two related topics – hedging of inflation-linked options and convexity adjustment of YoY swaps

and options – are addressed in chapters 3 and 4 respectively. Chapter 5 deals with calibration

schemes under the new model. It starts with a review of the original calibration scheme in Liang

(2010) and a discussion on parameterization. The next subsection focuses on how to maintain the

consistency between ZC and YoY option market and proposes an alternative scheme-

interpolation-based calibration scheme. Non-interpolation-based calibration scheme is presented

in the last subsection. Conclusions are presented in chapter 6. For self-contained purpose,

mathematical derivations and illustrative examples are presented in Appendices.

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2. Two Factor Stochastic Volatility LMM Model

2.1. Forward CPI and Forward Risk Neutral Measure

We denote the Consumer Price Index at time . In Kazziha (1999), - forward CPI at

time t, is defined as the fixed amount X to be exchanged at time for the CPI , so

that the swap has zero value at time t. With the description of zero coupon inflation linked swap

in Liang (2010), X at time zero is then written as

where denotes the swap breakeven of a zero-coupon inflation-linked swap. More

specifically, the fixed leg of the swap is priced at

which is equal to the floating leg, priced at

above denotes the price at time t of nominal zero coupon bond with maturity .

Since we note that the floating leg of the swap can be priced at

So we derive an important property of the forward CPI that it is a martingale under -

forward measure, i.e.

11

where defines the canonical filtration from time t.

2.2. Two Factor Model and Derivation of Pricing Formula

We present the model directly under - forward measure.

where we denote

the factor loading of ;

forward rate between times and as seen at time t, ignoring day count conventions;

;

volatility of and

where

will be defined shortly.

Before we proceed to define stochastic volatilities and the corresponding correlation structures,

we remark that terminates as at . To avoid the confusion in time, we define

Then, the two forward CPIs are matched in time dimension.

12

And the stochastic process of volatility V is

The correlation structures are defined as

With j = 1 and 2, k = i and i – 1 and < , > denotes the quadratic co-variation of stochastic

processes. The technical aspect of quadratic variation can be found in Brigo and Mercurio

(2006).

And at the same time, we also have, in the nominal market,

with

By applying the same drift-freezing technique and fast Fourier transformation as in Liang (2010)

and Mercurio and Moreni (2005)1, we derive the pricing formula of YoY caplet as

1 The two articles are on one-factor setting, but the extension to two-factor setting is straight-forward as

we have shown in Appendix I.

13

where

And functions A and B above are solutions of the following general Riccati equation:

14

which take the specific form in equations below

where

And the YoY caplet can thus be evaluated by computing numerically the integral. Detailed

derivation is presented in Appendix I.

Similarly but more easily, ZC caplet can be priced as

where

We will end this section with a few remarks, which serve to deepen our understanding of the

model. A more thorough discussion can be found in Christofferson (2007).

Since

15

thus,

and

So finally, we see that

The simple calculation above shows that the effective correlation is now stochastic. By adding

one more volatility factor, the model is not only extended, but is also fundamentally changed.

The richer volatility surface configurations result not only from a mere increase of the number of

parameters, but from a more complex model structure.

2.3. Implementation Issues

As pointed out in many articles, Heston integrals (1) and (3) involve a complex logarithm which

is inherently discontinuous. This discontinuity causes numerical integration to be difficult and

sometimes generates mispricing for long maturities, as documented in Albrecher et al. (2006). In

Liang (2010), the method proposed in Kahl and Jäckel (2006) is adopted to alleviate, if not

resolved the problem. However, it has been pointed out but left unresolved that the method is

difficult to extend to inflation – linked context. It turns out that Albrecher et al. (2006) provides a

more feasible correction for the purpose of this thesis.

16

They noted that complex root of has two possible values. By conventions, the

principal value is used in most formulas of Heston characteristic function and is also returned by

most software packages. But using the principal values causes a branch cut – a curve in the

complex across which a function is discontinuous. Function A in equation (2) above jumps

discontinuously each time the imaginary part of the argument of the logarithm crosses the

negative real axis. They followed by proving that choosing the second value of

would circumvent the problem. Contrary to Liang (2010), we adopt this

proposition for both YoY and ZC pricing formulas throughout the thesis.

Another benefit of this remedy is that the numerical integration becomes much more stable, in

the sense that it yields valid value for much wider range of parameters. Consequently, the

calibration process improves in efficiency. This is because when optimization algorithm scans

through the cost function across a range of parameter sets, many more points contain a valid

numerical value.

17

3. Hedging of Inflation - Linked Options

Although fast Fourier transformation enables us to derive a closed formula to price inflation-

linked options, this approach yields little information on hedging strategies. In this section, we

apply Itô‟s lemma to obtain a hedging strategy.

According to conventional wisdom, we normally dynamically hedge a derivative with its

underlying. In the present context, a YoY option should, thus, be hedged via a combination of

ZC swaps. However, in order to leverage on the synergy of a bank‟s trading book, which

contains both YoY and ZC options, we aim to hedge YoY options with ZC options. In this way,

we save transaction cost compared to usual strategies.

Let be the price of a YoY caplet, then by Itô‟s lemma,

where denotes as before the quadratic co – variation of stochastic processes and we

abbreviate further as

We propose a hedging portfolio, consisting of ZC swaps as well as ZC options with maturities of

and , i.e.

18

where

is the date of the reference CPI fixing, which is assumed to be identical for both swaps and

caps for simplification purpose;

and

denote ZC caps with maturities of and respectively;

represents ZC swap with reference CPI fixing at and maturity at

.

Again, by Itô‟s lemma, each component satisfies the following:

Putting everything together, we have

Thus, in order to hedge all the risks, we must have

19

Solving this simultaneous equation give us the hedging ratios:

We remark finally that important prerequisites of this section is that there is a liquid market for

ZC options and that the model must be able to price both YoY and ZC options accurately.

However, the latter is a question not entirely trivial and few articles explicitly address it. We will

discuss this in chapter 5 on calibration.

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4. Convexity Adjustment

A peculiar feature of inflation market is that there are two structures co-existing in both swap and

option markets, which brings us to the important notion of convexity adjustment.

When calculating YoY swap and YoY option, the main problem lies in calculating the YoY

forward value. A simplistic view would be to calculate the YoY forward ratio as the ratio of

forward CPIs, which is principally derived from ZC swaps. By doing so, we implicitly assume

that the forward value of a ratio is the ratio of the forward value. This is generally not true. Thus,

we need to compute convexity adjustment to correct this error.

More precisely, we want to compute u such that

Recall expression (8) from Appendix I that

Adopting the similar drift-freezing technique as before, but freezing further the volatility terms

too such that the drift is completely deterministic, we get

21

where we set

to simplify

the notation.

Define

, then is a martingale as below

As a result,

Having determined the convexity adjustment, we digress a little to the pricing of YoY swap and

the capturing of its convexity. As YoY swap is nothing but a strip of forward starting ZC

22

inflation linked swaps, here we only focus on forward starting ZC swaps for simplification‟s

sake.

Refer to Figure 4.1 below for a forward starting ZC swap. We enter the trade at time T0. It

involves exchanging a fixed payment with realized inflation from time T1 to T2. Contrary to

standard ZC swap which can be priced in a model – independent fashion, the pricing of forward

starting ZC swap is model-dependent.

By non-arbitrage argument and the definition of Ti - forward measure, we know that the inflation

leg is evaluated to

Letting

and by Itô‟s lemma, we obtain the following hedging strategy:

T0 T1

T2

Fixed leg

Figure 4.1. Structure of forward starting ZC swap

23

So to hedge a long position of a forward starting ZC swap, we short

units of ZC

swap and long

units of ZC swap.

In Table 4.1 and 4.2 below, we detail the hedging strategies under different scenarios and how

we capture the convexity along the way.

Position

Rebalancing P&L Rebalancing P&L

swap

0

0

swap

N/A

N/A

We note that the hedging shown in Table 4.1 is exact because the swap is linear in .

Table 4.1. Dynamic hedging when moves by 1.

24

Position

Rebalancing P&L

swap

swap

0

Position

Rebalancing P&L

swap

swap

0

We remark the “buy low sell high” pattern in the rebalancing as bolded in Table 4.2; this is

where we capture convexity just like how gamma is captured through dynamic hedging of

vanilla options. As a result, the greater the volatility of , the more convexity we can capture

through our dynamic hedging.

Table 4.2. Dynamic hedging when moves up (above) and down (below) by 1.

25

5. Calibration

Before we move on to calibration of two-factor model, we review first the boot – strapping

calibration scheme proposed in Mercurio and Moreni (2005) and Liang (2010)2. Suppose we

have already interpolated market quoted prices and a matrix of prices with no missing maturities

is available3. The first year caplet prices are used to obtain

. Next, with the

first six parameters fixed, the second year caplet prices are inputted to obtain

.

The next four parameters

are obtained similarly with third year caplet prices

and so on. The advantage of this algorithm is that each time, we only need to run an optimization

with four (six for the first) parameters. Assuming we have ten maturities, ten optimizations each

with four parameters (six for the first) will take much less time than one with 42 parameters. As

remarked in Liang (2010), this scheme works fine if the volatility surface displays similar pattern

across maturities. It encounters structural difficulty while handling more irregular volatility

surface, i.e. surface with both skew and smile configurations. A detailed analysis quoted from

Liang (2010) on this deficiency can be found in Appendix III.

As we have noted earlier, the model is extended so that it can generate richer volatility surface

configurations. The extension seems to be straight forward. However, it entails certain changes

to the calibration scheme.

Firstly, the original boot-strapping calibration scheme is no longer available. Should the boot-

strapping scheme still be applied and volatility parameters calibrated from the first year price, the

greater flexibility of the two factor model is lost. All the volatility parameters will still only

2 Parameters in this paragraph follow those in Liang (2010). They are the same as in this thesis, except not indexed

by k = 1 & 2 due to one-factor setting. 3 Interpolation will be dealt with specifically in the subsection of interpolation-based calibration

26

reflect the information of volatility surface captured in the first year price. As a result, calibration

is to be carried out with the global configuration of the volatility surface taken into consideration.

Secondly, the number of variables is an important factor in optimization. By extending the model

from one to two factors, we increase the total number of parameters from 42 to 84, if the final

maturity is taken to be 10 yr. Over – fitting should be avoided in modeling and running a global

optimization with 84 parameters is a computationally expensive. Thus, parameterization is

adopted to reduce the number of variables.

5.1. Parameterization

We parameterize correlation structures as proposed in Mercurio and Moreni (2010). In this

subsection, M denotes the total number of maturities against which we are calibrating.

For k = 1 and 2,

and

with

It is proven in Mercurio and Moreni (2010) that this parameterization is well defined in the sense

that the correlation matrix is positive semi-definite.

27

The main intuitions of such parameterization are: firstly in (4), decreases as i and j become

further apart and, thus, intuitively get less correlated while on the other hand, there will always

be an asymptotic correlation secondly in (5), maximum coupling is between and .

This is because monetary policy over period , summarized by , is considered as a

response to inflation behavior over period [ ].

is parameterized by

where

A more delicate task is to parameterize ‟s. Instead of viewing the model as one common

stochastic volatility process differentiated by different volatility coefficient, we propose to view

it as different stochastic volatility processes:

Similar to that we think that volatility surface is smooth; we also propose to parameterize the

volatility of volatility by a smooth function, e.g.

28

We remark that the proposed parameterization is by no means unique. For example,

was originally tested. It yielded satisfactory result though equation (6) produces an even more

accurate calibration and is therefore adopted.

This set of parameterization has a few repercussions: first and foremost, we reduce the total

number of parameters from 84 to 24; secondly, the smoothness of the parameter sets is

guaranteed. A calibration with smooth and regular parameter sets is more reliable than that with

an arbitrary and highly irregular parameter sets. However, as we will observe in the next

subsection, some of the parameterizations have to be loosened to achieve an accurate calibration

under certain circumstances.

5.2. Interpolation – Based Calibration

What we term „interpolated – based calibration‟ in this thesis is very similar to the existing

methodology in interest rate market. Flat volatility is backed out from market quoted prices and

then interpolated – linearly or in a more complicated fashion – to obtain volatilities and hence

prices for other missing maturities. Calibration is finally conducted against this matrix of market

quoted as well as interpolated prices. This scheme is reasonable when we only calibrate a single

instrument, e.g. YoY option. As a comparison benchmark for following contents, Appendix IV

illustrates the procedure on YoY caps with data from EUR HICP on 13 Jan 2010.

Nevertheless, the structure of inflation-linked market adds one complication. Market quoted

prices start from maturity of 2yr, 5yr and so on. If the focus is solely on YoY option market,

simple extrapolation from 2yr to 1yr suffices. As our scope encompasses both YoY and ZC

29

option markets, it then must be noted that first year YoY option is the same as first year ZC

option. So the first year prices of YoY options have to be imported from first year ZC options.

Furthermore, as we would like to price both YoY and ZC options, the interpolated prices of YoY

options must be consistent with the interpolated prices of ZC options. Belgrade et al. (2004)

represented so far the only attempt to address this issue of consistency. However, there is no

calibration against actual data to measure the accuracy. In this subsection, we improve on

Belgrade et al. (2004) to design a more accurate and robust interpolation scheme and,

consequently, a calibration algorithm based on the interpolated prices.

Based on market model approach, we present in the following parts an interpolation scheme that

will achieve theoretical consistency across two markets and at the same time taking into

consideration of market quoted prices. The core notion of „implied correlation‟ is introduced here

just like the notion of „implied volatility‟ was introduced to reconcile Black-Scholes model and

market quoted option prices.

Introduce a simplified market model with constant volatility as follows:

with

indicating the correlation structure.

a stochastic drift frozen to time 0.

30

Under this simplified model, ZC option can be easily priced with a Black – Scholes formula. To

price YoY option, we let

. Then by Itô‟s lemma,

By non – arbitrage argument, a YoY caplet is written as

This simple model provides us with an interpolation method. First of all, market quoted prices of

ZC options are used to obtain ‟s. We then interpolate to get the implied volatility surface of ZC

options. With formula (7), a consistent implied volatility of YoY option can be derived as a

function of correlation ‟s. we finally derive the „implied correlation‟ from market quoted

prices of YoY options.

31

We will illustrate the proposed approach with again the market data of EUR HICP on 13 Jan

2010. The ZC Cap prices are shown in Table 5.2.1 with highlighted being market prices and

others interpolated. Table 5.2.2 shows the corresponding implied volatilities.

1% 2% 3% 4% 5%

1 0.0101573 0.00536902 0.00285556 0.00168913 0.001103933

2 0.02119 0.0101 0.00434 0.00197 0.001

3 0.03372882 0.01563763 0.00606033 0.002350818 0.001017436

4 0.04723139 0.02190822 0.00807568 0.002844145 0.001109173

5 0.06074 0.02841 0.01021 0.00338 0.00123

6 0.07387372 0.03485188 0.01230405 0.003861447 0.001322455

7 0.08681 0.04141 0.01451 0.00439 0.00144

8 0.0990976 0.04767158 0.01651061 0.004794616 0.001485382

9 0.11069907 0.05366521 0.01843999 0.005182388 0.001531798

10 0.12209 0.05974 0.02048 0.00562 0.0016

1% 2% 3% 4% 5%

1 2.06% 2.02% 2.14% 2.36% 2.62%

2 2.28% 2.22% 2.33% 2.55% 2.83%

3 2.51% 2.42% 2.51% 2.74% 3.04%

4 2.73% 2.62% 2.70% 2.93% 3.25%

5 2.96% 2.83% 2.89% 3.12% 3.46%

6 3.08% 2.97% 3.04% 3.28% 3.63%

7 3.20% 3.12% 3.18% 3.43% 3.80%

8 3.25% 3.23% 3.30% 3.55% 3.93%

9 3.30% 3.34% 3.41% 3.67% 4.06%

10 3.35% 3.45% 3.53% 3.79% 4.19%

With Table 5.2.2 and formula (7), corresponding YoY option implied volatilities can be

expressed as a function of correlations. As for each strike, there are only four market-quoted

prices and nine correlations available. To avoid over – fitting and ensure regularity of result,

Table 5.2.1. EUR HICP ZC Cap prices, with maturities from 1yr to 10 yr and strikes from 1%

to 5%. Highlighted are market prices and others prices are interpolated (same below).

Table 5.2.2. EUR HICP ZC Cap implied volatilities, with maturities from 1yr to 10 yr and strikes from 1% to 5%.

32

correlations of each strike are parameterized by a cubic – spline. Finally, calibration against

market data with this cubic spline is presented below in Table 5.2.3 and 5.2.4.

1% 2% 3% 4% 5%

1 2.06% 2.02% 2.14% 2.36% 2.62%

2 1.33% 1.30% 1.35% 1.47% 1.62%

3 1.12% 1.07% 1.08% 1.16% 1.29%

4 1.06% 0.98% 0.98% 1.05% 1.16% 5 1.01% 0.92% 0.91% 0.96% 1.07% 6 0.90% 0.84% 0.83% 0.88% 0.97% 7 0.77% 0.73% 0.72% 0.76% 0.84% 8 0.64% 0.62% 0.59% 0.60% 0.64% 9 0.73% 0.67% 0.61% 0.64% 0.68%

10 0.81% 0.79% 0.83% 0.87% 0.96%

ρ(I, i, i - 1) 1% 2% 3% 4% 5%

1 118.38% 117.74% 118.12% 118.61% 118.72% 2 111.51% 111.76% 112.23% 112.59% 112.64% 3 107.61% 108.23% 108.71% 108.97% 109.01% 4 105.93% 106.52% 106.95% 107.16% 107.19% 5 105.15% 105.67% 105.94% 106.08% 106.12% 6 105.22% 105.53% 105.72% 105.83% 105.88%

7 104.99% 105.30% 105.56% 105.75% 105.88% 8 103.56% 104.20% 104.66% 104.75% 104.88% 9 102.34% 102.80% 102.64% 102.75% 102.78%

Table 5.2.4 shows correlation greater than one, which is certainly against mathematical intuition.

However, we would like to note that

1. The implied correlations only play an intermediary role, that lead to a smooth and

accurate calibration of YoY option prices. We do not seek to draw any conclusion on

correlation structure with it.

Table 5.2.3. EUR HICP YoY Cap spot vol, with maturities from 1yr to 10 yr and strikes from 1% to 5%.

Table 5.2.4. EUR HICP YoY Cap implied correlations, with maturities from 1yr to 9 yr and strikes from 1% to 5%.

33

2. The model set – up of constant volatility and frozen drift is simplistic and approximate;

the implied correlation thus does not purely reflect information on correlation structure

but, at the same time, contains information of other parameters not represented in the

model.

In consequence, despite its absurd appearance, we believe that when exercised with caution,

implied correlations do provide a reasonable and promising means to a consistent interpolation.

To further demonstrate the effects of the new interpolation, we compare the interpolated caplet

prices and spot volatilities from the new (Table 5.2.5) and the old (Table 5.2.6) interpolation.

Maturity 1% 2% 3% 4% 5%

1 0.010157 0.005369 0.002856 0.001689 0.001104 2 0.013712 0.007631 0.003915 0.002071 0.001206 3 0.015037 0.008588 0.004407 0.002270 0.001288 4 0.016078 0.009507 0.005067 0.002702 0.001587 5 0.016164 0.009724 0.005275 0.002868 0.001725 6 0.016016 0.009739 0.005369 0.002936 0.001757

7 0.015196 0.009161 0.004882 0.002514 0.001403 8 0.014394 0.008434 0.004070 0.001707 0.000703

9 0.014561 0.008760 0.004338 0.002121 0.001009 10 0.015104 0.009996 0.006613 0.004252 0.002988

4

4 This table can be directly derived from table 7.4.3 of Appendix.

Maturity 1% 2% 3% 4% 5%

1 0.009105135 0.004286642 0.001863572 0.000865558 0.000450534 2 0.014764865 0.008713358 0.004906428 0.002894442 0.001859466 3 0.016155617 0.009852065 0.005713865 0.003472092 0.002313825 4 0.016146946 0.00963961 0.005239501 0.002877243 0.001742613 5 0.014977 0.008328 0.003797 0.001491 0.000544

6 0.016024223 0.009817785 0.00546183 0.003027956 0.001843234

7 0.015185777 0.009082215 0.00478817 0.002422044 0.001316766 8 0.015564375 0.009763408 0.005567226 0.003137214 0.001917624 9 0.014595249 0.008994644 0.004960705 0.002665026 0.001550129

10 0.013900375 0.008431948 0.004492069 0.00227776 0.001232247

Table 5.2.5. EUR HICP YoY Caplet prices, with maturities from 1 to 10 yr and strikes from 1% to 5%. New Interpolation

Table 5.2.6. EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and strikes from1% to 5%. Old Interpolation4.

34

The first observation is that old interpolation does not take into consideration the ZC option

market. Hence, the first year prices do not correspond to the first year ZC option prices. New

interpolation scheme, on the other hand, corrects this error. Moreover, even though flat

volatilities are linearly interpolated and extrapolated in the case of Liang (2010), the resulted

spot volatility surface is still irregular, as we can see in the right of figure 5.2.1. But new

interpolation generates a much smoother volatility surface, e.g. figure 5.2.1 left.

With a full matrix of prices extracted, we are now ready to conduct the calibration, during which

the square sum of errors of market and model prices is minimized. Given the proposed

parameterization functional, the following algorithm is designed.

I. We first start with the most parsimonious structure, i.e. both volatility coefficients and

correlation parameters are parameterized by 24 variables as enumerated below:

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

1 2 3 4 5 6 7 8 9 10

0.00%

1.00%

2.00%

3.00%

1 2 3 4 5 6 7 8 9 10

Figure 5.2.1. EUR HICP YoY Caplet spot volatility, with maturities from 1yr to 10 yr and strikes from1% to 5%.

Left: new interpolation in this thesis; right: old interpolation (refer to figure 7.4.2 of Appendices).

35

II. If the above does not generate a good calibration, we will loosen some parameterization.

We start with volatility coefficients ‟s, as this condition appears to be the most

arbitrary. We will take the values of ‟s determined by

as the initial guess,

keep the rest of the parameters constant and run a calibration with the ten parameters.

III. Similarly, if this still does not generate satisfactory result, we loosen the regularity

condition imposed on

As

and ‟s are more important than

and

in terms of the impact on pricing, by loosening these two restrictions, we can

normally obtain a fairly good calibration.

IV. If the algorithm does not terminate at III, then in the final step, we optimize with :

There are altogether 56 parameters involved. Intimidating as it appears, this does not pose

great computational challenge as we are already close to the minimal and, thus, the

optimization terminates reasonably fast with few iterations.

We report the calibration result of YoY options in Figure 5.2.2 and parameters obtained in Table

5.2.7 and Figure 5.2.3.

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

1 2 3 4 5

2yr Market Vol

2yr Calibrated Vol

5yr Market Vol

5yr Calibrated Vol

7yr Market Vol

7yr Calibrated Vol

10yr Market Vol

10yr Calibrated Vol

Figure 5.2.2. Market and calibrated YoY implied volatility with x – axis representing the strikes (%)

36

α 4.5771823 0.8190538

σ1 σ2

ρ1(I, i, V) ρ2(I, i, V)

θ 0.2781674 0.0760959

0.013163 0.082623

-1 -0.06325

ε 0.9634653 1

0.017208 0.066601

-0.69911151 -0.02557

V(0) 0.2316799 0.0773175

0.005775 0.069499

1 -0.10504

0 0.079097

1 -0.11698

0.00598 0.077454

-1 -0.13001

0.007727 0.058275

-1 -0.14053

0.002213 0.038511

-1 -0.11102

0.00932 0.043456

1 -0.22586

0.009372 0.058504

1 -0.27116

0.006246 0.029766

-1 -0.09706

It is worth noting that the speed of reversion α is about 4.6 for one volatility process and about

0.8 for the other. Thus, we can interpret that the factor with smaller reversion is determining the

long – run variance while that with greater reversion is determining the short – run variance.

Also we note that the calibrated correlation structures are well – behaved.

The rational for using global optimization and the proposed algorithm are on the hypothesis that

information of the initial point is not available. This is so when there is a big shift to the market.

For example, when an inflation index turns out to be very different from expected. When market

0

0.5

1

1 3 5 7 9

0.8

0.85

0.9

0.95

1

1 3 5 7 9-0.8

-0.6

-0.4

-0.2

0

1 3 5 7 9

-0.5

-0.495

-0.49

-0.485

-0.48

1 3 5 7 9

Table 5.2.7. Model parameters. Left: volatility coefficients. Center: volatility factor loadings.

Right: forward CPI / volatility correlations

Figure 5.2.3. Model parameters. Left: . Center left:

. Center right: . Right:

.

37

model becomes more popular and regularly used and market is stable, simple local optimization

suffices.

The final step consists of calculating ZC options prices from the calibrated parameters. We

present the relative percentage errors in Table 5.2.8 below.

1% 2% 3% 4% 5%

1 -1.72% -0.72% 2.16% 2.44% -2.91%

2 -9.82% -25.17% -39.14% -44.87% -49.38%

3 -9.57% -35.80% -55.83% -60.99% -64.37%

4 -5.62% -28.16% -52.39% -58.46% -61.50%

5 -5.27% -28.02% -58.69% -68.40% -73.36%

6 -6.09% -35.37% -76.83% -87.98% -92.63%

7 -6.20% -42.76% -92.05% -98.06% -99.39%

8 -4.41% -34.99% -89.13% -97.94% -99.45%

9 -3.01% -28.36% -83.34% -96.39% -98.92%

10 -3.10% -35.17% -96.41% -99.77% -99.98%

Results are acceptable for short maturities and for at-the-money (ATM) caplets, but deteriorate

rapidly for longer maturities and greater strikes. We remark here a few points on how to further

assess the quality of the results:

1. Relative errors must be assessed with respect to bid-offer spreads. ZC option market is

rather illiquid compared to YoY option market and bid-offer spreads are much larger. An

accurate calibration against mid-prices might not be more significant compared to a

calibration with moderate relative error, when bid-offer spreads dominate.

2. Calibration should be carried out across a period. Illiquid market is always subject to

structural factors that result in consistent mispricing from model “fair” price. Such a

phenomenon is well documented in nominal bond market, though few articles tackle this

issue in inflation – linked market.

Table 5.2.8. Relative percentage error of ZC option prices with maturities from 1yr to 10 yr and strikes from 1% to 5%.

38

5.3. Non – Interpolation – Based Calibration

Non – interpolation – based calibration is introduced because of the observation that interpolated

YoY caplet prices are sensitive to implied correlation. By choosing an equally smooth but

slightly different implied correlation structure, we end up with a very different interpolated YoY

caplet prices. An example is presented below.

The implied correlation structures are almost identical but the resulted interpolated caplet prices

and spot volatility surfaces vary by a large extent as shown in Figure 5.3.2.

90.00%

100.00%

110.00%

120.00%

1 2 3 4 5 6 7 8 9

90.00%

100.00%

110.00%

120.00%

1 2 3 4 5 6 7 8 9

ρ(I, i, i - 1) 1 2 3 4 5

1 118.38% 117.74% 118.12% 118.61% 118.70%

2 111.50% 113.82% 114.26% 112.09% 112.16%

3 108.23% 108.23% 107.98% 109.16% 109.00%

4 105.50% 105.56% 106.57% 107.23% 107.37%

5 105.21% 105.50% 106.10% 106.30% 106.34%

6 105.18% 105.65% 105.60% 105.66% 105.71%

7 104.40% 104.70% 104.79% 104.71% 104.79%

8 103.48% 103.92% 104.16% 104.24% 104.38%

9 102.94% 103.60% 103.87% 104.12% 104.02%

Figure 5.3.1. YoY implied correlation. Left: perturbed. Right: original.

39

ρ(I, i, i - 1) 1 2 3 4 5

1 118.38% 117.74% 118.12% 118.61% 118.72%

2 111.51% 111.76% 112.23% 112.59% 112.64%

3 107.61% 108.23% 108.71% 108.97% 109.01%

4 105.93% 106.52% 106.95% 107.16% 107.19%

5 105.15% 105.67% 105.94% 106.08% 106.12%

6 105.22% 105.53% 105.72% 105.83% 105.88%

7 104.99% 105.30% 105.56% 105.75% 105.88%

8 103.56% 104.20% 104.66% 104.75% 104.88%

9 102.34% 102.80% 102.64% 102.75% 102.78%

It is thus logical to suspect that interpolation introduces some degree of arbitrary information and

contaminates calibration. This brings us naturally to the idea of non – interpolation – based

calibration. In this scheme, interpolation step is omitted and only market quoted cap prices are

used for calibration.

A crucial issue in non – interpolation – based calibration is how to ensure the regularity of

resulted caplet prices. But as it turns out, calibration with the most parsimonious structure, i.e.

0.0000%

1.0000%

2.0000%

3.0000%

1 2 3 4 5 6 7 8 9 10

0.00%

1.00%

2.00%

3.00%

1 2 3 4 5 6 7 8 9 10

Figure 5.3.2. EUR HICP YoY Caplet spot volatility. Left: perturbed; right: original.

Table 5.3.1. YoY implied correlation. Above: perturbed. Below: original.

40

the first step of the calibration algorithm proposed above, yields a satisfactory result.

Consequently, the regularity of parameters guarantees that final prices and implied volatility

surface is also smooth.

Finally, to avoid the problem of not taking into consideration of the information of ZC option

market, we calibrate against market quoted prices of both ZC and YoY options. As ZC option

market exhibits a much larger bid – offer spread, ZC option prices should not retain the same

significance as YoY option prices in terms of providing information to calibration. As a result,

we apply different weightings to different prices. Intuitively, the ratio of the two weights can be

inversely proportional to the ratio of the bid – offer spreads of the market. The optimization,

thus, becomes

where denotes market quoted ZC option prices. Other notations are interpreted similarly.

With 10% and 90%5, the result is reported in Table 5.3.2.

1% 2% 3% 4% 5% 6%

2 -0.39% 0.83% 1.05% 1.15% -2.02% -8.08%

5 -0.91% -0.49% -1.73% -2.26% 0.95% 4.24%

7 -1.21% 0.04% -0.18% -1.38% 0.53% 4.69%

10 -2.97% -0.96% 0.29% -0.68% -0.65% 2.49%

12 -4.82% -2.66% -0.86% -1.51% -2.21% 0.04%

5 We have tested various combinations such as 20% - 80%, 10% - 90% and 5% - 95%. 10% - 90% so far produces the

most accurate calibration and the result is therefore presented in this thesis.

Table 5.3.2. Relative percentage error of YoY cap prices. Bolded are relative error of extended caps.

41

To highlight the smoothness of parameters, obtained parameters and final spot volatilities are

reported in Figure 5.3.3 and 5.3.4 below.

0

0.05

0.1

0.15

0.2

0.25

0.028

0.03

0.032

0.034

0.036

1 2 3 4 5 6 7 8 9 10

-0.940.00

0.10

0.20

0.30

0.40

1 2 3 4 5 6 7 8 9 10

00.20.4

0.6

0.8

1

1 3 5 7 9

00.20.4

0.6

0.8

1

1 3 5 7 9

0

0.2

0.4

0.6

0.8

1 3 5 7 9-0.8

-0.6

-0.4

-0.2

0

1 3 5 7 9

0.000%

0.500%

1.000%

1.500%

2.000%

2.500%

1 2 3 4 5 6 7 8 9 10

Figure 5.3.4. Model parameters. Left: . Center left:

. Center right: . Right:

.

Figure 5.3.3. Model parameters. Left: . Right:

.

Figure 5.3.5. Calibrated YoY spot volatility.

42

Thus, by forcing the regularity of parameters and calibrating only against market quoted data, we

can still achieve an accurate calibration and generate a reasonably shaped implied volatility

surface as in Figure 5.3.5.

Finally, plugging the parameters to price ZC options yields Table 5.3.3.

1% 2% 3% 4% 5%

1 -1.82% 1.81% 0.02% -11.88% -29.45%

2 -1.02% 1.09% 3.37% 4.06% -0.55%

3 -0.54% 0.40% 1.15% 5.78% 11.63%

4 -0.33% 0.12% -1.28% 1.61% 11.96%

5 -0.29% -0.05% -2.96% -3.71% 6.78%

6 0.48% 1.21% -1.77% -5.46% 4.15%

7 0.95% 2.13% -0.60% -7.25% -0.57%

8 1.66% 3.63% 2.46% -5.47% -0.18%

9 2.15% 4.82% 5.24% -3.52% -0.52%

10 2.49% 5.76% 7.73% -1.59% -1.82%

We observe that the relative errors shown in Table 5.3.3 now are much reduced. However, this

does not necessarily mean that we should absolutely favor this method over the previous. Should

our goal be to arbitrage the two markets, interpolation-based calibration should still be employed

with certain supplementary statistical analysis on the relative errors as we have recommended.

On the other hand, if we were to price an exotic inflation-linked instrument, non-interpolation-

based calibration is more appropriate.

Another advantage of this approach is that as we keep all parameterized structures, we can easily

extend the parameters to price more extreme options. This is contrary to interpolation-based

calibration where we have to take all the options of interest upon calibration, some of which

might be thinly traded, upon calibration.

Table 5.3.3. Relative percentage error of ZC option prices with maturities from 1yr to

10 yr and strikes from 1% to 5%.

43

With non-interpolation-based scheme, we only need to calibrate with respect to more liquid

options and extend the calibrated parameters to price other options. In Table 5.3.2, we also

present the relative error of YoY caps with strikes of 6% and maturity of 12 years. Note that they

are not included in the calibration process.

Readers may notice that should we extend the parameters, it is inevitable to change the total

number of instruments M in formula (5) (from 10 to 12 in our case) and change the correlation

structure. While this is true, it is observed that total error generally changes little or decreases

when plugging in correlations from a new M. We believe that the correlation structure now

contains more complete information than before.

44

6. Conclusions

In this thesis, we have extended the inflation – linked stochastic volatility market model,

proposed in Mercurio and Moreni (2005) and modified in Liang (2010), to two – factor regime.

This extension provides greater flexibility in fitting volatility surface, while retaining the

tractability of the original model. Literature on two - factor Heston model has shed light on its

internal structure and why it works much better than single factor model. We still believe that it

will be constructive to conduct principal component analysis on the implied volatility of inflation

– linked market. It will tell us more on why two – factor model works better or not and how

many factors make a good modeling, etc.

Based on this extended model set – up, we have derived a theoretical hedging strategy, following

which, YoY options can be hedged with ZC options. This new perspective enables maximum

leverage of a trading book containing both options and saves transaction cost. Under the same

setting, the convexity adjustment of YoY swap rate is also approximated. We detail in the thesis

the process of how to capture the convexity through concrete trading activities.

The structure of inflation market where two product structures co-exist makes calibration unique

and few articles specifically tackle this issue. In this thesis, we fill the gap by proposing and

testing two methods: interpolation and non – interpolation – based calibration.

The interpolation – based calibration conforms more to the orthodox methodology. We devised a

simple market model where volatility is constant and introduced the notion of implied correlation

to interpolate YoY cap prices. In this way, the information of ZC option market is incorporated

into YoY cap prices. It is observed that the resulted implied volatility surface is smoother than

that resulted from a naïve interpolation of flat volatilities. This method produces a more accurate

45

calibration of YoY option prices though that of ZC options is reasonable only for short maturities

and ATM options.

On the other hand, non – interpolation based calibration is introduced so that no arbitrary/false

information is produced as a result of interpolation process. We assign different weight to the

calibration of ZC and YoY options, with the former certainly smaller than latter. Though we can

optimize the weights attached more systematically, our calibration experience shows that 10% -

90% works the best so far. The advantage of this methodology is that firstly, it ensures a smooth

parameter set; secondly, we obtain a more acceptable calibration of ZC options at the expense of

a slightly worse calibration of YoY options and finally the parameters can be extended to price

more extreme options.

Interesting and revealing, it is yet beyond the scope of the thesis to conclude which method is

superior or more applicable. In further research, we believe that more practical tests have to be

conducted. Areas of consideration can include:

1. Models can be tested across a period to further assess the quality of calibration.

2. Calibrated parameters should be tested across a period to assess its stability.

46

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49

Appendices

Appendix I Derivation of YoY caplet price under one factor stochastic volatility

Recall the following set-up as in section 2:

And the stochastic process of volatility V is

With j = 1 & 2 and k = i & i – 1,

And

50

with

Freezing the stochastic drift to time 0, we have

Let

Then by Itô‟s lemma,

We define further that

Note that

51

By standard option pricing theory, we have the price of a YoY caplet of period [ and

strike k under - measure:

Where we define

We then have

By we define

So we will evaluate firstly

52

Let , then

We note that

We assume that

where we define

With terminal conditions as

By Feynman – Kac formula, we have

53

Substituting the assumed form, we obtain

Putting everything together, we obtain

54

Or equivalently for j = 1 and 2,

So the partial differential equation is satisfied if the follows hold, for j = 1 and 2

Thus

More specifically, setting , we have for j = 1 and 2

55

Thus,

So we have at this stage

Where we recall that

We simplify further via the same drift – freezing technique,

Define the characteristic function as follows:

So by Feynman-Kac formula, we get

56

We suppose that the characteristic function takes the following form,

Where we have as terminal conditions

Substituting this into the original PDE, we obtain further that

57

Combining everything back together,

Or equivalently, for j = 1 and 2,

Thus, the differential equations satisfied by and

are

58

So we can solve explicitly,

By setting we obtain

59

Appendix II Riccati equation

Let A and B be solutions of the following general Riccati equation:

By setting

, where

and by rewriting (1) now as

We can then solve the above easily as

A can then be solved via integration

60

With

and

, so that finally we get

Appendix III Structural deficiency of one factor model

Recall that is introduced in the following manner:

Looking instead at the transformed volatility process:

As such, we see that the coefficient acts as a scaling factor, increasing both the volatility of

volatility and the long term variance. We see that the two have counter – effects against each

other. While increasing volatility of volatility increases the curvature of the smile, a greater long

term variance flattens it, while moving it parallel upwards. Furthermore, the correlation is not

affected by the factor loading.

The implication of this is that beside the volatility of volatility we do not possess much

flexibility in terms of fitting the curvature of the smile. We can, thus, anticipate a case where the

model will not generate an accurate calibration. If the volatility surface consists of skewed as

well as curved smiles, the one-factor model will find it hard to accommodate the two different

regimes.

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Appendix IV Interpolation based on flat volatilities

Shown in Table 7.4.1 is the market price of EUR HICP YoY caps on 13 Jan 2010. In this

appendix, we illustrate an interpolation scheme with this starting point.

Flat volatilities of 2, 5, 7 and 10 yrs are derived from the market quoted cap prices. We then linearly

interpolate and extrapolate the flat volatilities to obtain the flat volatility surface as presented in table

7.4.2 and figure 7.4.1.

1% 2% 3% 4% 5%

2 0.02387 0.013 0.00677 0.00376 0.00231

5 0.07115 0.04082 0.02152 0.0116 0.00691

7 0.10236 0.05972 0.03177 0.01705 0.01007

10 0.14642 0.08691 0.04679 0.02513 0.01477

1% 2% 3% 4% 5%

1 1.79% 1.74% 1.80% 1.94% 2.13%

2 1.65% 1.59% 1.63% 1.75% 1.92%

3 1.51% 1.44% 1.46% 1.56% 1.71%

4 1.36% 1.30% 1.30% 1.37% 1.49%

5 1.22% 1.15% 1.13% 1.18% 1.28%

6 1.15% 1.08% 1.09% 1.10% 1.19%

7 1.08% 1.01% 0.99% 1.02% 1.10%

8 1.03% 0.97% 0.94% 0.97% 1.05%

9 0.99% 0.93% 0.90% 0.93% 1.00%

10 0.94% 0.88% 0.86% 0.88% 0.95%

Table 7.4.1. EUR HICP YoY Cap prices

Table 7.4.2. EUR HICP YoY Cap flat vol. Bolded are quoted and others are linearly interpolated

62

Next, we compute the other YoY cap prices from the interpolated flat volatilities. The full matrix

of YoY cap prices is shown in table 7.4.3. Finally, the spot volatility surface, i.e. figure 7.4.2 can

be generated from the caplet prices.

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

1% 2% 3% 4% 5%

1 0.009105 0.004287 0.001864 0.000866 0.000451

2 0.02387 0.013 0.00677 0.00376 0.00231

3 0.040026 0.022852 0.012484 0.007232 0.004624

4 0.056173 0.032492 0.017723 0.010109 0.006366

5 0.07115 0.04082 0.02152 0.0116 0.00691

6 0.087174 0.050638 0.026982 0.014628 0.008753

7 0.10236 0.05972 0.03177 0.01705 0.01007

8 0.117924 0.069483 0.037337 0.020187 0.011988

9 0.13252 0.078478 0.042298 0.022852 0.013538

10 0.14642 0.08691 0.04679 0.02513 0.01477

Figure 7.4.1. EUR HICP YoY Cap flat vol surface

Table 7.4.3. EUR HICP YoY Cap prices. Bolded are quoted and others are from interpolated flat vols.

63

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

1 2 3 4 5 6 7 8 9 10

Figure 7.4.2. EUR HICP YoY Cap resulted spot vol