Singular Perturbations in Option Pricing

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Singular Perturbations in Option Pricing

J.-P. Fouque G. Papanicolaou R. Sircar K. Solna

March 4, 2003

Abstract

After the celebrated Black-Scholes formula for pricing call options underconstant volatility, the need for more general nonconstant volatility models

in financial mathematics has been the motivation of numerous works duringthe Eighties and Nineties. In particular, a lot of attention has been paid tostochastic volatility models where the volatility is randomly fluctuating drivenby an additional Brownian motion. We have shown in [2, 3] that, in the presenceof a separation of time scales, between the main observed process and thevolatility driving process, asymptotic methods are very efficient in capturingthe effects of random volatility in simple robust corrections to constant volatilityformulas. From the point of view of partial differential equations this methodcorresponds to a singular perturbation analysis. The aim of this paper is to dealwith the nonsmoothness of the payoff function inherent to option pricing. Wepresent the case of call options for which the payoff function forms an angle at

the strike price. This case is important since these are the typical instrumentsused in the calibration of pricing models. We establish the pointwise accuracy ofthe corrected Black-Scholes price by using an appropriate payoff regularizationwhich is removed simultaneously as the asymptotics is performed.

1 Introduction

Stochastic volatility models in financial mathematics can be thought of as a Brownian-type particle (the stock price) moving in an environment where the diffusion coefficientis randomly fluctuating in time according to some ergodic (mean-reverting) diffusion

process. We then have two Brownian motions, one driving the motion of the particleand the other driving the fluctuations of the medium. In the context of Physics thereis no natural correlation between these two Brownian motions since they do not live

Department of Mathematics, NC State University, Raleigh NC 27695-8205,

[email protected]. Work partially supported by NSF grant DMS-0071744.Department of Mathematics, Stanford University, Stanford CA 94305, papan-

[email protected] of Operations Research & Financial Engineering, Princeton University, E-Quad,

Princeton NJ 08544, [email protected]. Work partially supported by NSF grant DMS-0090067.Department of Mathematics, University of California, Irvine CA 92697, [email protected]

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in the same space. In the context of Finance they jointly define the dynamics of thestock price under its physical probability measure or an equivalent risk-neutral mar-tingale measure. Correlation between them is perfectly natural. There are economicarguments for a negative correlation or leverage effectbetween stock price and volatil-ity shocks, and from common experience and empirical studies, asset prices tend to godown when volatility goes up. The diffusion equation appears as a contingent claimpricing equation, its terminal condition being the payoff of the claim. We refer to [5]or [6] for surveys on stochastic volatility. When volatility is fast mean-reverting, ona time-scale smaller than typical maturities, one can perform a singular perturbationanalysis of the pricing PDE. As we have shown in [2], this expansion reveals a firstcorrection made of two terms: one is directly associated with the market price ofvolatility risk and the other is proportional to the correlation coefficient between thetwo Brownian motions involved. We refer to [2] for a detailed account of evidence ofa fast scale in volatility and the use of this asymptotics to parametrize the evolutionof the skewor the implied volatility surface. We also refer to [4] for a different type

of application, namely variance reduction in Monte Carlo methods.The present paper deals with the accuracy of such an expansion in presence of

another essential characteristic feature in option pricing, namely the nonsmoothnessof payoff functions. We present the case of call options since these are the liquidinstruments used in the calibration of pricing models. By inverting the Black-Scholesformula the price of a call option is given in terms of its implied volatility whichdepends on the strike and the maturity of the option. This set of implied volatilitiesform the term structure of implied volatility. For fixed maturity and across strikes itis known as thesmileor the skewdue to the observed asymmetry. These objects andtheir dynamics are what volatility models are trying to reproduce in order to price

and hedge other instruments.In [2] we have performed an expansion of the price in powers of the characteristicmean-reversion time of volatility, and we have shown that the leading order termcorresponds to a Black-Scholes price computed under a constant effective volatility.The first correction involves derivatives of this constant volatility price. When thepayoff is smooth we have shown that the corrected price, leading order term plus firstcorrection, has the expected accuracy, namely the remainder of the expansion is ofthe next order. The nonsmoothness of a call payoff which forms an angle at the strikeprice creates a singularity at the maturity time near the strike price of the option.

This paper is devoted to the proof of the accuracy of the approximation in thatcase. It is important because this is a natural situation in financial mathematics onehas to deal with. The proof given here relies on a payoff smoothing argument whichcan certainly be useful in other contexts.

In Section 2 we introduce the class of stochastic volatility models which we con-sider. They are written directly under the pricing equivalent martingale measure andwith a small parameter representing the short time-scale of volatility. We recall howoption prices are given as expected values of discounted payoffs or as solutions ofpricing backward parabolic PDEs with terminal conditions at maturity times. InSection 3 we recall the formal asymptotic expansion presented in [2]. In Section 4 weintroduce the regularization of the payoff and decompose the main result, accuracy

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of the price approximation, into three Lemmas. Section 5 is devoted to the proof ofthese Lemmas. Detailed computations involving derivatives of Black-Scholes pricesup to order seven are given in the appendices where we also recall the properties ofthe solutions of Poisson equations associated with the infinitesimal generator of theOrnstein-Uhlenbeck process driving the volatility.

2 Class of Models and Pricing Equations

The family of Ornstein-Uhlenbeck driven stochastic volatility models (St , Yt ) that

we consider can be written, under a risk-neutral probabilityIP, in terms of the smallparameter

dSt = rSt dt + f(Y

t )S

t dW

t,

dYt = 1

(m

Yt )

2

(Yt) dt +

2

dZt ,

where the Brownian motions (Wt,Zt ) have instantaneous correlation (1, 1):

dW, Zt=IE{dWt dZt } = dt,and

(y) =( r)

f(y) + (y)

1 2,

is a combined market price of risk. It describes the relationship between the physicalmeasure under which the stock price is observed, and the risk-neutral measure under

which the market prices derivative securities. See [2] for example. The price of theunderlying stock is St and the volatility is a function f of the process Yt . At the

leading order 1/, that is omitting the -term, Yt is an Ornstein-Uhlenbeck (OU)process which is fast mean-reverting with a normal invariant distributionN(m, 2).Notice that in this framework the volatility driving process (Yt ) is autonomous inthe sense that the coefficients in its defining SDE do not depend on the stock priceSt .

In this fast mean-revertingstochastic volatility scenario, the volatility level fluc-tuates randomly around its mean level, and the epochs of high/low volatility arerelatively short. This is the regime that we consider and under which we analyzethe price of European derivatives. A derivative is defined by its nonnegative payofffunctionH(S) which prescribes the value of the contract at its maturity time Twhenthe stock price is S. The payoff function must in general satisfy the integrabilitycondition

IE{H(ST)2} < ,with IE denoting expectation with respect to IP. Moreover, we assume:

1. The volatility is positive and bounded: there are constants m1 and m2 suchthat

0< m1 f(y) m2< y R.

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2. The volatility risk-premium is bounded:

|(y)| < l < y R.for some constant l.

It is convenient at this stage to make the change of variable

Xt = log St , t 0,

and write the problem in terms of the processes (Xt , Yt ) which satisfy, by Itos

formula the stochastic differential equations

dXt =

r 1

2f(Yt )

2

dt+ f(Yt ) dW

t, (2.1)

dYt =

1

(m Yt )

2

(Yt )

dt +

2

dZt . (2.2)

We also define the payoff function h in terms of the log stock price via

H(ex) =h(x), x R.The price at time t < Tof this derivative is a function of the present value of the

stock price, or equivalently the log stock price, Xt =x and the present value Yt =y

of the process driving the volatility. We denote this price by P(t,x,y). It is standardin finance to assume the price is given by (2.3) which is the expected discounted payoffunder the risk-neutral probability measure IP. See [1] for example.

P(t,x,y) =IE er(Tt)h(XT)|Xt =x, Y

t =y . (2.3)

We shall also write these conditional expectations more compactly as

P(t,x,y) =IEt,x,y

er(Tt)h(XT)

.

Under the assumptions on the models considered and the payoff, P(t,x,y) is theunique classical solution to the associated backward Kolmogorov partial differentialequation problem

LP = 0, (2.4)P(T, x , y) = h(x)

in t < T, x, y R, where we have defined the operatorsL = 1

L0+ 1

L1+ L2

L0 = 2 2

y2+ (m y)

y, (2.5)

L1 =

2f(y) 2

xy 2(y)

y, (2.6)

L2 = t

+1

2f(y)2

2

x2+

r 1

2f(y)2

x r . (2.7)

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The operatorL0 is the infinitesimal generator of the OU process (Yt) defined bydYt= (m Yt) dt +

2 dZt , (2.8)

L1contains the mixed partial derivative due to the correlation and the derivative dueto the market price of risk, andL2, also denoted byLBS(f(y)), is the Black-Scholesoperator in the log variable and with volatility f(y).

3 Price approximation

We present here the formal asymptotic expansion computed as in [2, 3] which leadsto a (first-order in

) approximation P(t,x,y)Q(t, x). In the next section we

prove the convergence and accuracy as 0 of this approximation which consists ofthe first two terms of the asymptotic price expansion:

Q

(t, x) =P0(t, x) + P1(t, x),which do not depend on y and are derived as follows. We start by writing

P = Q + Q2+ 3/2Q3+ =P0+

P1+ Q2+

3/2Q3+ , (3.1)Substituting (3.1) into (2.4) leads to

1

L0P0+ 1

(L0P1+ L1P0) (3.2)

+ (L0Q2+ L1P1+ L2P0) +

(L0Q3+ L1Q2+ L2P1) + = 0.We shall next obtain expressions for P0 and P1 by successively equating the four

leading order terms in (3.2) to zero. We let denote the averaging with respect tothe invariant distributionN(m, 2) of the OU process Y introduced in (2.8):

g = 1

2

R

g(y)e(my)2/22dy. (3.3)

Notice that this averaged quantity does not depend on .Below, we will need to solve the Poisson equationassociated withL0:

L0 + g= 0, (3.4)

which requires the solvability condition

g = 0, (3.5)in order to admit solutions with reasonable growth at infinity. Properties of thisequation and its solutions are recalled in Appendix C.

Consider first the leading order term:

L0P0 = 0.

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SinceL0 takes derivatives with respect to y, any function independent ofy satisfiesthis equation. On the other hand y-dependent solutions exhibit the unreasonablegrowth exp(y2/22) at infinity. Therefore we seek solutions which are independent ofy: P0= P0(t, x) with the terminal condition P0(T, x) =h(x).

Consider next:

L0P1+ L1P0 = 0,which corresponds to the second term in (3.2). SinceL1 contains only terms withderivatives in y it reduces toL0P1 = 0 and, as for P0, we seek again a functionP1=P1(t, x), independent ofy, with a zero terminal condition P1(T, x) = 0. Hence,Q = P0+

P1, the leading order approximation, does not depend on the current

value of the volatility level.The next equation

L0Q2+ L1P1+ L2P0= 0,

which corresponds to the third term in (3.2), reduces to the Poisson equation

L0Q2+ L2P0= 0, (3.6)sinceL1P1 = 0. Its solvability condition

L2P0 = L2P0= 0,is the Black-Scholes PDE (in the log variable) with constant square volatilityf2:

L2P0= LBS()P0 = P0t

+1

22

2P0x2

+

r 1

22

P0x

rP0= 0, (3.7)

where we define the effective constant volatility by

2 = f2.We choose P0(t, x) to be the classical Black-Scholes price, solution of (3.7) with theterminal condition P0(T, x) =h(x).

Observe thatQ2= L10 (L2L2)P0as a solution of the Poisson equation (3.6).This notation includes an additive constant iny which will disappear when hit by theoperatorL1 as follows. The fourth term in (3.2) gives the equation :

L0Q3+ L1Q2+ L2P1 = 0. (3.8)This is a Poisson equation in Q3, and its solvability condition gives

L2P1 = L1Q2 = L1L10 (L2 L2)P0,which, with its zero terminal condition, determines P1as a solution of a Black-Scholesequation with constant square volatility f2 and a source term. Using the expressionsforLi one can rewrite the source as:

L1L10 (L2 L2)P0 =L1L10 f(y)2 f212

2

x2

x

P0

=

v3

3

x3+ (v2 3v3)

2

x2+ (2v3 v2)

x

P0, (3.9)

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where

v2 =

2(2f )

v3 =

2f

, (3.10)

and is a solution of the Poisson equation:

L0(y) =f(y)2 f2. (3.11)We can therefore conclude:

1. The first termP0 is chosen to be the solution of the homogenized PDE prob-lem (3.7). In other words, P0 is simply the Black-Scholes price of the derivativecomputed with the effective volatility .

2. The second term, or correction to the Black-Scholes price, is given explicitly, asa linear combination of the first three derivatives ofP0, by

P1 = (T t)

V3

3

x3+ (V2 3V3)

2

x2+ (2V3 V2)

x

P0, (3.12)

with

V2,3=

v2,3, (3.13)

since it is easily seen, by usingL2P0 = 0, that equation (3.9) is satisfied, andthat, on the other hand, the terminal condition P1(T, x) = 0 is satisfied when

limtT(T t)iP0xi = 0 for i= 1, 2, 3.

Essential instruments in financial markets are put and call options for which thepayoff function H(S) is piecewise linear. We shall focus on call options:

H(S) = (S K)+ h(x) = (ex K)+,for some given strike price K > 0. Notice that h is onlyC0 smooth with a discon-tinuous first derivative at the kink x = log K, (at the money in financial terms).Nonetheless, at t < T, the Black-Scholes pricing function P0(t, x) is smooth andP1(t, x) is well-defined, but second and higher derivatives of P0 with respect to x

blow up as t T(at the money).Our main result on the accuracy of the approximation Q = P0 + P1 is asfollows:

Theorem 1 Under the assumptions (1) and (2) above, at a fixed pointt < T, x, yR, the accuracy of the approximation of call prices is given by

lim0

|P(t,x,y) Q(t, x)|| log |1+p = 0,

for anyp >0.

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Observe that this pointwise approximation is the sense of accuracy needed in financeapplications since option prices are computed at given values of (t,x,y).

Before giving in the next Section the proof of Theorem 1, we comment on theinterpretation of the approximation and on the validity of the result for more generalpayoffs.

Financial interpretation of the approximation.

In order to give a meaningful interpretation to the leading order term and thecorrection in our price approximation it is convenient to return to the variable S,the underlying price. With a slight abuse of notation we denote the call option priceapproximation byP0(t, S) +

P1(t, S). Indeed the leading order term P0(t, S) is the

standard Black-Scholes price of the call option computed at the effective constantvolatility . From (3.12), one can easily deduce that

P1(t, S) =

(T

t)V2S2

2P0

S2

+ V3S3

3P0

S3 , (3.14)

which shows that the correction is a combination of the two greeks Gamma andEpsilon, as introduced in [2]. This correction can alternatively be written in the form

P1(t, S) = (T t)

(V2 2V3)S2

2P0S2

+ V3S

S

S2

2P0S2

. (3.15)

Using the classical relation between Gamma and Vega for Black-Scholes prices ofEuropean derivatives

P0

= (T

t)S2

2P0

S2

,

which is easily obtained by differentiating the Black-Scholes PDE with respect to ,one can rewrite the correction as:

P1(t, S) = 1

(V2 2V3)

P0

+ V3S

S

P0

. (3.16)

Therefore the price correction is a combination of theVegaand the Delta-Vegaof theBlack-Scholes price. TheVegaterm corresponds simply to a volatility level correction.The Delta-Vegaterm is proportional to the correlation coefficient and captures themain effect of skewness in implied volatility as discussed in detail in [2].

Other payoff functions.

The main idea of the proof presented in the next Section is a regularization of thepayoff which does not rely on the particular choice of a call option. The only placewhere we use the explicit Black-Scholes formula for a call option is in the computation(B.1) of the successive derivatives nxP

0 carried out in the Appendix B. Note that if

we had started with a payoff function h which is continuous and piecewise smooth(a call option being a particular case), then P0 , the solution of the parabolic PDE(3.7), is an integral of the payoff function with respect to a normal density as in thecase of a call option. The first derivative with respect to x can be taken on the payoff

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function and the higher order derivatives can then be taken on the normal densityas detailed in Appendix B for a call option. Therefore Theorem 1 remains valid forgeneral European claims with continuous payoffs that have singular behavior in theirderivatives.

Numerical illustration. To illustrate the asymptotic approximation, we comparethe approximation

Q =P0+

P1

with a numerical solution of the PDE (2.4) for a particular stochastic volatility modeland a call option with strike price K= 100 and three months from expiration. (Inpractice, the asymptotic approximation is not used in this manner because of thedifficulties of estimating the volatility parameters precisely; instead the parametersof the approximation V2 and V

3 are estimated directly from observed options prices,

as described in [2]).We choose f(y) = ey, where this is understood to stand for a cutoff version of

the exponential function with the cutoffs (above and below) sufficiently large andsmall respectively so as not to affect the calculations within the accuracy of ourcomparisons. We use the parameter values

= 1/200, m= log 0.1, = 1/

2, = 0.2,= 0.2, r= 0.04,

and choose the volatility risk premium 0. It follows from explicit calculationsthat the parameters for the asymptotic approximation are

= 0.165, V2 = 3.30 10

4

, V3 = 8.48 10

5

.

Figure 1 shows the numerical solution from an implicit finite-difference approxi-mation at two levels of the current volatilityey, one at the long-run mean-level , andone far above it (0.607). These are compared to the asymptotic approximation whichdoes not depend on the current volatility level. In the range 0.95 K/S 1.04shown in the picture on the right, the maximum deviation of the asymptotic approx-imation from the price with the higher volatility is by 9% of the latter price, and themaximum deviation of the asymptotic approximation from the price with the lowervolatility is by 2.1% of this price.

4 Derivation of the accuracy of the price approxi-

mation

In order to prove Theorem 1, we introduce in the next section the regularized price,P,, the price of a slightly smoothed call option, withbeing the (small) smoothingparameter. We denote the associated price approximation Q,. The proof theninvolves showing that (i) P P,, (ii) Q, Q, (iii) P, Q,, and controllingthe accuracy in these approximations by choosing appropriately.

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80 85 90 95 100 105 110 115 1200

5

10

15

20

25

Stock Price S

CallOptionPrices

Asymptotics

ey=0.165

ey=0.607

96 97 98 99 100 101 102 103 104

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Stock Price S

CallOptionPrices

Asymptotics

ey=0.165

ey=0.607

Figure 1: Call option prices 3 months from maturity as a function of the currentstock price S. The strike price is K= 100 and the picture on the right focuses onthe region around the money.

4.1 Regularization

We begin by regularizing the payoff, which is a call option, by replacing it with theBlack-Scholes price of a call with volatility and time to maturity . We define

h(x) :=CBS(T , x; K, T; ),

where CBS(t, x; K, T; ) denotes the Black-Scholes call option price as a function ofcurrent time t, log stock price x, strike price K, expiration date T and volatility .It is given by

CBS(t, x; K, T; ) = P0(t, x; K, T; ) = exN(d1) Ker

2

2 N(d2) (4.1)

N(x) = 1

2

x

ey2/2 dy

d1 = x log K

+ b

d2 = d1

,

where we define

=

T t b= r2

+1

2.

For > 0, this new payoff isC. The price P,(t,x,y) of the option with theregularized payoff solves

LP, = 0P,(T, x , y) = h(x).

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4.2 Main convergence result

Let Q,(t, x) denote the first-order approximation to the regularized option price:

P, Q, P0 +

P1 ,

where

P0 (t, x) = CBS(t , x; K, T; ) (4.2)

P1 = (T t)

V33

x3+ (V2 3V3)

2

x2+ (2V3 V2)

x

P0 . (4.3)

We establish the following pathway to proving Theorem 1 where constants may de-pend on (t , T, x , y) but not on (, ):

Lemma 1 Fix the point(t,x,y) where t < T. There exist constants 1 >0, 1 >0andc1 > 0 such that

|P(t,x,y) P,(t,x,y)| c1for all0< < 1 and0< < 1.

This establishes that the solutions to the regularized and unregularized problems areclose.


|Q(t, x) Q,(t, x)| c2for all0< < 2 and0< < 2.

This establishes that the first-order asymptotic approximations to the regularized andunregularized problems are close.


|P,(t,x,y) Q,(t, x)| c3

| log | +

+

,

for all0< < 3 and0< < 3.

This establishes that for fixed , the approximation to the regularized problem con-verges to the regularized price as 0.

The convergence result follows from these Lemmas:Proof of Theorem 1. Take = min(1, 2, 3) and = min(1,2,3). Then using

Lemmas 1, 2 and 3, we obtain

|P Q| |P P,| + |P, Q,| + |Q, Q| 2 max(c1, c2)+ c3

| log | +

+

,

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for 0< < and 0< < , where the functions are evaluated at the fixed (t,x,y).Taking =, we have

|P Q| c5(+ | log |),for some fixed c5>0, and Theorem 1 follows.

A general conclusion to our work is given in Section 6 after the proofs of Lemmas1,2 and 3 given in the following section.

5 Proof of Lemmas

5.1 Proof of Lemma 1

We use the probabilistic representation of the price given as the expected discountedpayoff with respect to the risk-neutral pricing equivalent martingale measure IP.

P

,

(t,x,y) =IE

t,x,y

e

r(Tt)

h

(X

T)

.

We define the new process ( Xt ) by

dXt =

r 1

2f(t, Yt )

2

dt + f(t, Yt )

1 2 dWt + dZt

,

where (Wt) is a Brownian motion independent of (Zt ), (Y

t ) is still a solution of (2.2)

and

f(t, y) =

f(y) for t T for t > T.

Then we can write

P,(t,x,y) =IEt,x,y

er(Tt+)h(XT+)

,

andP(t,x,y) =IEt,x,y

er(Tt)h(XT)

.

Next we use the iterated expectations formula

P,(t,x,y) P(t,x,y) =

IE

t,x,y

IE

e

r(Tt+)

h(X

T+) er(Tt)

h(X

T) | (Z

s )tsT

,

to obtain a representation of this price difference in terms of the Black-Scholes functionP0 which is smooth away from the terminal date T. In the uncorrelated case itcorresponds to the Hull-White formula [7]. In the correlated case, as considered here,this formula is in [8], and can be found in [2](2.8.3). It is simple to compute explicitlythe conditional distributionD(XT|(Zs )tsT, Xt ) ofXTgiven the path of the secondBrownian motion (Zs )tsT. One obtains

D(XT|(Zs )tsT, Xt =x) =N(m1, v1),

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where the mean and variance are given by

m1 = x + t,T+ (r 1

22)(T t)

v1 = 2(T t)

and we define

t,T =

Tt

f(s, Ys) dZs

1

22 Tt

f(s, Ys)2ds (5.1)

2 = 1 2

T t Tt

f(s, Ys)2ds.

It follows from the calculation that leads to the Black-Scholes formula that

IEt,x,y{er(Tt)h(XT) | (Zs )tsT} =P0(t, Xt + t,T; K, T; ).Similarly, we compute

D(XT+| (Zs )tsT, Xt =x) =N(m2, v2),where the mean and variance are given by

m2 = x+ t,T+ r+ (r 1

22,)(T t)

v2 = 2,(T t),

and we define

2, = 2+ 2

T t .Therefore

IEt,x,y{er(Tt+)h(XT+) | (Zs )tsT} =P0(t, Xt + t,T+ r; K, T; ,),and we can write

P,(t,x,y) P(t,x,y) =IEt,x,y {P0(t, x + t,T+ r; K, T; ,) P0(t, x+ t,T; K, T; )} .

Using the explicit representation (4.1) and that is bounded above and below asf(y) is, we find

|P0(t, x+ t,T+ r; K, T; ,) P0(t, x + t,T; K, T; )| c1(et,T [|t,T| + 1] + 1)for somec1and forsmall enough. Using the definition (5.1) oft,Tand the existenceof its exponential moments, we thus find that

|P(t,x,y) P,(t,x,y)| c2for some c2 and for small enough.

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From the definition (3.12) of the correction

P1 and the corresponding definition(4.3) of the correction

P1 we deduce

Q,

Q

=

1 (T t)

V3

3

x3+ (V2 3V3)

2

x2+ (2V3 V2)

x

(P0 P0).

From the definition (3.10) of the vis, the definition (3.13) of the Vis and the boundson the solution of the Poisson equation (3.11) given in Appendix C, it follows that

max(|V2 |, |V3 |) c1

for some constant c1 > 0. Notice that we can write

P0 (t, x) =P0(t , x).

Using the explicit formula (4.1), it is easily seen thatP0 and its successive derivativeswith respect to x are differentiable in t at anyt < T. Therefore we conclude that for(t,x,y) fixed with t < T:

|Q(t, x) Q,(t, x)| c2

for some c2 >0 and small enough.


We first introduce some additional notation. Define the error Z, in the approxima-tion for the regularized problem by

P, =P0 +

P1 + Q2+

3/2Q3 Z,,

for Q2 and Q3 stated below in (5.3) and (5.4). Setting

L =1L0+ 1

L1+ L2,

one can write

LZ, = L P0 + P1 + Q2+ 3/2Q3 P, (5.2)=

1

L0P0 +

1

(L0P1 + L1P0 )+(L0Q2+ L1P1 + L2P0 ) +

L0Q3+ L1Q2+ L2P1

+L1Q3+ L2Q2+ L2Q3

= L1Q3+ L2Q2+ 3/2L2Q3 G,

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because P, solves the original equationLP, = 0 and we choose P0 , P1 , Q2 andQ3 to cancel the first four terms. In particular, we choose

Q2(t,x,y) = 1

2(y)

2P0x2

P0

x , (5.3)so that

L0Q2= L2P0 ,(with an integration constant arbitrarily set to zero) whereas Q3 is a solution ofthe Poisson equation

L0Q3 = (L1Q2+ L2P1 ), (5.4)where the centering condition is ensured by our choice ofP1 .At the terminal time T we have

Z,

(T, x , y) =

Q2(T, x , y) + Q

3(T, x , y)

H,(x, y), (5.5)where we have used the terminal conditions P,(T, x , y) = P0 (T, x) = h

(x) andP1 (T, x) = 0. This assumes smooth derivatives ofP

0 in the domain t T which

is the case because h is smooth. It is shown in Appendix A that the source termG,(t,x,y) on the right-side of equation (5.2) can be written in the form

G, =

4i=1

g(1)i (y)

i

xiP0 + (T t)

6i=1

g(2)i (y)

i

xiP0

+3/2 5

i=1

g(3)

i

(y)i

xiP

0

+ (T

t)7

i=1

g(4)

i

(y)i

xiP

0 . (5.6)

In Appendix A we also show that the terminal condition H,(x, y) in (5.5) can bewritten

H,(x, y) =

2i=1

h(1)i (y)i

xiP0 (T, x)

+ 3/2

3i=1

h(2)i (y)i

xiP0 (T, x)

. (5.7)

To bound the contributions from the source term and terminal conditions we need thefollowing two Lemmas that are derived in Appendix C and Appendix B respectively:

Lemma 4 Let=g(j)

i or =h(j)

i with the functionsg(j)

i andh(j)

i being defined in(5.6) and (5.7). Then there exists a constant c > 0 (which may depend ony) suchthatIE {|(Ys)|Yt =y} c < fort s T.Lemma 5 Assume T t > > 0 and IE {|(Ys)|Yt =y} c1 0 and > 0 such that for < andt s T

|IEt,x,y

ni=1

(Ys)i

xiP0 (s, X

s)

| c2[T+ s]min[0,1n/2], (5.8)

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and consequently

|IEt,x,y T

t

(T s)pni=1

er(st)(Ys)i

xiP0 (s, X

s) ds

| (5.9)

c2| log()| forn= 4 + 2pc2 min[0,p+(4n)/2] else .Proof of Lemma 3

We use the probabilistic representation of equation (5.2),LZ, = G, withterminal condition H,:

Z,(t,x,y) =IEt,x,y

er(Tt)H,(XT, Y

T)

Tt

er(st)G,(s, Xs , Ys)ds

.

From Lemma 5 it follows that there exists a constant c >0 such that

|IEt,x,y T

t

er(st)G,(Xs , Ys)ds

| c+ | log()| + / (5.10)|IEt,x,y

H,(XT, Y

T) | c+ / , (5.11)

and therefore also for (t,x,y) fixed with t < T:

|P, Q,| = |Q2+ 3/2Q3 Z,| c

+ | log()| +

/

. (5.12)

since Q2 and Q3 evaluated for t < Tcan also be bounded using (5.3) and (A.5).

6 Conclusion

We have shown that the singular perturbation analysis of fast mean-reverting stochas-tic volatility pricing PDEs can be rigorously carried out for call options. We foundthat the leading order term and the first correction in the formal expansion are cor-rect. The accuracy is pointwise in time, stock price and volatility level. It is preciselygiven in Theorem 1. The first correction involves higher order derivatives of theBlack-Scholes price which blow up at maturity time and at the strike price. To over-come this difficulty we have used a payoff smoothing method and we have exploitedthe fact that the perturbation is around the Black-Scholes price for which there is anexplicit formula. The case of call options is particularly important since the calibra-tion of models is based on these instruments. The case of other types of singularitiesis open. With some work one can certainly generalize the method presented hereto other European derivatives such as binary options. The case of path-dependentderivatives such as barrier options is more difficult due to the lack of an explicit for-mula for the correction. The situation with American contracts such as the simplestone, the American put, is much more involved due to the singularities at the exerciseboundary.

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Appendices

A Expressions for Source Term and Terminal Con-

ditionFrom (5.2), the source term in the equation for the error Z, is

G, = L1Q3+ L2Q2+ 3/2L2Q3. (A.1)

To obtain an explicit form for this source term, we consider the three terms separately.We first introduce the convenient notation:

D x

D2

2

x2

x.

Consider the termL2Q2 in (A.1). Using that

L2 = LBS(f(y)) = LBS() +12

f(y)2 2D2 (A.2)

LBS()D2P0 = 0,and (5.3), one deduces:

L2Q2 = 1

4 f(y)2 2(y)D2D2P

0 .

Consider next the termL1Q3 in (A.1). Using (3.8) we haveQ3 = L10

L1Q2+ L2P1 L1Q2+ L2P1 , (A.3)= L10

L1Q2 L1Q2 + (L2 L2) P1 .It follows from (5.3) that:

L1Q2 =

2f(y) 2

xy 2(y)

y

1

2(y)D2P0

=

1

2f(y)(y)

DD2P

0 +

1

2(y)(y)

D2P

0 .

Now let 1 and 2 be solutions of

L01 = f(y)(y) f , (A.4)L02 = (y)(y) ,

then we find using (3.11) and (A.2) that Q3 can be written:

Q3 =

21(y)DD2P0

2

2(y)D2P0

12

(y)D2P1

. (A.5)

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Substituting forL1 and expanding givesL1Q3 = 22f(y)1(y)DDD2P0 2f(y)2(y)DD2P0

2(y)1(y)DD2P0 + 2(y)2(y)D2P0

2 f(y)(y)

DD2P

1(y)(y)

D2P

1 .

Consider finally the termL2Q3 in (A.1), we find using (A.2) and (A.5)

L2Q3 = 1

2(f(y)2 2)

21(y)D2DD2P0

2

2(y)D2D2P01

2(y)D2D2P1

12

(y)D2

v3D3P0 + v2D2P0

,

with

D3= 3

x3 3

2

x2+ 2

x

and v2,3 defined in (3.10).To summarize, the source term is given by

G, =

22f(y)1(y)DDD2P0 2f(y)2(y)DD2P02(y)1(y)DD2P0 + 2(y)2(y)D2P0

2

f(y)(y)DD2P1 (y)(y)D2P1

1

4

f(y)2 2(y)D2D2P0 }

+3/21

2(f(y)2 2)

2

1(y)D2DD2P0

22(y)D2D2P0

1

2(y)D2D2P1

12

(y)D2(v3D3P0 + v2D2P0 )

By inspection, this can be written in the form (5.6).From (5.3) and (A.5) we can also see that the terminal condition H, in (5.5) can

be written in the form (5.7).

B Proof of Lemma 5

To prove Lemma 5 notice first that a calculation based on the analytic expression for

the Black-Scholes price in the standard constant volatility case gives

nxP0 (s, x) =

exN(u/+ b) for n= 1

exN(u/+ b) +n2

i=0b(n)i

euiue

(u/+b)2/2 for n 2 (B.1)

for some constants bi and with

T+ su x log(K)b (r/2 + 1/2).

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Assume first that T s (T t)/2 > 0, so that (T t)/2. SinceixP

0 (s, x) is uniformly bounded in , it follows that

|IEt,x,y

(Ys)ixP

0 (s, X

s)

| cIEt,x,y {|(Ys)|} (B.2)

for some constant c which may depend on x.Consider next the case 0< T s


20/20

Notice that in Lemma 4 satisfies

|(y)| c max(|(y)|, |(y)|, |1,2(y)|, |1,2(y)|)

for some constantcand withand1,2defined in (3.11) and (A.4) respectively. These

functions are solutions of Poisson equations with g = f2 f2org= f f org= which are bounded. Therefore(y) is at most logarithmically growingat infinity. The bound in Lemma 4 now follows from classical a priori estimates onthe moments of the process Yt which are uniform in . In the case = 0 this caneasily be seen by a simple time change t= t in (2.2). The case = 0 follows by aGirsanov change of measure argument.

References

[1] D. Duffie. Dynamic Asset Pricing Theory. Princeton University Press, 2nd

Edition, 1996.

[2] J.-P. Fouque, G. Papanicolaou, and K.R. Sircar. Derivatives in Financial Mar-kets with Stochastic Volatility. Cambridge University Press, 2000.

[3] J.-P. Fouque, G. Papanicolaou and K.R. Sircar.Mean-reverting stochastic volatil-ity. International J. Theor. and Appl. Finance, 13(2): 101-142. (2000).

[4] J.-P. Fouque and T. Tullie. Variance Reduction for Monte Carlo Simulation ina Stochastic Volatility Environment. Quantitative Finance 2: 24-30 (2002).

[5] R. Frey. Derivative asset analysis in models with level-dependent and stochasticvolatility. CWI Quarterly 10(1), pp 1-34, 1996.

[6] E. Ghysels, A. Harvey and E. Renault. Stochastic volatility, inG. Maddala andC. Rao (eds), Statistical Methods in Finance, Vol. 14 ofHandbook of Statistics,North Holland, Amsterdam, chapter 5, pp 119-191, 1996.

[7] J. Hull. and A. White. The Pricing of Options on Assets with Stochastic Volatil-ities. J. Finance XLII(2), pp 281-300, 1987.

[8] G. Willard.Calculating prices and sensitivities for path-independent derivative

securities in multifactor modelsPhD thesis, Washington University in St. Louis,MO (1996).

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Singular Perturbations in Option Pricing