simona svoboda - libor market model with stochastic volatility.pdf

8/14/2019 Simona Svoboda - Libor Market Model with Stochastic Volatility.pdf

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LIBOR Market Model with Stochastic Volatility.

First Year Transfer Report

Simona SvobodaMagdalen College, University of Oxford

November, 2005


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CONTENTS ii

2. Two-Regime Dynamics 53

3. Modelling Approaches 55

Chapter 6. Future Work 61

Bibliography 62


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Introduction

The LIBOR market model has become one of the most promising frameworks for modelling interest

rates and pricing of interest rate derivatives. This model is often attributed to Brace, Gatarek and

Musiela (BGM) [8] who were one of the first to publish this methodology. Early papers also include

those by Miltersen, Sandmann and Sondermann [39] and Jamshidian [32].

This modelling framework makes a break with previous approaches (see [59] for a comprehensive

study) which were based on the evolution of the continuously compounded short rate (e.g. Vasicek[60], CIR [12] and Hull-White [29] models) or instantaneous forward rates (HJM [27] approach).

None of these models were consistent with the market practice of using Blacks [6] formula to price

caps and swaptions and were fairly cumbersome to calibrate to market observables.

The market convention of pricing caps and swaptions using the Black formula is based on an

application of the Black and Scholes [7] formula for stock options with some simplifications and

assumptions about the distribution of the underlying interest rates i.e. that they are lognormally

distributed.

The LIBOR market model takes as its basic units, a set of spanning forward rates. Placing these

in a lognormal framework naturally leads to the Black cap formula for the pricing of interest rate

caps. Similarly, using a set of lognormally distributed spanning swap rates as the basic building

blocks leads to the Black formula for the pricing of swaptions.

Unfortunately the lognormal specification of forward LIBOR rates is incompatible with the simul-taneous lognormal specification of swap rates. Allowing each forward LIBOR rate to be lognormal

under its own measure, means that swap rates cannot be lognormal at the same time, even if each

is specified under its own measure (and vice versa). In practice, the forward swap rates derived

within the lognormal forward LIBOR framework do not deviate far from lognormality and suitable

approximations exist to allow consistent pricing of caps and swaption.

I look first at the lognormal forward LIBOR framework and then discuss how lognormal swap rates

can be incorporated within it.

1


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CHAPTER 1

Forward LIBOR Market Model

1. Model set up

Consider a trading horizon [0, T] and a set of times {Tk; k [0, n], k N} such that0 T0 < Tk < . . . < T n T. We form pairs of expiry-maturity times (Tk, Tk+1), k 0,for a set of spanning forward rates fk. Additionally, set k = Tk+1 Tk to be the year fractionrepresenting the tenor of forward rate fk.

Define the simple forward rate over period [Tk, Tk+1], fk, as:

(1.1) 1 + kfk(t) = P(t, Tk)

P(t, Tk+1), k= 0, . . . , n 1,

whereP(t, Tk) is the timet value of a risk-free zero coupon bond maturing at time Tk with nominal1.

The original derivation by BGM of the LMM was done within the HJM framework, making use of the

HJM result that no-arbitrage conditions impose a structure on the drift of the forward rate process.

Once the forward rate volatilities are fully specified, so are the forward rate drifts. This derivation

is under the risk-neutral measure, with a bank account as numeraire. Under this measure, none

of the forward rates are lognormal and one needs to change to a forward measure, corresponding

to the payout time of each forward rate, to obtain martingale dynamics and hence pricing of caps

via the Black formula. There are some products, such as Eurodollar futures, which do require therisk-neutral dynamics of forward rate (for example see Brigo and Mercurio [10] for an analysis). In

6 I derive the forward rate dynamics under this risk-neutral measure, the spot LIBOR measure,using a discretely compounded money market account as numeraire.

Elsewhere I use the forward measure approach since it is a natural framework for pricing of cap and

swaption based derivatives which constitute the foundation of the exotic interest rate derivatives

market.

For the original derivation of the LMM see [8] and [41]. I have examined this derivation in detail

in [59].

2. Martingale Dynamics

From equation (1.1) we have:

fk(t)P(t, Tk+1) = (P(t, Tk) P(t, Tk+1))/k.Since the right hand side is the price of a traded asset (difference between two zero coupon bonds,

each with nominal value 1k ), the left hand side must also be a traded asset and hence its value,expressed with respect to numeraire P(t, Tk+1), must be a martingale under the appropriate proba-bility measure. Let Qk be the measure associated with numeraire P(t, Tk+1), under which forward

2


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3. CHANGE OF MEASURE 3

rate fk(t) is a martingale, that is, it is driftless. Hence:

(2.1) dfk(t) = k(t)fk(t)dWk

k

(t), t < T k,

wherek(t) is the time t instantaneous volatility of forward rate fk,Wkk (t) is thekth element of the

n dimensional column vector Brownian motion associated with measure Qk. The vector Brownianmotion has correlation matrix , that is dWkdWk = dt, with element ij being the correlationbetween Brownian motions Wki and W

kj.

Expressed in terms of any other numeraire (and hence probability measure) the LIBOR rate ceases

to be a martingale and acquires a drift directly determined by the chosen numeraire. To determine

the form of this drift adjustment for the LIBOR rate, I first examine the generic numeraire change

methodology.

3. Change of Measure

Consider ann-vector of assetsXtwith dynamics under probability measureQS (probability measure

corresponding to numeraire S) defined by1:

(3.1) dXt = St(Xt)dt + t(Xt)CdZ

St ,

with2

St an (n 1) vector of instantaneous drifts,t an (n n) diagonal matrix of instantaneous volatilities,

ZS an n-dimensional standard Brownian motion under measure QS,C an (n n) matrix, such that CC = is the instantaneous correlation matrix. That is

CdZmay be interpreted as an n-dimensional Brownian motion with correlation matrix .

We wish to express the dynamics of Xt under probability measure QU. The diffusion coefficient

remains unchanged under a change of measure, hence the dynamics take the form:

(3.2) dXt= Ut (Xt)dt + t(Xt)CdZ

Ut ,

withUt as yet undetermined.

We use the Girsanov Theorem to define the new measure and its associated dynamics.

1Here, I follow the treatment in Brigo and Mercurio [10].2I acknowledge the somewhat inconsistent use of notation depicting time dependence, subscript and explicit

functional dependence on t (t (t)) are equivalent. This is done to ease the burden of notation.


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Girsanov Theorem. Let ZP be an n-dimensional standard Brownian motion on the prob-ability space (, F, P). Also let be any n-dimensional adapted column vector process. WithTfixed we define the process as:

(3.3) t = exp

12

t0

|s|2ds + t

0

sdZPs

.

Assume that3:

EP

exp

1

2

T0

|t|2dt

< ,

and define a new probability measure Q onFT such that:T =

dQ

dP.

Then Q is equivalent to P andZQ, a new Q-Brownian motion, is defined as:

(3.4) dZQt =dZPt

tdt.

The function t is known as the Radon-Nikodym derivative.

Returning to equation (3.1) we have:

dXt = St(Xt)dt + t(Xt)CdZ

St

= Ut (Xt)dt + t(Xt)CdZSt +

St(Xt)dt Ut (Xt)dt

= Ut (Xt)dt + t(Xt)C

dZSt (t(Xt)C)1

Ut (Xt) St(Xt)

dt

= Ut (Xt)dt + t(Xt)CdZUt

where:

dZUt =dZSt (t(Xt)C)1

Ut (Xt) St(Xt)

dt,

and by the Girsanov Theorem above, ZUt is ann-dimensional Brownian motion under the equivalentmeasureQU.

Set

(3.5) t = (t(Xt)C)1 Ut (Xt) St(Xt) ,

then the equivalent probability measureQU is defined by its Radon-Nikodym derivative with respecttoQS as:

(3.6) t = dQU

dQS

Ft

= exp

1

2

t0

|s|2ds + t

0

sdZSs

.

By (3.6) is an exponential martingale under measure QS and hence its dynamics are of the form:

(3.7) dt =

ttdZ

S

t .Geman et. al. (1995) [24] provide an invaluable tool for derivatives pricing by specifying the

Radon-Nikodym derivative in terms of the associated numeraires as:

(3.8) dQU

dQS =

UTS0U0ST

,

3This is the Novikov Condition (e.g. see [5]) which ensures is a martingale and EP [T] = 1.


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hence:

(3.9) T = dQU

dQS FT =

UTS0

U0ST

.

But, since is a QS martingale we have:

t = EQS

t [T] = EQS

t

UTS0U0ST

=

UtS0U0St

,(3.10)

dt = S0U0

d

UtSt

=

S0U0

U/St CdZ

St(3.11)

sinceU /Sis a martingale4under measure QS.

Comparing (3.7) and (3.11) and making use of (3.10) and the definition oft, we derive Ut , the

drift under the new probability measure, QU, as:

tt = S0U0U/St C

tUtSt

=U/St C

Ut (Xt) St(Xt)

(t(Xt)C)1 Ut

St=

U/St C

Ut (Xt) St(Xt)

= StUt

U/St CC

t(Xt)

Ut (Xt) =St(Xt) +

StUt

t(Xt)

U/St

.(3.12)

All that remains is to determine U/St , the instantaneous volatility of the assetU/S. We assume

the two numeraire assets have dynamics5:

dSt = (. . .)dt + StC dZ

St ,

dUt = (. . .)dt + Ut CdZ

St .

By the rules of stochastic differentiation we have:

d

UtSt

=

dUtSt

+ Utd

1

St

+ dUt d

1

St

,

and by Itos Formula:

d1

St = dSt

S2t+

1

S3tdSt dSt

= (. . .)dt St

S2tCdZSt ,

4An asset (U) in terms of its numeraire (S) is a martingale under the associated measure.5The dynamics are shown under measure QS. However any equivalent measure could be used, since we are only

interested in the diffusion coefficient, which remains unchanged under change of measure.


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hence:

dUt

St = (. . .)dt +Ut

StCdZSt

Ut

St

S2t

CdZSt

= (. . .)dt +

UtSt

UtSt

StSt

CdZSt ,

U/St =Ut

St Ut

St

StSt

.

Substituting into (3.12) gives the final form of the new drift coefficient:

Ut (Xt) = St(Xt) +

StUt

t(Xt)

UtSt

UtSt

StSt

=St(Xt) + t(Xt)

UtUt

St

St

.(3.13)

Allowing the process forXto be fully lognormal under measure QS, the drift and diffusion coefficienttake the form:

St(Xt) = diag(Xt)mSt,

t(Xt) = diag(Xt) diag(vXt ),

where:

mSt an (n 1) vector of instantaneous lognormal drift coefficients,diag(vXt ) an (n n) matrix with instantaneous lognormal volatility coefficients along the

diagonal and zeros elsewhere,

diag(Xt) an (n n) matrix with the elements ofXt along the diagonal and zeros elsewhere.

Consequently equation (3.13) may be expressed as:

Ut (Xt) = diag(Xt)mSt + diag(Xt) diag(v

Xt )

UtUt

St

St

Ut (Xt)dt = diag(Xt)mSt dt + diag(Xt)d ln Xt(d ln Ut d ln St)

= diag(Xt)

mSt dt + d ln Xt(d ln Ut/St)

,

hence Xt remains fully lognormal under QU and its drift takes the form Ut (Xt) = diag(Xt) m

Ut

with:

(3.14) mUt dt= mSt dt + d ln Xt(d ln Ut/St)

.

No assumptions have been made about the processes followed by the two numeraires, S and U. If

S and Ufollow lognormal processes, then starting with deterministic drift mSt, the resulting driftmUt will also be deterministic. For the case of more general processes describing the evolution ofSandU, a deterministic drift mSt will give rise to a stochastic driftm

Ut (sinced ln Xt(d ln Ut/St)

will

be a function ofUt andSt.)


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4. DRIFT SPECIFICATIONS 7

4. Drift Specifications

As specified in (2.1), fk(t) is a martingale under measure Qk with associated numeraireP(t, Tk+1),

hence:(4.1) dfk(t) = k(t)fk(t)dW

kk (t), t < T k.

To derive the dynamics under forward measureQi, i < k(associated with numeraire P(t, Ti+1)) weapply (3.14) with6:

Xt fk(t) = f(t, Tk, Tk+1),St P(t, Tk+1),Ut P(t, Ti+1),

hence:

mSt mkk(t) = 0; sincefk(t) is driftless under Qk,d ln Xt d ln fk(t) = (. . .)dt + k(t)dWkk (t),

d ln U/S d ln

P(t, Ti+1)

P(t, Tk+1)

,

By the definition of a zero-coupon bond price in terms of the discretely compounded forward rates,

we have:

ln

P(t, Ti+1)

P(t, Tk+1)

= ln

kj=i+1

(1 + jfj(t))

= kj=i+1

ln (1 + jfj(t)).

So (3.14) becomes:

mik(t)dt = mkk(t)dt + d ln fk(t)

d lnP(t, Ti+1)P(t, Tk+1)

=

kj=i+1

d ln fk(t)d ln (1 + jfj(t))

=k

j=i+1

j1 + jfj(t)

d ln fk(t)dfj(t)

=

kj=i+1

jfj(t)j(t)k(t)jk1 + jfj(t)

dt, t Ti+1.(4.2)

Similarly, the dynamics under forward measureQi

, i > k are derived as follows:

ln

P(t, Ti+1)

P(t, Tk+1)

= ln

ij=k+1

1

1 + jfj(t)

= ij=k+1

ln (1 + jfj(t)),

6Here we reduce the asset vector process Xt to a scalar valued process, being its k th element, the forward rate

fk(t) with reset time Tk.


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5. RELATIONSHIP BETWEEN BROWNIAN MOTIONS UNDER ADJACENT FORWARD PROBABILITY MEASURES8

and so:

mik(t)dt =

i

j=k+1 d ln fk(t)d ln (1 + jfj(t))=

ij=k+1


dt, t Tk.(4.3)

In summary, from (4.1), (4.2) and (4.3), the dynamics offk(t) under the forward measure Qi with

associated numeraireP(t, Ti+1), fort min(Ti+1, Tk) are:dfk(t) = m

ik(t)fk(t)dt + k(t)fk(t)dW

ik(t),

with

mik(t) =k

j=i+1jfj(t)j(t)k(t)jk

1 + jfj(t)

, i < k;

mik(t) = 0, i= k;

mik(t) = i

j=k+1


, i > k.

5. Relationship between Brownian Motions under adjacent forward probability

measures

By the Girsanov Theorem we have:

(5.1) CdZU(t) = C dZS(t) Ctdt,where ZS(t) and ZU(t) are Brownian motions under equivalent probability measures QS and QU

respectively and t is defined in (3.5) as:

t = (t(Xt)C)1 Ut (Xt) St(Xt) .

For lognormal assetXt we have:

t dt =

diag(Xt) diag(vXt )C

1diag(Xt)(m

Ut mSt) dt

=

diag(vXt )C1

d ln Xt(d ln Ut/St),

which follows from (3.14), and so:

CdZU(t) = C dZS(t)

C diag(vXt )C

1d ln Xt(d ln Ut/St)

=C dZS(t) diag(vXt )1 d ln Xt(d ln Ut/St).

Applying this to the LMM case, CdZU(t) dWk+1(t) and CdZS(t) dWk(t) with correlation7matrix. Here, the asset vectorXt, is reduced to a 1-dimensional asset, being its k

th elementfk(t),

7dZU(t) and dZS(t) are standard Brownian motion vectors, premultiplication by C imposes the required corre-

lation structure, while dWk+1(t) and dWk(t) are specified as correlated Brownian motions.


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6. SPOT LIBOR MEASURE DYNAMICS 9

and hence:

dWk+1k (t) =dWkk (t)

k(t)

1 d ln fk(t)d lnP(t, Tk+2)

P(t, Tk+1)=dWkk (t) + k(t)

1 d ln fk(t)d ln (1 + k+1fk+1(t))

=dWkk (t) + k+1k(t)

1

1 + k+1fk+1(t)d ln fk(t)dfk+1(t)

=dWkk (t) + k+1fk+1(t)

1 + k+1fk+1(t)k+1(t)kk+1dt.

The above specification allows some observations. The drift adjustment is a function of the forward

rate with natural payoff at the maturity of the numeraire associated with the new probability

measure, but also the correlation between forward rates with natural payoffs at the maturity times of

the old and new numeraires. For non-adjacent forward measures, the drift adjustment is a function of

all forward rates spanning the interval between them, hence the further apart the forward measures,

the greater the drift adjustment.

6. Spot LIBOR Measure Dynamics

The numeraire associated with the risk-neutral measure is the continuously rebalanced bank account

B(t) defined in terms of the instantaneous short rate as:

B(t) = exp

t0

r(s)ds

, dB(t) = r(t)B(t)dt.

This continuously rebalanced numeraire does not fit naturally into the set-up of the LMM, as defined

in

1, where we have a preassigned maturity and tenor structure. Instead, we introduce a discretely

rebalanced bank account, where rebalancing only takes place at the predefined maturity dates.Define(t) such that T(t)1 t < T(t), then the discretely rebalanced bank account is defined as:

(6.1) B(t) = P(t, T(t))(t)1j=0 P(Tj , Tj+1)

=P(t, T(t))

(t)1j=0

(1 + jfj(Tj)).

At each time Tj , starting at T0 = 0, the bank account accrues interest at the currently resettingforward rate fj(Tj), paying at Tj+1 (once reset, this forward rate becomes the tenor j spot rate).The accrued amount is then reinvested at the forward rate resetting at that time, fj+1(Tj+1) and

so on. The final ratef(t)1((t) 1) pays out at time T(t) > t, hence the final discounting termP(t, T(t)) gives the time t bank account value.

We determine the drift of fk under the spot LIBOR measure using (3.14) and the Qk

forwardmeasure lognormal martingale dynamics in (2.1). Hence:

(6.2) mk(t)dt= mkk(t)dt + d ln fk(t)(d lnB(t)/P(t, Tk)),

where mk(t) is the drift coefficient under the spot measureQ; mkk(t) = 0, since the forward rate isdriftless under forward measure Qk;B(t) and P(t, Tk) are the numeraires associated with the spotand forward measures respectively.


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7. SWAP RATE 10

In terms of forward rates, P(t, Tk) is defined as:

P(t, Tk) = P(t, T(t))

k1

j=(t)

1

1 + jfj(t) ,

hence using the definition ofB(t) in (6.1), we have:lnB(t)/P(t, Tk) = ln

(t)1j=0 (1 + jfj(Tj))k1j=(t)

11+jfj(t)

= ln

(t)1j=0

(1 + jfj(Tj))k1j=(t)

(1 + jfj(t))

=

(t)1

j=0

ln (1 + jfj(Tj)) +

k1

j=(t)

ln (1 + jfj(t)).

Hence

mk(t)dt = d ln fk(t)

d lnB(t)/P(t, Tk)=

k1j=(t)

d ln fk(t) (d ln (1 + jfj(t)))

=k1j=(t)

j1 + jfj(t)

d ln fk(t) dfj(t)

=k1j=(t)

j1 + jfj(t)

fj(t)j(t)k(t)jk dt,

and the forward rate dynamics under the spot LIBOR measure are:

(6.3) dfk(t) =fk(t)k1j=(t)


dt + k(t)fk(t)dWk(t),wheredWk(t) is a Brownian motion under the spot LIBOR measureQ.

7. Swap Rate

7.1. Definition. An interest rate swap is a contract to exchange a series of floating interest

payments in return for a series of fixed rate payments. Hence, consider a series of payment dates

betweenT+1and T , > . The fixed leg of the swap paysjKat each timeTj+1, j = , , 1where j = Tj+1 Tj. In return, the floating leg pays jfj(Tj) at time Tj+1 where fj(Tj) is thetenor j rate, set at time Tj for payment Tj+1. Hence given the set of forward rate reset datesTj , j = , , 1 and the series of payment dates Tj , j = + 1, , , the time t, t T value


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7. SWAP RATE 11

of the interest rate swap is8:1

j=P(t, Tj+1)j(fj(t) K) .The par/fair forward swap rate S,(t) is the value of the fixed rate K, such that the present valueof the contract is zero, hence:

(7.1) S,(t) =

1j= P(t, Tj+1)jfj(t)1

j= P(t, Tj+1)j=

P(t, T) P(t, T)1j= P(t, Tj+1)j

,

where the second equality is due to the definition of the forward rate fj(t) in (1.1).

7.2. Dynamics. As mentioned in the introduction, the market convention is to price swaptions

by means of the Black formula. This implies the assumption that swap rates are driven by lognormal

dynamics. In an analysis closely matching that of forward rates in

2, we rearrange (7.1) to give:

S,(t)

1j=

P(t, Tj+1)j =P(t, T) P(t, T).

Since the right hand side is the price of a traded asset (the difference between two different maturity

bonds), the left hand side is also the price of a traded asset and its value, expressed in terms of an

appropriate numeraire, is a martingale under the associated measure. Therefore we can write:

dS,(t) =,(t)S,(t)dW,(t),

where dW,(t) is a Brownian motion under probability measure Q, with associated numeraire

1j= P(t, Tj+1)j , being the annuity stream with payment dates Tj, j = + 1, , . Under this

annuity measure, Q, , the swap rate has a lognormal distribution, making it consistent with theBlack swaption pricing formula.

Despite this neat, concise set up, the incompatibility of the swap based LIBOR market model and

the forward LIBOR market model is problematic. Exotic interest rate derivatives tend to display

dependence on both forward and swap rates; hence we need a single coherent framework within

which both forward rates and swap rates can be expressed. Various approximations have been

developed to allow analytic swaption prices within the forward LIBOR model. See Rebonato [51]

and Brigo and Mercurio [10] on the relative merits of using either the forward or swap rate LIBOR

market model.

7.3. Swaption Volatilities in the Forward LIBOR Model. The swap rate is naturally

expressed in terms of forward rates, so in order to value swaptions in the forward LIBOR market

model we need to express the Black swaption volatility in terms of forward rate volatilities. As in thecase of forward rates and caplets, the Black swaption volatility is the square root of the integrated

swap rate variance over the life of the swaption, that is:

(7.2)

Black, (T)2

T =

T0

2,(t)dt=

T0

dS,(t)

S,(t)

dS,(t)

S,(t)

.

8See appendix for a derivation.


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7. SWAP RATE 13

shows the integrated volatility to be a path dependent integral10 which must be evaluated by Monte

Carlo simulation.

By rewriting formula (7.3) as:

2,(t) =

1j=

1i=

ij(t) j(t)i(t)ij(t),

with:

ij(t) = wj(t)wi(t)fj(t)fi(t)1

i=wi(t)fi(t)2 ,

we represent the swap rate volatility as a weighted sum of the covariance terms j(t)i(t)ij(t). Asa second approximation we set the values of these weights, ij(t), to their time t= 0 values. So ineffect, we set the forward rates fj, and the swap rate weights wj , to their initial values. Rebonato

[51] performs an extensive analysis as to the validity (and hence accuracy) of this approximation.

Therefore the swap rate instantaneous volatility is approximated as:

(7.4) 2,(t) =

1j=

1i= wj(0)wi(0)fj(0)fi(0)j(t)i(t)ij(t)

S,(0)2 ,

and from (7.2) the Black swaption volatility becomes:

(7.5)

Black, (T)2

T =

1j=

1i=wj(0)wi(0)fj(0)fi(0)

S,(0)2

T0

i(t)j(t)ij(t)dt.

This two-part approximation allows us to value swaptions analytically in the forward LIBOR market

model, making Monte Carlo superfluous.

10This is incompatible with the path independent Black implied volatility. This highlights the incompatibility

of simultaneously lognormal forward and swap rates.


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Appendix

Given the set of forward rate reset dates Tj, j = , , 1 and the series of payment datesTj , j = + 1, , , the payoff discounted to time t, t T, of a swap paying fixed and receivingfloating interest is:

1

j=D(t, Tj+1)j(fj(Tj) K) ,

whereD(t, Tj+1) = B (t)/B(Tj+1) is the stochastic discount factor. The time t value of this swapcontract can be evaluated by taking expectations:

Swap,(t, K) = Et

1j=

D(t, Tj+1)j(fj(Tj) K)

=

1j=

Et[D(t, Tj+1)j(fj(Tj) K)]

=

1j=

Ejt

d QdQj

Ft

D(t, Tj+1)j(fj(Tj) K)

=

1j=

P(t, Tj+1)j(fj(t) K) .

14


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CHAPTER 2

Covariance Structure

1. Structure of the Forward Rate Dynamics

Consider the forward rate dynamics of Chapter 12 expressed in matrix notation and with respectto some measure QU:

(1.1) df(t) = (f, t)f(t)dt + (t)f(t)dWU(t),

where:

f(t) (n 1) column vector of forward rates,(f, t) (n 1) column vector of drifts, which may be functions of the forward rates

themselves and time,

dWU(t) (n 1) column vector of correlated standard Brownian motions under a chosenmeasureQU. The Brownian motions have correlation matrix , that is,

dWU(dWU) = dt,

(t) (n n) diagonal matrix, where the ith element, i is equal to the instantaneouspercentage volatility of the ith forward rate.

Within this specification, each forward rate is modelled via its own Brownian motion and associated

instantaneous volatility function. The Brownian motions are specified under a probability measure

QU associated with numeraire U which uniquely determines the drift vector. If the correlationmatrix,, has full rank, the above equations provide a specification of dynamics of the term structurewith as many factors as forward rates.

A second specification proves more useful:

(1.2) df(t) = (f, t)f(t)dt + (t)f(t)dZ(t),

wheref(t) and(f, t) are unchanged from above, but now dZ(t) is an (m 1) vector of orthogonalBrownian motions1 and (t) is an (n m) real matrix where the ij , the (i, j)th element, is theloading on the ith forward rate of the j th orthogonal Brownian motion (source of uncertainty).

The number of orthogonal (independent) sources of uncertainty, m, may now differ from the numberof forward rates, n. Preferably m < < n. Each of the n forward rates is affected by each ofthe m Brownian shocks. For each forward rate i, i = 0, . . . , n 1 the level of responsiveness toBrownian shock j, j = 1, . . . , m is determined by factor ij . Hence the volatility of forward ratei,

i= 0, . . . , n 1 is decomposed among m Brownian motions.1Here I omit the specific reference to the pricing measure QU and it should be taken as given, unless explicitly

indicated otherwise.

15


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1. STRUCTURE OF THE FORWARD RATE DYNAMICS 16

For m = 1 we have a one-factor model where all forward rates are driven by a single source ofuncertainty and hence have perfect instantaneous correlation. Generally for m= n (specificallym < n) the resulting correlation matrix will differ from the full-rank correlation matrix above.

The first formulation in (1.1) describes the yield curve evolution in terms of forward rate specificBrownian shocks, while the second formulation makes use of shocks affecting the entire yield curve

and specifying the extent to which each forward rate is affected by each Brownian shock. The

relationship between the two formulations, that is the relationship between matrices and is:

(1.3) 2i (t) =mk=1

2ik,

whereiis theith diagonal element of matrix . Using this equality, the dynamics of the ith forward

rate in (1.2) may be expressed as:

dfi

fi

= idt +m

k=1 ik dZk= idt + i

mk=1

ikmj=1

2ij

dZk

= idt + i

mk=1

bik dZk, i= 0, . . . , n 1,(1.4)

where

bik = ikm

j=1 2ij

,

and elementbik is the proportion of total volatility of forward ratei attributable to Brownian shock

k. We have decomposed the responsiveness of the forward rates to the Brownian shocks into twodistinct components:

Instantaneous volatility componenti, which gives the total level of volatility of forwardrate i. This is obtained from the Black implied volatility used to price caplets on theforward rate with expiry Ti via the relationship

(1.5) 2Black(Ti)Ti =

Ti0

2i (u)du.

. Correlation component. The components ({bik}mk=1), contain information about the cor-

relation structure. In fact bb =, where b is the (n m) matrix of elements bik and isthe correlation matrix of the Brownian motions specified in (1.1).

The two sets of components may be specified in a more-or-less independent manner.

For the case m < n, where the number of factors is smaller than the number of forward rates, weneed to ensure that the total volatility of each forward rate is fully recovered (this ensures correct

pricing of caplets). Hence we require:

(1.6)

mk=1

b2ik = 1, i= 0, . . . , n 1,


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3. THE INTERPLAY BETWEEN VOLATILITIES AND CORRELATIONS 18

Say we require the terminal correlation between rates fi() andfj() at timeT, i < j. We needto evaluate:

Corr

(fi(T), fj(T))

= E

fi(T) E [fi(T)]

fj(T) E [fj(T)]

E

(fi(T) E [fi(T)])2

E

(fj(T) E [fj(T)])2 ,(3.1)

where the expectations are taken under forward measure Q, . Since the drifts of the forwardrates are functions of forward rates themselves, and hence stochastic, these expectations need to be

evaluated by means of a short stepped Monte Carlo simulation.

If we were able to obtain a closed form, exact expressions for the unconditional and conditional

distributions of the forward rates, we would not need to evolve the underlying process by means

of short stepped Monte Carlo to determine the terminal distribution. The distribution would be

known a priori and fully characterised by a small number of descriptive factors.

A generalised Brownian process, as used to model the forward rate process in (1.1) and (1.2), isboth Gaussian and conditionally Gaussian, however only in the case when the drift and volatility

vectors of the logarithm of the forward rates, () and (), respectively, are deterministic i.e. atmost functions of time.

Introducing approximations to the drift vector, such that it becomes deterministic (fully determined

by the volatilities and correlations), enables us to make use of the closed form solution for the

terminal distribution of the forward rates, thereby evaluating the terminal correlation.

Going back to Chapter 1,4 we allow a partial freezing (with respect to time) of the drift in theforward rate dynamics. Hence the dynamics offk(t) under the forward measure Q

with associated

numeraireP(t, T+1), for t min(T+1, Tk) are:dfk(t) = m

k(t)fk(t)dt + k(t)fk(t)dW

k (t),

with

mk(t) =k

j=+1

jfj(0)j(t)k(t)jk1 + jfj(0)

, < k;

mk(t) = 0, = k;

mk(t) =

j=k+1

jfj(0)j(t)k(t)jk1 + jfj(0)

, > k;

where the dependence on time in the forward rate component of the drift has been eliminated by

freezing it at its time t = 0 value. This allows us to write:

fk(T) = fk(0)exp T0

mk(t) k(t)22 dt + T0 k(t)dWk (t).So evaluating (3.1) as the correlation coefficient of bivariate lognormal variables produces:

(3.2) Corr(fi(T), fj(T)) =exp

T0

i(t)j(t)ijdt

1exp

T0

i(t)2dt

1

expT

0 j(t)2dt

1

.


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4. TERM STRUCTURE OF VOLATILITIES AND THE INSTANTANEOUS VOLATILITY 19

Alternatively, as in Rebonato [51], [54] consider the correlation coefficient of the logarithm of the

forward rates as2:

(3.3) Corr(ln fi(T), ln fj(T)) = T0 i(t)j(t)ij dtT0

i(t)2dtT

0 j(t)2dt

.

By the Cauchy-Schwarz inequality3 we know:

Corr(ln fi(T), ln fj(T)) ij , ij 0,Corr(ln fi(T), ln fj(T)) ij , ij


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4.1. Functional Form. The instantaneous volatility can be either piecewise constant or of

some parametric form.

Piecewise constant. The piecewise constant specification implies that the volatility of for-

ward ratefkhas a constant value between each pair of expiry-maturity dates (Ti, Ti+1), i=0, , k 1.

Parametric form. The instantaneous volatility is specified as some function of current timet, expiry time Tk and some set of parameters{i}.

Within these two basic specifications we can assign an underlying structure on the resulting volatil-

ities to reduce the degrees of freedom and impose a desirable future evolution on the volatility term

structure.

Let m(t) be such that m(t) = i for Ti < t Ti+1, m(0) = 0 and define m(t) R for t [0, T];k R for k = 0, . . . , n 1; km(t) R fork = 0, . . . , n 1, t [0, T]. Also define functionsh: R R, g : R R, f : R R.

4.1.1. inst(t, Tk) = m(t), inst(t, Tk) = h(t). The instantaneous volatility is a function of

current time only. This specification implies that all forward rates exhibit the same responsivenessto random shocks. Although popular at times, mainly for numerical reasons, this functional form

does not have a very sound financial justification nor does it allow the exact recovery of all possible

market observed term structures.

4.1.2. inst(t, Tk) = k, inst(t, Tk) = g(Tk). The instantaneous volatility is purely forwardrate specific. Although this specification allows us to fit any exogenously specified volatility term

structure6, it is not financially appealing since it implies each forward rate has the same (constant)

volatility over its life, regardless of its term to maturity. Forward rates with the same remaining term

to maturity will display different volatilities, and so the term structure of volatilities will change

shape over time.

4.1.3. inst(t, Tk) =km(t), inst(t, Tk) = g (Tk)h(t). Responsiveness to the random shocksis split into a time dependent part and a forward rate dependent component. To ensure correct

pricing of a set of market caplets we require:

2Black(Tk)Tk =g(Tk)2

Tk0

h(u)2du.

Hence for any chosen function h(t), we can always find a g(Tk) such that the caplets are correctlypriced by setting7:

g(Tk)2 =

2Black(Tk)TkTk0

h(u)2du.

The functional dependence on the specific forward rate introduces the same characteristic observed

in specification 4.1.2, the shape of the volatility term structure changes through time.

6In the simplest case, by setting inst(t, Tk) =k = g(Tk) =Black(Tk).

7Equivalently in the piecewise continuous case, the caplet pricing requirement is:

2Black(Tk)Tk =2k

Tk0

2m(u)du= 2k

n1i=0

2i i,

and constant k can be found as:

2k =2Black(Tk)Tkn1

i=0 2i i

.


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4.1.4. inst(t, Tk) = km(t), inst(t, Tk) = f(Tk t). The instantaneous volatility is purelya function of the residual term to maturity. This functional form is financially appealing since the

volatility term structure will maintain its shape through time; this is easily demonstrated by the

equality T0

2inst(u, T)du=

T+

2inst(u, T+ )du.

However, it is no longer possible to fit any exogenous set of caplet implied volatilities. The time

homogeneous structure of the instantaneous volatility function requires that 2Black(Tk)Tk, as ob-tained from todays term structure of volatilities, must be a strictly increasing function ofTk. Soif 2Black(Tk)Tk is not monotonically increasing in Tk, we will be not be able to exactly recoverobserved caplet prices with a purely time homogenous instantaneous volatility function. Therefore,

only if we observe 2Black(Tk)Tk to be a strictly increasing function are we always able to find aninstantaneous volatility function of the form f(Tk t) such that all caplets are correctly priced.The market observed 2Black(Tk)Tk, is usually not strictly increasing and hence a time homogenous

solution cannot be found. Since empirical evidence suggests time homogeneity to be a desirablecharacteristic, the subsequent functional forms use a time homogenous specifications, with a multi-

plicative augmentation that is either time or forward rate dependent.

4.1.5. inst(t, Tk) = kkm(t), inst(t, Tk) = g(Tk)f(Tk t). This functional specificationcomprises a forward rate specific component g(Tk) (similarly k) and a component depending onthe residual term to maturity Tk t. As in 4.1.3 above, we can impose correct pricing of todaysmarket observed caplets, for any function f(Tk t) (similarly km(t)), by setting:

g(Tk)2 =

2Black(Tk)TkTk0

f(Tk u)2du.

Ifg (Tk) = , constant for all Tk, k= 0, . . . , n

1 (similarly k = i for all k , i), we have a time

homogeneous term structure of volatilities. By first determining f() (similarly{}) such that wehave as close a fit as possible to caplet prices and then determining g() (similarly {}) to fine tunethe fit, we allow the instantaneous volatility to be as time homogenous as possible (see [51]). Ideally

g() (similarly{}) should be as close to constant as possible, across forward rate maturities.4.1.6. inst(t, Tk) =m(t)km(t), inst(t, Tk) =h(t)f(Tk t). The time homogenous instan-

taneous volatility component is augmented by a purely time dependent one. The relative respon-

siveness of equal maturity forward rates remains the same over time, but is augmented by the time

dependent functionh(t) (similarlym(t)). While this specification is financially appealing, it is moredemanding from a numerical/computational perspective. As before, the correct pricing of caplets

is ensured by8:

2Black(T)T = T

0

g(u)2h(T

u)2du.

8In the piecewise constant case, by:

2Black(T)T =

n1i=0

2i 2kii.


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We are unable to split the integral as in 4.1.3 and 4.1.5 making it more difficult to jointly specify

the two functions. Introducing simplifications, such as allowing h() to be piecewise constant9 couldproduce unexpected results when calculating the covariance terms.

4.1.7. inst(t, Tk) =m(t)kkm(t), inst(t, Tk) =h(t)g(Tk)f(Tk t). One could implementthis hybrid function which contains features of all the approaches described above. Rebonato [51]

recommends a three stage procedure where a best fit optimisation is first implemented for h(Tk t)and any residual mispricing of todays caplets is eliminated by the time dependent and forward rate

dependent components, g() and f() respectively (equivalently for km(t),{} and{}.).The choice of volatility function and its specific form it should be financially intuitive, with associated

parameters having a transparent econometric interpretation. Therefore we discuss some empirical

observations of the qualitative shape of the volatility term structure.

4.2. Qualitative Shape of the Volatility Term Structure. In general, the volatility term

structure displays a humped shape, with relatively low volatility on the short end, a peak around the

18 month maturity and then decreasing volatility towards the long end of the maturity spectrum,with a possible flattening out of volatilities. The general shape of the volatility term structure

remains the same through time, that is volatilities of caps with different terms to maturity maintain

the same relative magnitudes through time. There may, however b e short periods when the term

structure displays anomalous behaviour, taking on a generally monotonically decreasing shape.

Rebonato [51] posits a financial explanation for these observations. In major financial markets (US,

UK and Euro area) monetary authorities set the level of the short term rate. Under normal market

conditions, their actions are fairly transparent to market players and unexpected actions are not

common, hence the instantaneous volatilities of rates with short term to maturity tend to be fairly

low. There may, however, be periods of market turmoil when uncertainty is high regarding the

actions of monetary authorities. Rebonato refers to this as excited conditions, characterised by

high uncertainty regarding the future level of the short rates; and hence high volatility of the short

maturity forward rates.By contrast, the long end of the maturity spectrum is affected mainly by expectations regarding

inflation and the level of real rates. Day to day economic news has a limited impact, hence volatilities

tend to be fairly low and stable.

It is the intermediate maturities that tend to be most influenced by the arrival of new economic

information regarding the state of the economy and the actions of the monetary authorities; leading

to higher volatilities in the area of 6 to 18 months maturity.

A note on time dependence of the instantaneous volatility and the terminal decorre-

lation. The above description indicates the strong time dependence of the instantaneous volatility

function. As discussed in3 the covariance (correlation) elements defined in (3.3) are central tothe evolution of forward rates as well as prices of LIBOR products. The observed time dependence

plays a central role in achieving terminal decorrelation among forward rates. The greater the timedependence of the instantaneous volatility function, the lower the terminal correlation, even in the

presence of perfect instantaneous correlation. This time dependence becomes increasingly important

in determining the level of terminal correlation when instantaneous correlation is high, as is usually

the case between same currency forward rates.

9This amounts to using a hybrid instantaneous volatility specification with both a continuous and piecewise

constant component.


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4.3. Specific Functional Form. Here we focus on the parametric versions of the functions

assessed in4.1. Two general functional forms stand out as being financially and computationallyfeasible, allowing the term structure to maintain its general shape through time, yet having enough

flexibility to calibrate to exogenous caplet prices:

inst = h(t)g(Tk)f(Tk t),inst = g(Tk)f(Tk t).

We need to assign a specific parametric form to functions f(), g() and h().4.3.1. f(Tk t). Since we could like the term structure of volatilities to be as time-homogenous

as possible, the time-homogenous component dominates in determining the instantaneous volatili-

ties. We would like this function to allow both the empirically observed humped and monotonically

decreasing shapes.

Pricing of caplets (via the variance) and the evaluation of covariance terms requires integration of

the square of this function; we would like a parametric form that allows a simple analytic solution

for the integral of its square.Additionally the specific parametric form should also be financially intuitive with parameters that

can be related to market observables. Rebonato [51], [54] argues that the functional form:

(4.2) f(Tk t) = (a + b(Tk t))exp(c(Tk t)) + dsatisfies these criteria10. It is able to produce both a monotonically decreasing and a humped

instantaneous volatility curve, providing a fair degree of flexibility as to the exact shape of the

humped curve. The parameters{a,b,c,d} have the following financial interpretation: As =Tk t , f(Tk t) d, hence d is associated with the volatility at very long

maturities. Therefore we require d >0. Similarly, = Tk t 0, f(Tk t) a+d and hence a+d should be approximately

equal to the shortest maturity implied volatilities11. Again a + d >0.

The location (on the term to maturity axis) of the hump is determined by evaluatingf() = 0. The hump is therefore at = bcacb , which is a local maximum for b > 0. Forb < 0 no maximum occurs. Additionally, empirical evidence shows that the hump in thevolatility curve is found around 1-year, hence bca

cb 1.

In the light of the above observation, consider the curve in the normal state. At theshort maturity end, we require a positive slope i.e. f(0)> 0. This is satisfied by the jointconstraintsa < b/cand b >0, where the second constraint is the same one that guaranteesthe presence of a local maximum.

4.3.2. h(t). The choice of this function is more subjective, since empirical observations of cal-endar time dependence are difficult to isolate. Rebonato [51] recommends a function of the type:

(4.3) h(t) = N

i=1

isin tiM + i+1 exp(N+1t),where N is the number of free parameters (recommended to be as small as 2 or 3) and M is thematurity of the longest caplet. This function is a linear combination of sine waves, (where the

10This same functional form is supported and analysed by Brigo and Mercurio [10].11For 0 the time interval over which one integrates to get from instantaneous to implied (average) volatility

also tends to zero and so the two quantities will converge. Hence the instantaneous and implied volatilities are equal.


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5. INSTANTANEOUS CORRELATION FUNCTION 24

phases and amplitudes are optimised) multiplied by a decay factor (with optimised decay constant).

The intuition behind this functional form is to allow a sufficient number of frequencies to pick

up the localised time dependence, but not too many so that excessive market noise obscures the

specification.4.3.3. g(Tk). The forward rate dependent function is the final smoothing function ensuring

perfect pricing of todays market caplets (perfect fitting to todays volatility term structure). If we

set:

(4.4) g(Tk) = 1 + k, 0 k n 1,where n is the total number of forward rate maturities, we should not find any is significantlydifferent from zero.

5. Instantaneous Correlation Function

As with the instantaneous volatility function, we may allow a parametric or non-parametric specifi-

cation of the instantaneous correlation. The correlation is a long-run characteristic of forward ratesand is not expected to change significantly over time. This leads to a time independent specifica-

tion of correlation between each pair of forward rates of maturity Ti and Tj respectively, makingthe correlation matrix stationary through time. The non-parametric form involves determining the12 n(n 1) unique elements of the (n n) correlation matrix from market prices. Alternatively,

having decided on a parametric form, the free parameters are determined to best fit market prices.

A general functional form for the instantaneous correlation function between forward rates with

maturity Ti and Tj is:

(5.1) ij =(t, Ti, Tj).

Since the instantaneous correlation enters the evaluation of the covariance elements specified in (3.3)

we require (together with a square-integrable volatility function) that ij(t) should be integrable

over time interval [Tk, Tk+1]. Additionally to be a valid correlation function we require 1 ij 1.Rebonato [51] proposes the time-homogeneous correlation functions:

ij = (Ti t, Tj t) orij = (Ti Tj).

Determining the correlations (regardless of having chosen a non-parametric or parametric specifica-

tion) from market prices is a particularly difficult task. Rebonato [51] discusses three main reasons

for this difficulty. While caplet prices depend directly on the instantaneous volatility (via the root

mean square) of individual forward rates, there are no vanilla instruments depending purely on the

correlation between forward rates. In fact the only vanilla instruments depending on the instanta-

neous correlation are European swaptions, where the price depends on the covariance over the life

of the swaption: Texp0

i(u)j(u)ij(u)du.

Since the instantaneous correlation always appears coupled to the (only partially known) instan-

taneous volatility functions, it is particularly difficult to estimate the correlation functions based

on swaption prices. Secondly, it is not the instantaneous correlation directly that determines the

swaption prices but rather the terminal correlation (as shown in (3.3)). This is determined by both

the time dependence of the instantaneous volatility and the instantaneous correlation. Different


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CHAPTER 3

Deviations From Lognormality

1. Caplet Prices and Lognormality

In the model set-up of Chapter 1 forward rates are assumed to follow a lognormal process. We

determine a terminal probability measure, for each forward rate, such that it becomes an exponential

martingale with deterministic volatility. The caplet prices obtained from these forward rates are

consistent with the lognormal distribution (under the terminal measure) so using the Black formula

to calculate implied volatilities corresponding to these prices, results in, for each maturity, a flatcurve as a function of caplet strike.

However, it is only under the terminal measure that full lognormality1is a prerequisite for obtaining

Black consistent prices. Under other measures, the forward rate process obtains a non-deterministic

drift term and is no longer lognormal. Under this new measure, the value of the discounting term

in the caplet pricing formula (no longer the zero coupon bond maturing at time of caplet payout)

takes on different values in different states of the world (hence for different values of caplet payoff).

The additional drift term in the forward rate dynamics compensates for the covariation between

caplet payout and discounting factor.

2. Deviation from Lognormality

In the mid 1990s, the caplet market started to exhibit implied volatilities that were monotonicallydecreasing as a function of strike, i.e. a skew. The reason for this observation was posited to be a

deviation from lognormality of the forward rate process. In the late 1990s the skew started to assume

an upward sloping profile for higher strike values taking on a hockey stick shape. The reason for

this additional feature is seen to be an additional deviation from lognormality superimposed on that

producing the monotonically decreasing smile.

3. The Smile in Equity vs. Interest Rate Market

Rebonato [51] proposes that the interest rate skew has arisen for different reasons to those explaining

the appearance and persistence of a skew in the equity and foreign exchange markets. Hence, it

would not be correct to apply the modelling methodologies applied in the equity markets, where

both the skew and smile effects are explained using one mechanism. This view is supported by

Gatarek [23].In the equity markets, a negative correlation is observed between changes in share prices and changes

in volatilities. That is, as share prices fall, the volatility increases and vice versa. This negative

1By full lognormality I refer to both drift and volatility terms taking on the lognormal form. Under alternate

measures the diffusion coefficient remains unchanged (hence lognormal), however the drift becomes a function of the

forward rates and hence we are no longer able to find a closed form solution for the distribution of the forward rate

at caplet expiry. This is the prerequisite for Black prices under the terminal measure.

26


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4. ALTERNATE INTEREST RATE DYNAMICS 28

Rewriting (4.2) as:

(4.3) dfk(t) = fk(t) +1

fk(0) ,k(t)dWkk (t),

the diffusion coefficient may be interpreted as a weighted sum of lognormal and normal components,

with the lognormal process recovered for= 1 (or = 0) and the normal process for = 0.

From (4.1) we may write a closed form solution for the value offk(t), t < t Tk as2:

fk(t) = (fk(t) + )exp

1

2

tt

,k(u)2du +

tt

,k(u)dWkk (u)

,

hence:

Ekt [ln(fk(t

) + )] = ln(fk(t) + ) 12

tt

,k(u)2du;

and we express the distribution offk(t

), conditional on fk(t), as a shifted lognormal distributionwith density function:

(4.4) Pk(fk(t)|fk(t)) = 1

(fk(t) + )(t, t)

2exp

12

lnfk(t

)+fk(t)+

+ 1

2(t, t

)2

(t, t)

2,

wherex > and (t, t)2 =tt

,k(u)2du.

The density function in (4.4) indicates one of the great advantages of the displaced diffusion ap-

proach. Analytical tractability is maintained since we have

P(t, Tk+1) Ek

[fk(Tk) K]+ |Ft

=P(t, Tk+1) Ek

[(fk(Tk) + ) (K+ )]+ |Ft

,

and so we may evaluate a caplet on forward rate fk(), with strikeK, using the Black caplet formulawith input spot fk() +, strike K+ and volatility parameter (t, t). Repeating this processfor a series of strikes{K}i and using the Black caplet formula a second time to retrieve the Blackvolatility implied by these prices, will produce a series of caplet implied volatilities with a dependence

on K i.e. imp imp(, K).A note of caution is required when dealing with displaced diffusion dynamics. Since fk(t) + is anexponential martingale (see (4.1), we have fk(t) + > 0 and hence fk(t) >, and the forwardrate can take on negative values.

4.1.2. Determining. Although we are able to use the Black formula to price caplets within thedisplaced diffusion framework, we need to determine the correct input volatility, (0, t

). Specifi-

cally we observe the lognormal implied volatility, imp from the market and need to determine the

corresponding for the choice of, such that the caplet prices match. Marris [38] applies numer-ical methods to solve this for any level of moneyness by inversion of the Black pricing formula. To

2Alternatively, in terms of the parameterisation in (4.2) the value offk(t), 0< t Tk is expressed as:

fk(t) =

fk(0)

exp

1

2

t0

2,k (u)2du +

t0

,k (u)dWkk(u) (1 )

, = 0.


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obtain an analytic approximation, he matches caplet prices that are close-to-the-money, specifically

with strike K=fk(0)exp

12

2t

(which gives = 1

2). A first order approximation yields:

(0, Tk) = fk(0)

fk(0) + imp(Tk).

An analysis by Rebonato [54] follows a similar methodology, matching at-the-money caplets (K=fk(0)), and using a Taylor Series expansion of the cumulative normal distribution to obtain thesame first order approximation and a further refinement of the form:

imp

f(0)f(0)+

1 1

242impT

1 1

24

f(0)f(0)+

imp2

T.

This extra degree of accuracy proves necessary for longer maturities (see the analysis in Rebonato

[54]).

4.2. CEV Dynamics. CEV dynamics refers to a general family of diffusions of the form:

(4.5) dfk(t) =,k(t)fk(t)dW

kk (t), 0 1,

where the lognormal and normal diffusions may be seen as limiting cases for = 1 and = 0respectively. This formulation may be classed as a local volatility model since the percentage

volatility is a function of the forward rate itself, specifically,k(t)f1k (t) and hence for 0, (4.7) becomes:

(4.8) dfk(k(0, t)) = fk(k(0, t)))dW

kk (k(0, t)).

Andersen and Andreasen make the following observations:

For 0< < 12

SDE (4.8) does not have a unique solution unless a boundary condition is

specified at fk = 0 (equivalently fk = 0) i.e. fk = 0 is set to be an absorbing boundary, For 1

2, (4.8) has a unique solution,

For 0<


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diffusion dynamics:

dfk(t) = (fk(t) + (1 )fk(0)) ,k(t)dWkk (t)

and the CEV dynamics of (4.5):

dfk(t) =,k(t)fk(t)dW

kk (t), 0 1.

By letting = , the displaced diffusion process may be used as a numerical/computational proxyfor the CEV dynamics6.

Surprisingly, the evidence as to why this agreement in prices is found is somewhat sketchy. Marris

provides graphical evidence of the coincidence of caplet implied volatilities over a range of strikes,

but does not provide a concrete analytical proof. He comments that the somewhat unnatural

parameterisation of the displaced diffusion in (4.2) was reverse engineered to yield the coincidence

with the CEV dynamics7.

Muck [40] notes the lack of proof of the admissibility of this approximation, especially for the

case of derivatives depending on joint distributions of forward rates, rather than just the terminal

distribution of a single rate as is the case for the caplets (and to some extent Barrier Options)

considered by Marris. He provides an analysis of swaptions and barrier swaptions, finding good

agreement between prices obtained in the two frameworks.

As a starting point to obtaining some clarity as to why the two processes yield coincident caplet

prices for a specific parameterisation, we note that for = and ,k() = ,k()f1k (0), theinstantaneous volatilities, quadratic variations and slopes of the quadratic variations

8

match forfk() = fk(0). Hence, at-the-money, we match both the level and slope of the implied volatilitycurve.

Consider the instantaneous volatility in (4.2) for = :

(fk(t) + (1 )fk(0)) ,k |fk(t)=fk(0)= (fk(t) + (1 )fk(0)) ,kf1k (0)|fk(t)=fk(0)=fk(0),k

=fk(t),k |fk(t)=fk(0) which is the instantaneous volatility in (4.5).

6In terms of the displaced diffusion parameterised by , this implies = 1 fk(0).7Unfortunately he does not provide any details of these calculations.8This was also note by Muck [40].


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5. ALTERNATE WAYS TO INTRODUCE SMILES 32

The slope, with respect to fk(t) of the quadratic variation (alternatively consider the slope of theinstantaneous volatility function) is expressed as:

The displaced diffusion case: (fk(t) + (1 )fk(0))2

2

,kfk(t)

fk(t)=fk(0)

= 2 (fk(t) + (1 )fk(0)) 2,k |fk(t)=fk(0)= 2 f(2

1)k (0)

2,k .

The CEV case:

fk(t),k

2fk(t)

fk(t)=fk(0)

= 2 f(21)k (0)

2,k .

Earlier in this chapter, I noted that a strong dependence between the level of forward rates andthe (lognormal) Black implied volatilities may be observed in the market. If we switch from a

lognormal process to CEV or displaced diffusion dynamics, hence expressing the implied volatility

in the appropriate co-ordinates, much of this dependence is eliminated. See Rebonato [54] for the

CEV case and [52], [53] for the displaced diffusion case.

5. Alternate ways to introduce smiles

5.1. Jumps. Another way of introducing skew and smile characteristics to the volatility curve

is via jumps in the dynamics of the underlying. An application of this approach to the LMM was

developed by Glasserman and Kou [25] and Glasserman and Merener [26]. This model has not

gained much acceptance for several reasons. It presents some technical complications, requiring

strongly time dependent parameters of the jump process (specifically the jump frequency and jumpdistribution) which gives rise to non-time-homogenous volatility term structures. Additionally the

jump component needs to be improbably high to calibrate to observed smiles.

5.2. Uncertain Volatility and Mixture of Models. Another method introduced in the past

few years involves prices obtained as a weighted average of prices from different simple models.

Supporters suggest this to be an easy way to add stochastic volatility to a pricing approach and

incorporate a market observed smile. Gatarek [23] and Johnson and Lee [33], among others, propose

using a conventional model for each of a predetermined set of input volatilities i, i = 1, , N.The final instrument price then becomes a weighted average of these prices, where the weights are

chosen so as to calibrate to some market instruments. In fact this general approach is based on an

early result by Hull and White [28], see Chapter 4,3 for a discussion of this result. AdditionallyJohnson and Lee [33] propose to use this to calibrate to volatility smile dynamics contained in exotic

options such as barriers and cliquets.

Piterbarg [44] opposes this modelling approach, claiming it to be under-specified and self-inconsistent.

He does not dispute its applicability and validity for European options, where only the average

volatility (term volatility) over the term to expiry has a bearing on the option price; the actual path

taken is immaterial. The problem arises when pricing exotics (which depend on the volatility path).

There may be multiple dynamics capable of producing the same mixture, yet giving rise to different

derivative prices.


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5. ALTERNATE WAYS TO INTRODUCE SMILES 33

An alternate mixture of models approach taken by Brigo and Mercurio [ 9], [10] is more thorough

and avoids the problems highlighted by Piterbarg. Brigo and Mercurio propose a LMM where each

forward rate follows a diffusion process uniquely determined by a mixture of lognormal density

functions. Specifically they determine a local volatility function (, fj()) such that the SDE9:

dfj(t) = (t, fj(t)) fj(t)dWj(t)

has a unique, strong solution with marginal density given by a mixture of lognormals as:

pt(y) = d

dyQj{fj(t) y} =

Ni=1

ipit(y), t Tj ,

withN

i=1 i = 1 andpit(y) being lognormal densities associated with a deterministic volatility i(t)

of the ith lognormal dynamics:

dfj,i(t) =i(t)fj,i(t)dWj(t), i= 1, , N.

9As before dWj is a Brownian motion under the Qj forward measure.


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CHAPTER 4

Stochastic Volatility Approach

1. Introduction

The approaches discussed in the previous two chapters go a long way to accounting for the variability

observed in time series of implied volatilities. Additionally they incorporate a skew generating

mechanism within the pricing framework. However, as noted by Andersen and Andreasen [ 1], they

are unlikely to account for all the pertinent dynamics. Rebonato [50], [54] discusses empirical data

of implied volatility dynamics, finding support for the inclusion of stochastic volatility to improveLMM realism.

Once a stochastic volatility model has been specified, the pricing of derivative instruments (especially

in the LMM framework) is almost exclusively by Monte Carlo. This is due to the additional source

of uncertainty introduced by the stochasticity of the volatility, as well as the complexity of the payoff

function that characterises the types of derivatives for which such a model is required. In general

no closed-form solutions exist, even for the simple vanilla instruments used for calibration (caps,

swaptions). Calibration by means of Monte Carlo is prohibitively time consuming, so the main work

in developing a stochastic volatility model is finding computationally efficient approximations for

pricing of vanilla instruments to facilitate calibration.

2. Modelling techniques

Lewis [37] is an extensive text detailing an array of volatility modelling approaches with reference

to equity and FX. Many of these approaches have subsequently been applied, by the authors listed

below, to the LMM case. Among others, Lewis deals with transform based solutions (Fourier and

Laplace transforms), asymptotic expansions - specifically volatility of volatility expansions and the

mixture approach, already mentioned in the previous chapter.

In another recent text by Fouque et. al. [21], volatility is assumed to follow a fast mean-reverting

process and call prices are approximated by expansion of this fast mean reversion parameter. The

literature on asymptotics expansions for stochastic volatility is extensive including many papers

by Fouque et. al. such as [19], [22], [20], Sircar and Papanicolaou [58], Lee [36] (who compares

slow-variation, small-variation and Fouque et. al.s fast-variation asymptotics) and Rasmussen and

Wilmott [48].

Three main published approaches to the inclusion of stochastic volatility in the LMM are: Joshi

and Rebonato [35], [34], [50]; Andersen and Brotherton-Ratcliffe [3] and Andersen and Andreasen

[1]; and Piterbarg [45], [46], [47]. Piterbarg builds on the work by Andersen, Brotherton-Ratcliffe

and Andreasen, I discuss their approach first.

However, before looking at stochastic volatility extensions of the LMM, I make a short detour

to examine a result by Hull and White [28] which has become prevalent in stochastic volatility

applications.

34


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3. UNCORRELATED ASSET AND VARIANCE PROCESSES 35

3. Uncorrelated Asset and Variance Processes

Here we consider pricing problem as in Hull and White [28]. Consider a derivative security

contingent on an underlying asset S. Let V = 2

be the instantaneous variance of this asset.Additionally, S andV are driven by the dynamics:

dS= rSdt + (t)SdW,

dV =(, t)V dt + (, t)V dZ,

whereris the risk free rate, the drift and diffusion coefficients of the variance process are independentofSand the Brownian motions1 are uncorrelated, i.e. dWdZ= 0.

Within this framework, the time t derivative price must be the present value of the expectation ofthe time T derivative value2. Hence:

(3.1) (St, 2t , t) = exp(r(T t))

(ST,

2T, T)p(ST|St, 2t )dST,

where (ST, 2

T, T) = max[0, ST K] is the payout of a European call option and p(ST|St, 2

t )is the conditional distribution of ST given time t asset price and instantaneous variance. Thishighlights that the distribution ofSTis dependent on the processes driving the asset price, S, andthe variance, 2. Hull and White use the following relationship between three dependent randomvariables:

p(x|y) =

g(x|z)h(z|y)dz

to simplify (3.1). Defining V, the average or term variance as:

V = 1

T t Tt

2udu,

the conditional density function of the asset price may be simplified as:

p(ST|2t ) = g(ST|V)h(V|2t )dV .The conditional distributions ofSon either side remain dependent on the starting asset value St,but this notation is suppressed for simplicity. Hence, (3.1) becomes:

(St, 2t , t) = exp(r(T t))

(ST,

2T, T)g(ST|V)h(V|2t )dSTdV

=

exp(r(T t))

(ST,

2T, T)g(ST|V)dST

h(V|2t )dV

=

(V)h(V|2t )dV ,(3.2)

where is the Black Scholes call price with input variance V. Hence, the derivative price is theBlack Scholes price integrated over the distribution of the term variance. This last step is onlyvalid in the case that the asset price and variance processes are instantaneously uncorrelated and of

course, only applicable to European style derivatives where the valuation depends on term variance

only; that is the path taken by the variance is immaterial.

1These are risk-neutral dynamics, hence W and Z are Brownian motions in the risk-neutral world.2Where time Tis usually taken to be expiry time and hence the derivative value is the terminal payout.


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4. ANDERSEN, BROTHERTON-RATCLIFFE APPROACH 36

To fully determine the price of the European derivative (St, 2t , t) we need an analytic form for thedistribution of the term variance, i.e. an analytic form for its conditional pdfh(V|2t ). This is notpossible for a general process for V. Determining the first few moments of this distribution, Hull

and White use a Taylor series expansion of(V) around V, the mean value of the term variance,to approximate (St, 2t , t) as follows:

(V) =( V) + (V)V

V

(V V) +12

2 (V)V2

V

(V V)2 +16

3 (V)V3

V

(V V)3 + ,

(St, 2t , t) =(V)h(V|2t )dV

=( V) + (V)V

V

(V V) h(V|2t )dV +

1

2

2 (V)V2

V

(V V)2 h(V|2t )dV

+ 16 3 ( V)V3 V(V V)3 h(V|2t )dV + =( V) +1

2

2 (V)V2

V

(V V)2 h(V|2t )dV

+1

6

3 (V)V3

V

(V V)3 h(V|2t )dV +

=( V) +12

2 (V)V2

V

Var(V) +1

6

3 (V)V3

V

Skew(V) + .(3.3)

The last two steps follow since the derivative terms are constants and the integrals represent the

second and third central moments of the distribution ofV.

4. Andersen, Brotherton-Ratcliffe Approach

4.1. Forward rate and volatility dynamics. Andersen and Andreasen [2] extended the

classical (lognormal) LMM to allow volatility functions with a freely specifiable level dependence,

with arbitrage free dynamics of the forward rates expressed as3:

(4.1) dfk(t) =(fk(t))k(t)dW

k(t), k= 0, , n 1,where k is an m-dimensional

4 volatility function and Wk is an n-dimensional Brownian motionunder the Qk forward measure. Alternatively, under the spot pricing measure Q, we have:

dfk(t) = (fk(t))

k

(t) [k(t)dt + dW(t)] , k= 0,

, n

1,(4.2)

k(t) =k

j=n(t)

j(fj(t))j(t)

1 + jfj(t) , Tn(t)1< t Tn(t),(4.3)

3See Chapter 3 4.2 for the detailed analysis.4In previous sections fk is driven by a single Brownian motion, hence k was a scalar function. This extension

to m Brownian sources represents a simple change of co-ordinates.


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where the correlation between the kth and jth forward rates is implied in the volatility loadingsconstitutingj and k. Andersen and Brotherton-Ratcliffe [3] extend this set-up by specifying ascalar mean-reverting variance process of the form:

(4.4) dV(t) = ( V(t))dt + (V(t))dZ(t),whereZ(t) is a Brownian motion under Q;, and are positive constants and is a well behavedfunction such that : R+ R+. This positivity is required to preclude negative variance, henceset5 (0) = 0. V(t) becomes a multiplicative scaling factor on the diffusion coefficient of the forwardrate process, that is:

(4.5) dfk(t) =(fk(t))

V(t)k(t)

V(t)k(t)dt + dW(t)

, k= 0, , n 1,

wherek is unchanged from (4.3) and dZ(t) is uncorrelated with the Brownian motions driving theforward rates, i.e. dZ(t)dWi(t) = 0, i= 1, , m.This assumption of independence of the sources of randomness is contrary to that assumed in

stochastic volatility models for equity and FX. In equity/FX models the stochastic volatility processis overlaid on a lognormal process for the underlying; the correlation between sources of uncertainty

is required to fit the asymmetry observed in volatility smiles. In this implementation, the stochastic

volatility factor is overlaid on a forward rate process with level dependent volatility specified by

function (fk(t)). This function is perfectly (positively or negatively) dependent on the forwardrate and introduces asymmetry (skewness) into the volatility profile. The remaining uncertainty in

the volatility specification can then be introduced by means of an uncorrelated Brownian motion.

This is the two-component set-up supported by Rebonato [51] and discussed in Chapter 33.The independence of the Brownian motions means we can express the forward rate dynamics in

(4.5) under the forward risk neutral measure Qk, leaving the process for V(t) unchanged. Hence

(4.5) becomes:

(4.6) dfk(t) = (fk(t))V(t)k(t)dWk(t), k= 0, , n 1,with V(t) unchanged as in (4.4). This can easily be shown by revisiting to the change of measureanalysis in Chapter 1. Specifically from equation (3.14) of Chapter 1, the drift under the new

probability measure may be specified as6:

kt (V(t)) = t(V(t)) +dV(t)d ln (P(t, Tk+1)/B(t))

dt=t(V(t)).

As detailed in Chapter 14, ln (P(t, Tk+1)/B(t)) is a function of forward rates and hencedV(t)d ln (P(t, Tk+1)/B(t)) is a sum of covariances between dV(t) and dfj(t), j = n(t), , k forTn(t)1< t Tn(t), which are all zero (since we defined the Brownian motion driving the variance

to be independent to those driving the forward rates).As mentioned above, approximations allowing fast and accurate pricing of caps and swaptions are

key to efficient calibration to market prices. Andersen and Brotherton-Ratcliffe develop asymptotic

expansions for the pricing of these instruments.

5When variance becomes zero, (4.4) becomes a deterministic process with positive drift, ensuring a return to

positivity.6SinceV(t) is not lognormal, unlike the asset in the original analysis, I substitute the correct co-ordinates.


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4.2. Caplet Pricing Approximations. A cap comprises a series of caplets, each paying at

timeTk+1 an amountk(fk(Tk) X)+ whereX is the strike. The time t price is:

C(t) = P(t, Tk+1)k EQk

t (fk(Tk) X)+ .Let G(t, fk(t), V(t)) = E

Qk

t [(fk(Tk) X)+]. For European caplets we require only the total termvolatility hence we may replace k(t)dW

k(t) in (4.6) withk(t)dY(t), where dY(t) is a scalarBrownian motion such that dZ(t)dY(t) = 0. Dynamics ofV(t) in (4.4) remain unchanged and weapply the Keynmann-Kac Theorem to derive the PDE for G(t,f,V) as:

G

t + ( V) G

V+

1

222(V)

2G

V2 +

1

22(fk)Vk2

2G

fk2

= 0,

s.t. G(Tk, fk(Tk), V) = (fk(Tk) X)+.Given the above PDE with associated dynamics offk(t) andV(t), there are two points of deviationfrom the standard Black formulation:

The forward rate dynamics (4.6) are not lognormal under the forward risk neutral mea-sure associated with their payoff time. Instead the level dependence of the volatility is

determined by a general function of the forward rate (fk(t)). The variance is not deterministic, but driven by a mean reverting stochastic process.

To determine a solution to the above PDE, and hence an analytic caplet pricing formula, Andersen

and Brotherton-Ratcliffe first consider the deviation from lognormality of the forward rate dynamics.

They develop an asymptotic solution for this PDE which takes the form of a small-time expansion

around(fk) =fk, the lognormal dynamics of the underlying for which a known solution, the Blackcaplet formula, exists.

Then, allowing volatility to be stochastic, they use the Taylor series expansion developed by Hull

and White [28] (see

3) which requires the central moments of the integrated volatility distribution

to be calculated.

To ease notation we letk(t) k(t) in what follows.

4.3. Asymptotic Expansion of the Pricing PDE. Andersen and Brotherton-Ratcliffe first

consider the case where V and (t) are constant i.e. V(t)(t)2 = c, a constant. This is used as abase which is extended for stochastic V and time dependent .

For this special case ofV (t)2 =c, the above PDE becomes:

(4.7) G

t +

1

22(fk)c

2G

fk2 = 0 s.t. G(Tk, fk(Tk)) = (fk(Tk) X)+.

Let =Tk t, the asymptotic solution to PDE (4.7) is G(t, fk) = g(t, fk; c) withg(t, fk; c) =fkN(d+) XN(d),(4.8)

d =ln (fk/X) 12 (t, fk, c)2

(t, fk, c) ,

whereN() is the cumulative normal distribution function and takes the form:(t, fk, c) = 0(fk)c

1/21/2 + 1(fk)c3/23/2 + O(5/2),(4.9)


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with:

0(fk) = ln (fk/X)

fkX (u)1du ,(4.10)1(fk) = 0(fk)fk

X (u)1du

2 ln

0(fk)

(X)fk(fk)X

1/2.(4.11)

The details of this solution are presented in the Appendix.

Slightly relaxing our volatility assumptions to allow V(t) and (t) to be a functions of time, theasymptotic solution (4.8) remains largely unchanged if we set c to be the integrated variance, thatis:

c= 1 Tkt

(u)2V(u)du.

For the fully general case of stochastic V(t) and time-dependent (t), the asymptotic solution

becomes:

G(t, fk, V) = Et

g(t, fk; 1U(Tk))

=

0

g(t, fk; 1U(Tk))P(U(Tk)|V(t))dU(Tk),

U(Tk) =

Tkt

(u)2V(u)du,

where the expectation is over the distribution ofU(Tk) under theQk measure, henceP(U(Tk)|V(t))

is the pdf of the integrated volatility, conditional on V(t).

To evaluate the above expectation using a Taylor series expansion requires the central moments of

the distribution ofU(Tk). From the definition ofU(Tk), the first central moment is:

Et[U(Tk)] =

Tk

t (u)2Et[V(u)] du=

7 Tk

t (u)2 + (V(t) )e(ut) du U(t, V(t)).

4.4. Finding the Higher Moments. Andersen and Brotherton-Ratcliffe follow the ideas in

Lewis [37], who shows that for the case of uncorrelated Brownian motions driving the processes for

the underlying and the volatility, the density function of the term volatility and the fundamental

transform are Laplace transform-inversion pairs. That is:

(4.12)

LTkP(U(Tk)|V(t))(s) = Et

esU(Tk)

=

0

esU(Tk)P(U(Tk)|V(t))dU(Tk) = H(t, V(t); s),

7From the deterministic part of the dynamics ofV(t) are:

dV(t) =( V(t))dt,

which may be solved as a simple ODE: ut

dV(s)

V(s) =

ut

ds,

V(u) = + (V(t) ) e(ut).


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whereLTk denotes the Laplace transform of the time Tk variable, s is the transform variable andH(t, V(t); s) is the fundamental transform8 of derivative H(t, fk, V).

If we are able to determine this fundamental transform of a derivative, given our specific forward

rate and volatility processes, we can use a Laplace transform-inversion to determine the pdf of theterm volatilityU(Tk). Alternatively, the Laplace transform above may be interpreted as the moment

generating function ofU(Tk) with parameters. This allows us to determine the raw moments ofthe pdf by taking progressively higher order derivatives of the moment generating function.

Following Lewis [37] we derive the PDE for Hand attempt to solve it9.

The derivative price G(t, fk, V), is an expansion around the know solution for the case (fk) = fk,

hence in determining the PDE for Hwe consider this special case of the fk dynamics (4.6), namely:

dfk(t) = fk(t)

V(t)k(t)dWkk (t), k= 0, , n 1,

d ln fk(t) = 12 V(t)k(t)2dt +

V(t)k(t)dWkk (t).

Applying Itos lemma, the PDE for H(t, fk, V) in terms of ln fk(t) is:

(4.13) H

t 1

2 V(t)(t)2 H

ln fk+ ( V(t)) H

V + 12 V(t)(t)

2 2H

(ln fk)2 +

12

2(V(t))22H

V2 = 0.

Introducing the Fourier transform ofH(t, fk, V):

H(t , V , q ) =

eiq lnxH(t,x,V) d ln x,

H

t =

eiq lnxH

t d ln x.

Substituting for H

t

from (4.13) yields:

H

t =

eiq lnx

12

V(t)(t)2 H

ln x ( V(t)) H

V 1

2V(t)(t)2

2H

(ln x)2

12

2(V(t))22H

V2

d ln x

= ( V(t)) H

V 1

22(V(t))2

2H

V2 + 1

2V(t)(t)2

eiq lnx H

ln xd ln x(4.14)

12

V(t)(t)2

eiq lnx 2H

(ln x)2d ln x.

8The fundamental transform is the Fourier transform of a derivative price, such that it satisfies the terminal

boundary condition H(T, V; s) = 1. It can be used as a fundamental derivative from which other derivative prices

are determined.9Of course our ultimate goal is a solution to the caplet pricing problem, G(t, fk, V). However, we use the

Fourier transform technique as a tool to determine the moments of the integrated variance distribution, rather than

a final solution to G(t, fk, V). These required moments are a function of the forward rate and variance specifications

rather than a specific derivative payoff, hence we use the simplest possible derivative which will allow us to apply the

relationship (4.12).


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whereh1(t, V; s) andh2(t, V; s) are found by substituting the above expansion into (4.17), groupingterms with coefficients i, i = 2, 4 and solving the resulting ODEs. h1(t, V; s) and h2(t, V; s) arespecified in terms of double integrals which need to be computed numerically in most cases. I do

not show these results.Having found the solutionh(t, V; s) we combine the definitions of the transforms in (4.12) and (4.16)as:

h(t, V; s) = Et

es(U(Tk)U(t,V))

=

0

es(U(Tk)U(t,V))P(U(Tk)|V(t))dU(Tk).

Interpreting this representation as a moment generating function, we can determine the central

moments by taking derivatives ofh(t, V; s) with respect to s, hence:

Et[(U(Tk) U(t, V))n] = (1)n nh

sn

s=0

, n= 2, 3, .

Here the second and third moments Et (U(Tk) U(t, V))2

and Et (U(Tk) U(t, V))3

are the

variance and skew used in the Taylor expansion (see equation (3.3)) to approximate the derivativeprice. The first term of the expansion is derivative price evaluated at the mean integrated variance,

i.e. g(t, fk; 1U(t, V)).

4.5. Swaption Pricing Approximation. Treating the swap rate as the weighted sum of its

constituent forward rates, and using approximations similar to those discussed in 7 of Chapter1, its dynamics may be expressed in terms of the forward rate volatilities, while maintaining the

martingale structure of the forward rate dynamics in (4.6). This means the caplet and swaption

pricing problems are identical (although they are formulated under different numeraires), and the

results developed above may be applied to swaptions as well.

An application of Itos Lemma and a change of measure yields:

(4.18) dS,(t) = V(t) 1j=

S,(t)

fj(t) (fj(t))j(t)dW,(t).

We would like the swap rate dynamics to assume the same form as that of forward rates in (4.6),

hence:

dS,(t) = (S,(t))

V(t),(t)dW,(t).

Equating these two we have:

dS,(t) = (S,(t))

V(t)

1j=

wj(t)j(t)dW

,(t),(4.19)

with:

wj(t) = S,(t)

fj(t)

(fj(t))

(S,(t)) .(4.20)

Now, assuming that S,(t)fj(t)

and (fj(t))(S,(t))

vary little with time, freeze the weights at their time

t= 0 values and the swap rate dynamics are approximated as:

dS,(t) =(S,(t))

V(t)

1j=

wj(0)j(t)dW

,(t),


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5. PITERBARG APPROACH 43

or in terms of the swap rate volatility:

dS,(t) =(S,(t))V(t),(t)dW,(t).

The validity of the assumptions underlying this approximation is dependent on reasonable shifts

in the yield curve and the specific form of. See Andersen and Andreasen [2] and Andersen andBrotherton-Ratcliffe [3] on these points.

4.6. Asymptotic Expansions. The pricing solution discussed above relies on two sets of

asymptotic expansions. The first is a small time (specifically term to expiry) expansion for the

solution of PDE (4.7), second is the volatility-of-variance expansion to solve the PDE for the centred

transform (4.17). This brings into question the performance of these solutions when time is not small

or when volatility-of-variance becomes large. Andersen and Brotherton-Rat

simona svoboda - libor market model with stochastic volatility.pdf

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