Phase Transition in Ln Ir Model Pirjol_D_FE_Seminar_10!17!2011

8/3/2019 Phase Transition in Ln Ir Model Pirjol_D_FE_Seminar_10!17!2011

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MotivationBlack, Derman, Toy model

Model with log-normal rates in the terminal measurePhase Transition

Phase Transition in a Log-normal Interest RateModel

Dan Pirjol1

1J. P. Morgan, New York

17 Oct. 2011

Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
http://find/http://goback/


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Outline

Introduction to interest rate modeling

Black-Derman-Toy model

Generalization with continuous state variable Binomial tree

BDT model with log-normal short rate in the terminal measure

Analytical solution Surprising large volatility behaviour

Phase transition

Summary and conclusions



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[1] F. Black, E. Derman, W. ToyA One-Factor Model of Interest Rates

Fin. Analysts Journal, 24-32 (1990).[2] P. Hunt, J. Kennedy and A. Pelsser,Markov-functional interest rate modelsFinance and Stochastics, 4, 391-408 (2000)

[3] A. PelsserEfficient methods for valuing interest rate derivativesSpringer Verlag, 2000.

[4] D. Pirjol,Phase transition in a log-normal Markov functional model

J. Math. Phys 52, 013301 (2011).[5] D. Pirjol,Nonanalytic behaviour in a log-normal Markov functional model,arXiv:1104.0322, 2011.



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Interest rates

Interest rates are a measure of the time value of money: what is thevalue today of $1 paid in the future?

0 5 10 15 20 25

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

USD Zero Curve

27Sep2011

ExampleZero curve R(t) for USD as of 27-Sep-2011. Gives the discount curve as

D(t) = exp(R(t)t).Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model

M i i


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Interest rates evolve in time

0 5 10 15 20 25

0.01

0.02

0.03

0.04

0.05

USD Zero Curve

22Jun to 27Sep

ExampleThe range of movement for the USD zero curve R(t) between22-Jun-2011 and 27-Sep-2011.


M ti ti


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Yield curve inversion

0 5 10 15 20 25 30

4.25

4.5

4.75

5

5.25

5.5

5.75

6

USD Zero Rate

1Nov07

ExampleBetween 2006 and 2008 the USD yield curve was inverted - rates for 2years were lower than the rates for shorter maturities


Motivation


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Interest rate modeling

The need to hedge against movements of the interest ratescontributed to the creation of interest rate derivatives.

Corporations, banks, hedge funds can now enter into many types ofcontracts aiming to mitigate or exploit/leverage the effects of theinterest rate movements

Well-developed market. Daily turnover for1:

interest rate swaps $295bn forward rate agreements $250bn interest rate options (caps/floors, swaptions) $70bn

This requires a very good understanding of the dynamics of theinterest rates markets: interest rate models

1The FX and IR Derivatives Markets: Turnover in the US, 2010, Federal Reserve

Bank of New YorkDan Pirjol Phase Transition in a Log-Normal Interest Rate Model

Motivation


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Setup and definitions

Consider a model for interest rates defined on a set of discrete dates(tenor)

0 = t0 < t1 < t2 tn1 < tnAt each time point we have a yield curve.

Zero coupon bonds Pi,j: price of bond paying $1 at time tj, as of time ti.Pi,j is known only at time ti

Li = Libor rate set at time ti for the period (ti, ti+1)

Li =1

1

Pi,i+1 1

...0

Li

1 i i+1 n...


Motivation


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Simplest interest rate derivatives: caplets and floorlets

Caplet on the Libor Li with strike K pays at time ti+1 the amount

Pay = max(Li(ti) K, 0)

Similar to a call option on the Libor Li

Caplet prices are parameterized in terms of caplet volatilities i via theBlack caplet formula

Caplet(K) = P0,i+1CBS(Lfwd

i , K, i, ti)

Analogous to the Black-Scholes formula.


Motivation


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Black, Derman, Toy modelModel with log-normal rates in the terminal measure

Phase Transition

Analogy with equities

Define the forward Libor Li(t) at time t, not necessarily equal to ti

Li(t) =1

Pt,iPt,i+1

1

This is a stochastic variable similar to a stock price, and its evolution canbe modeled in analogy with an equity

Assuming that Li(ti) is log-normally distributed one recovers the Blackcaplet pricing formula

Generally the caplet price is the convolution of the payoff with (L), theLibor probability distribution function

Caplet(K) =

0

dL(L)(L K)+


Motivation


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Phase Transition

Caplet volatility - term structure

t

ATM

Volatility hump at the short end


Motivation


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Phase Transition

ATM caplet volatility - term structure

0 2 4 6 8 10

20

40

60

80

100

1203m caplet vols

27Sep11

Actual 3m USD Libor caplet yield (log-normal) volatilities as of

27-Sep-2011Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model

MotivationBl k D T d l


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Phase Transition

Caplet volatility - smile shape

Fwd

ATM

(K)

Strike


MotivationBlack De man To model


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Phase Transition

Modeling problem

Construct an interest rate model compatible with a given yield curve P0,iand caplet volatilities i(K)

1. Short rate models. Model the distribution of the short rates Li(ti) at

the setting time ti. Black, Derman, Toy model

Hull, White model - equivalent with the Linear Gaussian Model(LGM)

Markov functional model

2. Market models. Describe the evolution of individual forward LiborsLi(t)

Libor Market Model, or the BGM model.


MotivationBlack Derman Toy model


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Phase Transition

The natural (forward) measure

Each Libor Li has a different natural measure Pi+1

Numeraire = Pt,i+1, the zero coupon bond maturing at time ti+1The forward Libor Li(t)

Li(t) =1

Pt,iPt,i+1

1

is a martingale in the Pi+1 measure

Li(0) = Lfwdi = E[Li(ti)]

This is the analog for interest rates of the risk-neutral measure for equities


MotivationBlack Derman Toy model


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Phase Transition

Simple Libor market model

Simplest model for the forward Libor Li(t) which is compatible with agiven yield curve P0,i and log-normal caplet volatilities iLog-normal diffusion for the forward Libor Li(t): each Li(t) driven by itsown separate Brownian motion Wi(t)

dLi(t) = Li(t)idWi(t)

with initial condition Li(0) = Lfwdi , and Wi(t) is a Brownian motion inthe measure Pi+1

Problem: each Libor Li(t) is described in a different measure.

We would like to describe the joint dynamics of all rates in a commonmeasure.This line of argument leads to the Libor Market Model. Simplerapproach: short rate models




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Phase Transition

The Black, Derman, Toy model

Describe the joint distribution of the Libors Li(ti) at their setting times ti

n1

0

Li

1 i i+1 n... ...

L L L L0 1 n2

Libors (short rate) Li(ti) are log-normally distributed

Li(ti) = Lieix(ti) 12

2i ti

where Li are constants to be determined such that the initial yield curveis correctly reproduced (calibration)

x(t) is a Brownian motion. A given path for x(t) describes a particularrealization of the Libors Li(ti)




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, , yModel with log-normal rates in the terminal measure

Phase Transition

Black, Derman, Toy model

The model is formulated in the spot measure, where the numeraireB(t) is the discrete version of the money market account.

B(t0) = 1

B(t1) = 1 + L0

B(t2) = (1 + L0)(1 + L1)

Model parameters:

Volatility of Libor Li is i Coefficients Li

The volatilities i are calibrated to the caplet volatilities (e.g. ATM vols),and the Li are calibrated such that the yield curve is correctly reproduced.




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Black, Derman, Toy model - calibration

Price of a zero coupon bond paying $1 at time ti is given by anexpectation value in the spot measure

P0,i = E 1

B(ti)

= E

1(1 + L0)(1 + L1(x1))

(1 + Li1(xi1))

The coefficients Lj can be determined by a forward induction:

P0,1 =1

1 + L0 L0

P0,2 = P0,1E

11 + L1(x1)

= P0,1+

dx2t1

e1

2t1 x2

11 + L1e1x

12

21t1

L1and so on... Requires solving a nonlinear equation at each time step




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BDT tree

The model was originally formulated on a tree.Discretize the Markovian driver x(t), such that from each x(t) it can

jump only to two values at t = t +

x(t)

xup(t + )

xdown(t + )

d

dd

Mean and variance

E[x(t + ) x(t)] = 12

1

2

= 0

E[(x(t + )

x(t))2] =

1

2

+1

2

=

Choose xup = x +

, xdown = x

with equal probabilities such thatthe mean and variance of a Brownian motion are correctly reproduced



M d l i h l l i h i l


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BDT tree - Markov driver x(t)

0

+1

1

+2

0

2

E

0 1 2

t

Inputs:

1. Zero coupon bonds P0,iEquivalent with zero rates Ri

P0,i =1

(1 + Ri)i

2. Caplet volatilities i



M d l ith l l t i th t i l


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BDT tree - the short rate r(t)

r0

r1e1 12

21

r1e1 12

21

r2

e22

222

r2e222

r2e22222

dd

d

dd

d

dd

d

E

0 1 2

t

Calibration: Determiner0, r1, r2, such that thezero coupon prices arecorrectly reproduced

Zero coupon bonds P0,iprices

P0,i = E 1

Bi

Money market account B(t)- node and path dependent

B0 = 1

B1 = 1 + r(1)

B2 = (1 + r(1))(1 + r(2)) .Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model


Model with log normal rates in the terminal measure
http://goback/http://find/http://find/http://goback/


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Calibration in detail

t = 1: no calibration needed

P0,1 =1

1 + r0

t = 2: solve a non-linear equation for r1

P0,2 =1

1 + r0

12

1

1 + r1e1 1221

+1

2

1

1 + r1e1 12 21

and so on for r2, .

Once ri are known, the tree can be populated with values for the shortrate r(t) at each time t

Products (bond options, swaptions, caps/floors) can be priced by working

backwards through the tree from the payoff time to the presentDan Pirjol Phase Transition in a Log-Normal Interest Rate Model


Model with log-normal rates in the terminal measure


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Pricing a zero coupon bond maturing at t = 2

P0,2

Pup1,2 =

11+rup1

Pdown1,2 =1

1+rdown1

1

1

1

dd

d

dd

d

ddd

We know rup1 = r1e1 12

21 and rdown1 = r1e

1 12 21 can find the bond

prices Pt,2 for all t





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Model with log normal rates in the terminal measurePhase Transition

BDT model in the terminal measure





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gPhase Transition

BDT model in the terminal measure

Keep the same log-normal distribution of the short rate Li as in the BDTmodel, but work in the terminal measure

Li(ti) = Lieix(ti) 12

2i ti

Li(ti) = Libor rate set at time ti for the period (ti, ti+1)

Numeraire in the terminal measure: Pt,n, the zero coupon bond maturingat the last time tn

...0

L i

1 i i+1 n...





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gPhase Transition

Why the terminal measure?

Why formulate the Libor distribution in the terminal measure?

Numerical convenience. The calibration of the model is simpler thanin the spot measure: no need to solve a nonlinear equation at eachtime step

The model is a particular parametric realization of the so-calledMarket functional model (MFM), which is a short rate model aimingto reproduce exactly the caplet smile. MFM usually formulated inthe terminal measure.

More general functional distributions can be considered in theMarkov functional model Li(ti) = Lif(xi), parameterized by anarbitrary function f(x). This allows more general Libor distributions.





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Phase Transition

Preview of results

1. The BDT model in the terminal measure can be solved analytically forthe case of uniform Libor volatilities i = . Solution possible (inprinciple) also for arbitrary i, but messy results.

2. The analytical solution has a surprising behaviour at large volatility:

The convexity adjustment explodes at a critical volatility, such thatthe average Libors in the terminal measure (convexity-adjustedLibors) Li become tiny (below machine precision)

This is very unusual, as the convexity adjustments are supposed tobe well-behaved (increasing) functions of volatility

The Libor distribution function in the natural measure collapses tovery small values (plus a long tail) above the critical volatility

Caplet volatility has a jump at the critical volatility, after which itdecreases slightly



Model with log-normal rates in the terminal measurePh T i i


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Phase Transition

What do we expect to find?2

Convexity adjusted Libor Li = related to the price of an instrumentpaying Li(ti, ti+1) set at time ti and paid at time tn

Price = P0,nEn[Li] = P0,nLi

Floating payment with delay: the convexity adjustment depends on thecorrelation between Li and the delay payment rate L(ti+1, tn)

...0

Li

1 i i+1 n...

2

Argument due to Radu Constantinescu.Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model


Model with log-normal rates in the terminal measurePh T iti


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Phase Transition

Convexity adjustment

Consider an instrument paying the rate Lab at time c. The price isproportional to the average of Lab in the c-forward measure

Price = P0,cEc[Lab]

ab

0

L

a b

ab

c

L bcL

Can be computed approximatively by assuming log-normally distributedLab and Lbc in the b-forward measure, with correlation

Ec[Lab] Lfwdab

1 Lfwdbc (c b)(eabbc

ab 1) + O((Lfwdbc (c b))2)





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Phase Transition

Convexity adjustment

The convexity adjustment is negative if the correlation between Lab andLbc is positive

Ec[Lab] Lfwdab

1 Lfwdbc (c b)(eabbc

ab 1) + O((Lfwdbc (c b))2)

0 2 4 6 8

0.2

0.4

0.6

0.8

1

1.2

1.4

a

cbexp1 Convexity-adjusted Libors Lifor 3m Libors (b = a + 0.25 )

Parameters:

ab = bc = 40% = 20%, c = 10 years

Naive expectation: The convexity adjustment is largest in the middle ofthe time simulation interval, and vanishes near the beginning and the end.


MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure

Phase Transition


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Phase Transition

Convexity adjusted Libors - analytical solution

The solution for the convexity adjusted LiborsLi for several values of thevolatility

Simulation parameters: Lfwdi = 5%, n = 40, = 0.25

5 10 15 20 25 30 35 40

1

2

3

4

5

=0.2

=0.3

=0.4



Phase Transition


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Phase Transition

Surprising results

For sufficiently small volatility , the convexity-adjusted Libors Liagree with expectations from the general convexity adjustmentformula

For volatility larger than a critical value cr, the convexityadjustment grows much faster.

The model has two regimes, of low and large volatility, separated bya sharp transition

Practical implication: the convexity-adjusted Libors Li become very

small, below machine precision, and the simulation truncates themto zero



Phase Transition


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Explanation

The size of the convexity adjustment is given by the expectation value

Ni = E[Pi,i+1ex 12

2ti]

Recall that the convexity-adjusted Libors are Li = Lfwdi /Ni

0 0.1 0.2 0.3 0.4 0.5

2

4

6

8

10

Plot of log Ni vs the volatility

Simulation with n = 40 quarterlytime steps

i = 30, t = 7.5, r0 = 5%

Note the sharp increase after a critical volatility cr 0.33



Phase Transition


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Explanation

The expectation value as integral

Ni = E[Pi,i+1ex 12

2ti] =

+

dx2ti

e 12ti x

2

Pi,i+1(x)ex 12

2ti

5 10 15 17.50

0.5

1

1.5

2

x

The integrand

Simulation with n = 20 quarterlytime steps i = 10, ti = 2.5

=

0.4 (solid)0.5 (dashed)

0.52 (dotted)

Note the secondary maximum which appears for super-critical volatilityat x

10

ti. This will be missed in usual simulations of the model.



Phase Transition


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More surprises: Libor probability distribution

Above the critical volatility > cr, the Libor distribution in the naturalmeasure collapses to very small values, and develops a long tail

0 0.02 0.04 0.06 0.08 0.1

10

20

30

40

50

60

70

0.050.35

0.1

0.2

= =

=

=

L

Simulation with n = 40 quarterly time steps = 0.25. The plot refers tothe Libor L30 set at time ti = 7.5.



Phase Transition


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Caplet Black volatility

0 0.1 0.2 0.3 0.4

0

0.25

0.5

0.75

1

1.25

1.5Black curve: ATM caplet volatility

Red curve: equivalent log-normalLibor volatility

2LNti = logE[L21]

E[L1]2

Simulation with n = 40 quarterly time steps = 0.25. The plot refers tothe Libor L30 set at time ti = 7.5

For small volatilities, the ATM caplet vol is equal with the Libor vol i.Above the critical volatility, the ATM caplet vol increases suddenly



Phase Transition


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Caplet smile

Above the critical volatility the caplet implied volatility develops a smile

0.01 0.02 0.03 0.04 0.05 0.06

0.2

0.4

0.6

0.8

1

1.2

1.4

0.10.20.3

=0.34

0.35=

=

==

Simulation with n = 40 quarterly time steps = 0.25. The plot refers tothe Libor L30 set at time ti = 7.5.



Phase Transition


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Conclusions

For sufficiently small volatility, the model with log-normallydistributed Libors in the terminal measure produces a log-normalcaplet smile

The probability distribution for the Libors in the natural (forward)measure is log-normal

Above a critical volatility cr the Libor probability distributioncollapses at very small values, and develops a fat tail

The caplet Black volatility increases suddenly above the criticalvolatility, and develops a smile

These effects are due to a coherent superposition of convexityadjustments

In practice we would like to use the model only in the sub-critical regime.Under what conditions does this transition appear, and how can we findthe critical volatility?



Phase Transition


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Phase Transition





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The generating function

We would like to investigate the nature of the discontinuous behaviourobserved at the critical volatility, and to calculate its value

Define a generating function for the coefficients c(i)j giving the one-step

zero coupon bond

f(i)(x) =ni1j=0

c(i)j x

j

This was motivated by a simpler solution for the recursion relation

The expectation value which displays the discontinuity is simply

Ni =

ni1j=0

c(i)j e

j2ti = f(i)(e2ti)





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Properties of the generating function

f(i)(x) is a polynomial in x of degree n i 1

f(i)(x) = 1 + c(i)1 x + c

(i)2 x

2 + + c(i)ni1xni1

where the coefficients are all positive and decrease with j

Can be found in closed form in the zero and infinite volatility limits 0,

It has no real positive zeros, but has n i 1 complex zeros. They arelocated on a curve surrounding the origin.





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Complex zeros of the generating function

Example: simulation with n = 40 quarterly time steps = 0.25, flatforward short rate r0 = 5%

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.3

The zeros of the generating functionat i = 30, corresponding to the

Libor set at ti = 7.5 years

Number of zeros = n i 1 = 9

Volatility = 30%

Key mathematical result: The generating function f(i)(x) is continuousbut its derivative has a jump at the point where the zeros pinch the realpositive axis





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Complex zeros - volatility dependence

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.3

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.31

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.32

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.33

0 0.1 0.2 0.3 0.4 0.5

2

4

6

8

10

Solid curve: plot off(i)

(e

2ti

)

Criterion for determining the critical volatility: The critical point at whichthe convexity adjustment increases coincides with the volatility where thecomplex zeros cross the circle of radius R1 = e

2ti





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Complex zeros - effect on caplet volatility

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.3

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.31

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.32

-4 -2 0 2 4

-4

-2

0

2

4

i30, psi0.33

0 0.1 0.2 0.3 0.40

0.25

0.5

0.75

1

1.25

1.5

Black curve: ATM caplet volatility

vs

Simulation with n = 40 quarterly time steps, at i = 30The turning point in ATM coincides with the volatility where thecomplex zeros cross the circle of radius R1 = e

2ti





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Phase transition

The model has discontinuous behaviour at a critical volatility cr

The critical volatility at time ti is given by that value of the modelvolatility for which the complex zeros of the generating functionf(i)(z) cross the circle of radius e

2ti

The position of the zeros and thus the critical volatility depend on

the shape of the initial yield curve P0,i For a flat forward short rate P0,i = e

r0ti the zeros are zk = er0xk,

where xk are the complex zeros of the simple polynomial

Pn(x) =1

1

e

r0+ x + x2 + + xni1

Approximative solution for the critical volatility at time ti

2cr 1

i(n i 1) log 1

r0





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Phase transition - qualitative features

The critical volatility decreases as the size of the time step isreduced, approaching a very small value in the continuum limit

The critical volatility increases as the forward short rate r0 isreduced, approaching a very large value as the rate r0 becomes very

small the applicability range of the model is wider in the smallrates regime

The phenomenon is very similar with a phase transition in condensedmatter physics, e.g. steam-liquid water condensation/evaporation, orwater freezing

The Lee-Yang theory of phase transitions relates such phenomena to theproperties of the complex zeros of the grand canonical partition function.



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Motivation


Phase Transition


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Conclusions

The Black, Derman, Toy model with log-normally distributed Liborin the terminal measure can be solved exactly in the limit ofconstant and uniform rate volatility

The analytical solution shows that the model has two regimes atlow- and high-volatility, with very different qualitative properties

The solution displays discontinuous behaviour at a certain criticalvolatility cr

Low volatility regime

Log-normal caplet smile Well-behaved Libor distributions

High volatility regime

The convexity adjustment explodes Libor pdf is concentrated at very small values, and has a long tail A non-trivial caplet smile is generated


Motivation


Phase Transition


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Comments and questions

A similar behaviour is expected also in a model with non-uniformLibor volatilities (time dependent), but the form of the analyticalsolution is more complicated

Is this phenomenon generic for models with log-normally distributedshort rates, e.g. the BDT model, or is it rather a consequence ofworking in the terminal measure?

Are there other interest rate models displaying similar behaviour?


Motivation


Phase Transition


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BDT model in the terminal measure - calibration

and exact solution


Motivation


Phase Transition


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Calibrating the model by backward recursion

The non-arbitrage condition in the terminal measure tells us that the zerocoupon bonds divided by the numeraire should be martingales

Pi,j

Pi,n= E[

1

Pj,n|Fi] , for all pairs (i,j)

Denote the numeraire-rebased zero coupon bonds as

Pi,j =Pi,j

Pi,n

Choose the two cases (i,j) = (i, i + 1)

Pi,i+1(xi) = E[Pi+1,i+2(1 + Li+1)|Fi] (i,j) = (0, i)

P0,i = E[Pi,i+1(1 + Li)]


Motivation


Phase Transition

R f P ( ) L


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Recursion for Pi,i+1(xi), Li

The two non-arbitrage relations can be used to construct recursivelyPi,i+1(xi) and Li working backwards from the initial conditions

Pn1,n(x) = 1 , Ln1 = Lfwdn1

No root finding is required at any step.

The calculation of the expectation values requires an integration overxi+1 at each step. Usual implementation methods:

1. Tree. Construct a discretization for the Brownian motion x(t)

2. SALI tree. Interpolate the function Pi,i+1(xi) between nodes andperform the integrations numerically

3. Monte Carlo implementation


Motivation


Phase Transition

A l i l l i f if


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Analytical solution for uniform

Consider the limit of uniform Libor volatilities i = The model can be solved in closed form starting with the ansatz

Pi,i+1(x) =

ni1j=0

c(i)j e

jx 12j22ti

Matrix of coefficients c(i)j is triangular (e.g. for n = 5)

c =

c(n1)0 0 0 0 0

c(n2)0 c

(n2)1 0 0 0

c(n

3)0 c

(n

3)1 c

(n

3)2 0 0

c(1)0 c

(1)1 c

(1)2 c

(1)3 0

c(0)0 c

(0)1 c

(0)2 c

(0)3 c

(0)4


Motivation


Phase Transition

R i l i f h ffi i(i )


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Recursion relation for the coefficients c(i)

j

The coefficients c(i)j and convexity adjusted Libors Li can be determined

by a recursion

c(i)j = c

(i+1)j + Li+1c

(i+1)j

1 e(j1)2ti+1

Li =P0,i P0,i+1

ni1j=0 c(i)j e

j2ti= Lfwdi

P0,i+1

ni1j=0 c(i)j e

j2ti

starting with the initial condition

Ln1 = P0,n1 1 , Pn1,n(x) = 1


Motivation


Phase Transition

R i f th ffi i t(i )


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Recursion for the coefficients c(i)

j

Linear recursion for the coefficients c(i)j . They can be determined

backwards from the last time point starting with c(n1)0 = 1

c(i)j = c

(i+1)j + Li+1c

(i+1)j1 e

(j1)2ti+1

i = n 1 :

i = n 2 :

i = n 3 :

c(n1)0

c

(n

2)

0

c(n3)0

c

(n

2)

1

c(n3)1 c

(n3)2

T

dd

ds

T T

dd

ds

dd

ds


Motivation


Phase Transition

S l ti f th d l


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Solution of the model

Knowing Pi,i+1

(x) and Li

one can find all the zero coupon bond prices

Pi,j(x) =Pi,j(x)

Pi,i+1(x)(1 + Liiex 12 2ti)

where

Pi,j(x) = E 1

Pj,n|Fi

= E[Pj,j+1(1 + Ljjexj 12

2tj))|Fi]

=

nj1k=0

c(j)k e

kx 12 (k)2ti

+Ljj

nj1k=0

c(j)k e

(k+1)x 12 (k2+1)2ti+k

2(tjti)

All Libor and swap rates can be computed along any path x(t)


Phase Transition in Ln Ir Model Pirjol_D_FE_Seminar_10!17!2011

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Transcript of Phase Transition in Ln Ir Model Pirjol_D_FE_Seminar_10!17!2011