Stochastic Volatility Surface Estimation

Suhas Nayak and George Papanicolaou

May 9, 2006

Abstract

We propose a method for calibrating a volatility surface that matches option prices using anentropy-inspired framework. Starting with a stochastic volatility model for asset prices, we castthe estimation problem as a variational one and we derive a Hamilton-Jacobi-Bellman (HJB)equation for the volatility surface. We study the asymptotics of the HJB equation assumingthat the stochastic volatility model exhibits fast mean-reversion. From the asymptotic solutionof the HJB equation we get an estimate of the stochastic volatility surface. We also incorporateuncertainty in quoted derivative prices through a penalty term, i.e. by softening the constraintsin the HJB equation. We present numerical solutions of our estimation scheme. We find that,depending on the softness of the constraints, certain parameters of the volatility surface relatedto the implied volatility smile can be calibrated so that they are stable over time. Theseparameters are essentially the ones found in previous fast mean-reversion asymptotics papersby Fouque, Papanicolaou and Sircar. We find that our procedure provides a natural way ofinterpolating between the prior parameters and the parameters of Fouque, Papanicolaou andSircar.

1 Introduction and Review

Parameter identification for systems governed by partial differential equations is a well-studiedinverse problem. In mathematical finance, finding the volatility of a risky asset from options withmultiple strikes and maturities is an example. The information contained in the market is notenough to identify a pricing model and so many sets of model parameters and many types ofmodels could potentially be compatible with the observed option prices. We study the problem ofestimating a volatility surface from option prices in an incomplete market setting.

1

1.1 Complete Markets

In the complete markets case, there have been several approaches to estimating volatilities fromobserved option prices. One may try to use the Black-Scholes partial differential equation

Ct +1

2σ(S, t)2CSS + rCS − rC = 0, t < T

C(S, T ) = h(S)

directly to estimate σ(S, t). This was done in Andersen and Brotherton-Ratcliffe ([2], [3]). Here,h(S) is the payoff of the option. Penalization criteria like smoothness norms were used by Lagnadoand Osher [18] and Jackson et. al. [16] to regularize the volatility extraction procedure.

Another possible approach in the complete market setting is to use a dual of the option-pricingPDE. Dupire’s equation (see [10], [11]) determines the dependence of option prices on strikes andexpiration dates. It has the form

∂T C − 1

2σ2(T, K)K2 ∂2C

∂K2+ rK

∂C

∂K= 0, T > t

C(t, K) = h∗(K)

Here, h∗(K) has the form as h(S). Since this equation has derivatives in K and not in S, it is easierto handle because derivative prices are quoted for certain expiration times and strikes. Achdou andPironneau [1] used this equation in combination with least-squares and a regularization (penalty)term to solve the inverse problem of estimating σ(T, K). Their objective was to minimize

J(σ) =∑

i

|C(ti, Ki) − ci|2 + Jr(σ)

over σ subject to C solving Dupire’s equation. Here the {ci} are observed derivative prices andJr(σ) is an appropriate Tychonoff regularization functional that involves the L2-norm of σ and theL2-norm of its derivatives with respect to K and T .

1.2 Entropy-based methods

Regularization may also be achieved through the use of entropy. Entropy minimization for calibrat-ing one-period asset pricing models was used by Buchen and Kelly [7], by Gulko [15], by Jackwerthand Rubinstein [17] and by Platen and Rebolledo [19].

Relative entropy-based methods can be motivated as follows. There is a range of strategies availableto hedge an option. If the hedger is fully confident that the option will follow dynamics given bya known model, then it would be worthwhile to ignore observed market volatilities and hedgeusing the model. If, on the other hand, the hedger believes that current prices of the option fullydetermine the future evolution of the stock, then it would be a good idea to hedge according to

2

these current option prices. Relative entropy-based methods help bridge the gap between these twopossible strategies. They provide estimates for model parameters that are close to prior informationwhile still matching current option prices.

Within the class of relative entropy methods there are two general approaches. In Avellaneda [4],a probability law is found for the risky asset that satisfies certain moment constraints, namely thatit matches observed market prices of options, and is close to a a prior probability law, which canarise from historical or other econometric information. The closeness to the prior is measured bythe relative entropy, which is given by

H(P |P0) = EP

[

ln

(

dP

dP0

)]

where P0 is the prior distribution, and dPdP0

is the Radon-Nikodym derivative of P with respect toP0. For H(P |P0) to be finite, the measure P has to be absolutely continuous with respect to P0.So, in particular, if under P0, we assume our asset prices follow

dXi(t) =ν∑

i=1

σ(0)ij dBj(t) + µ

(0)i dt

where {Bj} are standard Brownian motions, then, by Girsanov’s theorem, we must have that underP , the asset follows the price process

dXi(t) =ν∑

i=1

σ(0)ij dBj(t) + µidt.

Here µi = µ(0)i +

∑

j σ(0)ij mj and mj is a market price of risk. In other words, once the volatility

of the asset under the prior probability law is specified, the volatility of the asset under the newmeasure must be the same. Only the market price of risk can be adjusted to give a consistentpricing measure.

In this framework we assume that many different, perhaps correlated, shocks drive the evolution ofeach asset. Moreover, we assume that there is a good prior that describes the effect of fluctuations

through the volatilities σ(0)ij . It is unlikely that we will have an accurate assessment of these

volatilities. Yet once they are fixed in the model we are unable to deviate from them in thenew probability law because of the absolute continuity that is assumed. In order for the marketprice of risk to correct for possible misspecification of volatilities, wild swings in its value may benecessary. The work of Carmona and Xu [8], who introduce a stochastic volatility model in anentropy framework, suffers from the same problems. There, the market price of risk associatedwith the stochastic volatility process is the only degree of flexibility and all other parameters arefixed prior to the estimation procedure.

It is desirable, therefore, to use a different relative entropy approach, one that was introduced in[5]. They considered processes for equity of the form

dSt

St= σ0,tdBP0

t + µP0

t dt, under P0

3

anddSt

St= σtdBP

t + µPt dt, under P.

Unless σ0 = σ under P , the relative entropy of the two measures is infinite, since P and P0 are thenmutually singular. Avellaneda et. al. in [5] extended the concept of relative entropy to probabilitymeasures under which the processes do not necessarily have the same volatility. They consideredthe most singular part of the relative entropy using a time discretization that is based on trinomialtrees. With this discretization and a small σ − σ0 expansion, they found that to highest order therelative entropy looks like (σ2 − σ2

0)2 over each time step.

Both approaches to relative entropy calibration have the same general objective. Once the form ofthe relative entropy is determined the objective is to minimize it over all possible P subject to theconstraint that the prices of options under P match observed prices.

1.3 Stochastic volatility surface estimation

Finding volatilities across strikes and expiration dates for incomplete markets is a very difficulttask. We focus our attention on stochastic volatility models. These models have a large number ofparameters that need to be known for pricing purposes and options can be quite sensitive to them.There are, however, several papers that deal with this issue. Broadie et al. [6], for example, fit theparameters of various stochastic volatility models. They first obtain parameters using a long-runtime series of underlying asset returns. They then use the information found in option prices toestimate volatility and risk premia. This second stage involves the minimization of an objectivefunction that is just the sum of the squares of the difference in the model-derived Black-Scholesimplied volatilities and the implied volatilities that correspond to the data. Their method forcesconsistency in parameters between the data for the underlying asset and the data for the optionprices.

The problem of option pricing in a stochastic volatility setting was tackled in a series of papers byFouque, Papanicolaou and Sircar (see [12] and the references therein). They developed a methodthat reduced the number of parameters that were needed for pricing and hedging to just three. Twowere derived directly from the smile associated with options prices, and the other was some estimateof the underlying’s volatility. They introduced a type of model where the stochastic volatility factorfollowed a fast mean-reverting process. The data set of options they then considered was chosen sothat the asymptotics would be valid. For example, the strikes of the options were near at-the-moneyand the options had expiration dates that were neither close nor far away. With this data set theyfound that their parameters were stable over time. Importantly, the option pricing equations theyobtained within the fast mean-reverting regime were independent of the stochastic volatility drivingfactor. This was not the case in the entropy-based estimation procedure of Carmona and Xu [8].A subsequent paper of Fouque et al. [14] extended the domain of applicability by introducingstructure to the maturity cylces of options. We will not pursue that method here.

In this paper, we introduce a method for calibrating volatility surfaces in a stochastic volatility

4

environment. We use an entropy-inspired framework that allows flexibility in volatilities whenpassing from the prior probability law to the pricing one. Because we are unable to observe theunderlying stochastic volatility process, we adopt the framework of Fouque, Papanicolaou andSircar and consider a regime where fast mean-reversion is apparent. We also relax the pricingconstraints so that our volatility surface is less sensitive to unreliable out-of-the-money optionprices. We formulate and solve a Hamilton-Jacobi-Bellman (HJB) equation associated with thevariational problem. An HJB equation in a fast mean-rversion setting was studied by Sircar andZariphopoulou [20] in the context of portfolio optimization. Our problem is different and resultsin a different HJB equation. From this equation, we obtain a volatility surface that consistentlyincorporates the Fouque-Papanicolaou-Sircar stochastic volatility parameters. These parametersare found to be stable over time. Our method, in fact, permits a natural interpolation between themodel parameters of Fouque, Papanicolaou and Sircar and our own prior. It is an interpolationthat depends on the level of pricing uncertainty.

The rest of this paper is organized as follows. In Section 2 we formulate the problem and introduce amethod for obtaining the volatility surface. We review the fast mean-reversion regime and simplifythe corresponding HJB equation in Section 3. We reduce the complexity of the problem by studyingfast mean-reversion asymptotics in Section 4. In Section 5 we describe the iterative procedure thatwas used to estimate the surface, while in Section 6 we detail the numerical methods we used tosolve the resulting partial differential equations. In Section 7 we present the results of our method.We discuss the sensitivity of the method to the various parameters in Section 8. We end with abrief summary and conclusion.

2 Stochastic volatility in the relative entropy framework

2.1 A simplified form for the relative entropy

In the relative entropy framework that we follow, we allow the functional form of the volatility tochange from P0, our prior probability law, to P , the probability law we wish to determine, becauseotherwise we would only have freedom to pick a risk premium. We therefore adapt the method ofAvellaneda et. al. to the case of stochastic volatility.

In order to do this, we discretize in time the stock and stochastic volatility processes so that wemay determine the most singular part of the relative entropy. We assume the following dynamicsunder P0:

dSt

St= rdt + σ0(St, Yt)dWt

dYt = γ0,tdt + κdZt

where, as before, we assume the correlation between the two Brownian motions is ρ. In other words,

dZtdWt = ρdt.

5

Here, r is the riskless interest rate process and σ0 is our prior estimate of the volatility surface. Yt

is the stochastic volatility driving factor, which has a drift of γ0 and has its own volatility κ. UnderP , we assume that the dynamics are given by:

dSt

St= rdt + σ(St, Yt)dW ∗

t

dYt = γtdt + κdZ∗t (1)

Here, γ and γ0 are functions that may depend on both t and Yt, and again we assume the samelevel of correlation between the Brownian motions.

Details of the procedure used to get the relative entropy given the dynamics above are provided inAppendix A. The result is that the relative entropy generated per unit of time is just

η(σ) = (σ2 − σ20)

2

when σ2 is not very far from σ20. This particular form is an unsurprising consequence of approxi-

mating a distance function.

2.2 Formulation of the variational problem

Since we wish to minimize the relative entropy between our prior and a model that fits the data,we consider the optimization problem

supσ

−EP

[∫ T

0η(σs)ds

]

subject to the constraints:∣

∣EP[

e−rTihi(STi)]

− Ci

∣

∣ ≤ βi,

for each i and where βi ≥ 0. Here, hi(STi) is the payoff of option i, which has expiration date Ti.

Ci is the price of this option that is quoted in the market.

The constants βi may be considered to be a measure of our confidence in the market data. Wenote that in previous works involving relative entropy minimization, these constants were taken tobe zero (hard price constraints). By allowing deviation from market prices under our new measureP , we have softened the pricing constraints, since it is not practical for us to consider the marketdata as being exact. Some data points may be the result of only one trade and hence would notbe reliable. Imposing hard constraints also leads to kinks in the smile near the data points.

We simplify the objective above by forming a single Lagrangian. The following analysis is similarto the one in [9] for a different problem. Our objective may be rewritten as

infσ

EP

[∫ T

0e−rsη(σs)ds

]

subject to

∀i : EP[

e−rTih(STi)]

− Ci ≤ βi (λ+i )

∀i : Ci − EP[

e−rTih(STi)]

≤ βi (λ−i ).

6

Here, Lagrange multipliers are written next to the constraints. From this, we obtain the augmentedLagrangian which we try to minimize over both sets of λ:

supσ

−EP

[∫ T

0e−rsη(σs)ds

]

+∑

i

(λ−i − λ+

i )(EP[

e−rTih(STi)]

− Ci) +∑

i

βi(λ+i + λ−

i ). (2)

Without loss of generality, we may assume that, for each i, at most one of λ+i and λ−

i is nonzero.This is because we are minimizing over λi, and so we could otherwise decrease both λi’s by an equalbut positive amount. We would then obtain a new expression where the third term in Equation(2) is decreased but the other two are unchanged.

Now suppose that λ+i is nonzero (at the minimum). Then our solution is the same as the one

for an optimization with the first constraint replaced by equality for that value of i. In this case,the second constraint cannot simultaneously be an equality and the solution must therefore be aninterior one. Similarly, if the λ−

i is nonzero, then the second constraint is an equality. These are theonly two possible cases. If we now write for each i, λi = λ+

i − λ−i , we may consider the objective:

supσ

−EP

[∫ T

0e−rsη(σs)ds

]

+∑

i

λi(EP[

e−rTih(STi)]

− Ci) +∑

i

βi|λi|. (3)

If we have to minimize this over λi (for each i), then there are two cases. First, the λi, forsome i, may be positive, in which case we have solved the original optimization problem with thefirst inequality replaced by equality. Or, the λi, for that same i, is negative, in which case theoriginal problem with the second constraint replaced by equality has been solved. Hence, the twoproblems given in Equations (2) and (3) are equivalent, as they both reduce to the same cases onceminimization over λi has been performed.

We may write an indirect value function

V λ(t, s, y) = supσ

−EPt

[∫ T

t

e−r(s−t)η(σs)ds

]

+∑

i

λiEPt

[

e−r(Ti−t)h(STi)]

.

We note that our value function discounts the entropy to make the subsequent expressions simpler.Our objective in Equation (3) amounts to minimizing the expression

V λ(0, S, y) −∑

i

λiCi +∑

i

βi|λi| (4)

over λ.

We know that this V λ solves the HJB equation that is derived in Appendix B and has the form

Vt − rV + rSVS + γVy +1

2κ2Vyy + Φ(

1

2S2VSS , ρκSVSy) = −

∑

i

λiδ(t − Ti)hi(S).

V (T, S, y) = 0

7

Here,Φ(X1, X2) = sup

σX1σ

2 + X2σ − η(σ).

Before describing how to obtain a volatility surface from these equations, it is worthwhile notingsome features of the HJB equation. With ρ = 0, the HJB equation becomes much simpler becauseΦ(X, Y ) may be directly evaluated. The equation decouples and the value function would solve thesame PDE that was found in Avellaneda et al. [5].

The volatility surface may be derived from the value function as follows. Once we solve for theλ = λ∗ that minimizes Equation (4), we may write the desired volatility surface as

σ∗(S, y, t) = arg supσ

1

2S2V λ∗

SSσ2 + ρκSV λ∗

Sy . (5)

We have thus described a way to incorporate stochastic volatility in a relative entropy framework.We did this by first determining an appropriate form for the relative entropy. Once we did this,however, encoding our objective into a value function is standard, and this value function mustsolve an HJB equation. Apart from the form of the relative entropy, our other main contributionso far is the incorporation of pricing uncertainty through the parameters βi.

Although this methodology seems reasonable, there are some issues that need to be addressed.The HJB equation as written is highly dependent on many prior parameters. We need to knowγ, κ and ρ accurately in order to proceed. Since these are all parameters associated with theunobservable stochastic volatility driving factor, determining them is a difficult task. The volatilitysurface, moreover, is dependent on the actual value of the stochastic volatility driving factor. Thenext section develops a way to get around these difficulties by considering a regime in which thestochastic volatility driving factor is fast mean-reverting. This represents the point of departurefrom the standard HJB equation methodology.

3 The fast mean-reversion setting

3.1 A special form for the dynamics of the stochastic volatility driving process

We have so far dealt with the presence of stochastic volatility in a fairly general way. We nowconsider some special models for the dynamics of the volatility driving factor, Y . One feature thatmost models of stochastic volatility incorporate is mean reversion. This refers to the tendency ofthe process to go back to its invariant or long-run distribution. A particularly tractable model thatexhibits mean reversion is the Ornstein-Uhlenbeck process

dYt = α(m − Yt)dt + κdZt.

8

Here, α is the rate of mean reversion, while m is the long-run mean of the process. We suppose thatα, κ and m are constants. Moreover, Zt is a Brownian motion that is correlated to the Brownianmotion that drives the stock price process, Bt, with correlation ρ.

The invariant distribution for this process is a Y0 that satisfies

E[Lg(Y0)] = 0

for any smooth and bound g, where

L = α(m − y)∂

∂y+

1

2κ2 ∂2

∂y2.

If we let Ψ(y) be the density function of the invariant distribution, it is easily seen that

Ψ(y) =1√

2πν2exp

(

−(y − m)2

2ν2

)

, (6)

where

ν2 =κ2

2α.

This is exactly the normal density with mean m and variance ν2, which tells us that the parameterν controls the size of the equilibrium fluctuations from the mean, m.

The fast mean reversion limit was considered in Fouque, Papanicolaou and Sircar [12]. Theyfound empirical evidence for fast mean reversion in options data. Analytically, fast mean reversioncorresponds to the limit α → ∞. Care, however, must be taken to ensure that the invariantdistribution remains the same as we take this limit. In other words, we would like our model tohave fluctuations of the same size regardless of how quickly the volatility reverts to its mean. Wetherefore take

κ = ν√

2α

for some constant ν.

If this is the specification of Y under the physical measure, then to price derivatives we need toconsider the process under the risk-neutral law. Suppose that under some physical probability lawthe stock and volatility driving process follow

dSt = µStdt + σ0(St, Yt)dB0,t

dYt = α(m − Yt)dt + ν√

2αdZ0,t.

Then, under a risk-neutral law, the processes must follow, by Girsanov’s theorem,

dSt = rStdt + σ0(Yt)dB∗0,t

dYt =

(

α(m − Yt) − ν√

2αρµ − r

σ0− ν

√2αγ0,t

√

1 − ρ2

)

dt + ν√

2αdZ∗0,t. (7)

9

Here, γt is a risk-premium factor or the market price of volatility risk that parametrizes the spaceof equivalent martingale measure. By taking, γ0,t = γ0(t, St, Yt), we specifically restrict ourselvesto a Markovian setting. Equation (7) is the specification of the dynamics of the processes S and Yunder our prior risk-neutral probability law.

We now just need to find the relative entropy between another risk-neutral probability law and ourprior. We assume that under P the price of the stock evolves as

dSt = rStdt + σ(St, Yt)dB∗t

dYt =

(

α(m − Yt) − ν√

2αρµ − r

σ− ν

√2αγt

√

1 − ρ2

)

dt + ν√

2αdZ∗t . (8)

With these particular assumptions on the dynamics of Yt, the form of η(σ) to first order in ρ isunchanged from the analysis in Appendix A.

It is worth noting one important aspect of the dynamics under the prior probability law. We haveassumed that

σ0(St, Yt) = σ0(Yt).

In other words, our prior estimate of the volatility is dependent only on the stochastic volatilitydriving factor and not on the stock price. With this choice of the functional form σ0 now correspondsto function f in the work of Fouque, Papanicolaou and Sircar [12]. Although this particular formseems like a heavy restriction, it is not. The reason is that the fast mean-reversion setting doesnot need an explicit functional form for the prior volatility estimate. Instead, this setting replacesexplicit functional forms with observable quantities and averages. This will be made clear later.

3.2 Simplifying the HJB equation

The HJB equation we need to solve is now a little different to the one we derived earlier. Becauseof the appearance of the σ in the denominator of one of the diffusive terms for Yt in Equation (8),Φ must now be a function of three variables. Specifically, we define Φ as:

Φ(X1, X2, X3) = supσ

[

σ2X1 + ρX2σ + ρX31

σ− η(σ)

]

. (9)

The HJB equation thus becomes:

Vt − rV + rSVS + (α(m − y) − ν√

2αγ√

1 − ρ2)Vy + ν2αVyy

+ Φ

(

1

2S2VSS , ν

√2αSVSy,−ν

√2α(µ − r)Vy

)

= −∑

i

λiδ(t − Ti)hi(S) (10)

Before we carry out the asymptotic analysis we simplify Φ for σ close to σ0 and for ρ small. Wetherefore expand σ in ρ (a small ρ expansion). Let:

σ = σ + Aρ

10

where A is a coefficient to be determined. Here, σ is taken to be the maximizer when ρ = 0, whichis just given by σ2 = X1/2 + σ2

0. The first-order condition for a maximum gives:

2σX1 + ρX2 − ρX3

σ2− 4σ(σ2 − σ2

0) = 0.

Upon substituting for σ, we find that the coefficient for ρ is

2AX1 + X2 −X3

σ2− 4A(σ2 − σ2

0) − 8Aσ2 = X2 −X3

σ2− 8Aσ2.

Equating the coefficient of ρ to zero yields:

X2 −X3

σ2− 8Aσ2 = 0,

which implies that

A =X2 − X3

σ2

4X1 + 8σ20

.

So, to first order in ρ:

Φ(X1, X2, X3) =X2

1

4+ σ2

0X1 + ρX2σ + ρX3

σ

which, upon expanding σ, leads to:

Φ(X1, X2, X3) =X2

1

4+ σ2

0X1 + ρX2

(

X1

2+ σ2

0

) 1

2

+ ρX3

(

X1

2+ σ2

0

)− 1

2

.

Given this form of Φ, we note that the maximizing value of σ is clearly approximated by σ.

We remark also that the small ρ expansion may be justified a posteriori. We will find that they-dependence in our value function only occurs at the 1

αscale. This implies that, since we find

X2 ∝ √αVxy, X2 scales like 1√

α, which is small for large α. A similar argument holds for X3.

Because ρ always premultiplies X2 or X3, the small ρ expansion of Φ is reasonable.

We now have a suitable form of the HJB equation, which uses the small ρ approximation for Φ.

Vt − rV + rSVS + (α(m − y) − ν√

2αγ√

1 − ρ2)Vy + ν2αVyy +1

4

(

1

2S2VSS

)2

+ σ20

1

2S2VSS

+ ρν√

2αSVSy

(

1

4S2VSS + σ2

0

) 1

2

− ρν√

2α(µ − r)Vy

(

1

4S2VSS + σ2

0

)− 1

2

= −∑

i

λiδ(t − Ti)hi(S)

V (0, S, y) = 0 (11)

11

4 Fast mean-reversion asymptotics applied to the HJB equation

In order to apply fast mean-reversion asymptotics, we suppose that α is large. For book-keepingpurposes, we replace α by α

ε. We consider the value function corresponding to this situation in the

limit as ε → 0.

We first let x = log S (which means that 12S2VSS = 1

2(Vxx − Vx)). Substituting this into Equation(11) yields

Vt − rV + (r − 1

2σ2

0)Vx +1

2σ2

0Vxx + (α(m − y) − ν√

2αγ√

1 − ρ2)Vy + ν2αVyy

+1

16(Vxx − Vx)2 + ρν

√2αVxy

(

Vxx − Vx

4+ σ2

0

) 1

2

− ρν√

2α(µ − r)Vy

(

Vxx − Vx

4+ σ2

0

)− 1

2

= −∑

i

λihi(x)δ(t − Ti). (12)

Here,σ0 = σ0(Yt).

We now replace α by αε

so that we may apply some asymptotic techniques. We therefore write ourequation as

(

L2 +1√εL1 +

1

εL0

)

V +(Vxx − Vx)2

16+

1√ερν

√2αVxy (g(Vxx, Vx, y) − 1)

− 1√ερν

√2αVy

(

1

g(Vxx, Vx, y)− 1

)

−∑

i

λihi(x)δ(t − Ti) (13)

where

L2 =∂

∂t− r + (r − 1

2σ0(y)2)

∂

∂x+

1

2σ0(y)2

∂2

∂x2

L1 = ρν√

2α∂2

∂x∂y− ν

√2αγ

√

1 − ρ2∂

∂y− ρν

√2α(µ − r)

∂

∂y

L0 = α(m − y)∂

∂y+ ν2α

∂2

∂y2

g(Vxx, Vx, y) =

(

Vxx − Vx

4+ σ2

0

) 1

2

. (14)

If we suppose thatV0,xx−V0,x

4 is small in relation to σ0, which is reasonable as long as our prior issomewhat close to the actual σ, then we may approximate g as just

g(y) = σ0(y). (15)

This approximation is consistent with the other approximations we have made in deriving a formof the relative entropy. We note that this approximation for g is also what we would have if wehad happened to pick a prior that matched the prices within the tolerance desired.

12

We also expand V in powers of ε

V = V0 +√

εV1 + εV2 + ε√

εV3 + ....

4.1 The leading order term

In order to solve for the V0 term we equate the expansions of the HJB equation at the first fewscales. At the 1

εscale

L0V0 = 0,

which implies that V0 = V0(t, x) (it is independent of y).

At the 1√ε

scale

L0V1 + L1V0 + ρν√

2αV0,xy(g(V0,xx, V0,x, y) − 1) − ρν√

2α(µ − r)V0,y(g(V0,xx, V0,x, y)) = 0

but since V0 is independent of y, the equation becomes

L0V1 = 0

and this implies, as before, V1 = V1(t, x).

At the order 1 scale

L2V0 + L1V1 + L0V2 +1

16(V0,xx − V0,x)2 = −

∑

i

λihi(x)δ(t − Ti),

where, again, we have used that V1 and V0 is independent of y to get rid of the Vxy part of thenonlinearity. Similarly, we may get rid of the L1V1 = 0. Solvability for V2 requires that theother terms have average 0. But since V0 is independent of y, this means that the equation thatdetermines solvability yields the following equation for V0

LBS(σ0)V0 +1

16(V0,xx − V0,x)2 = −

∑

i

λihi(x)δ(t − Ti). (16)

Here, LBS is the Black-Scholes operator after the log transformation has been performed.Theterminal condition is V0(T, x) = 0. This is exactly the equation that is solved in Avellaneda et. al.

[5]. It appears here as the zero order approximation in our method.

4.2 The correction term

Once we have solved the equation for V0, we may remove the average (which we have set to zero)to get an equation for V2

L0V2 +1

2(σ0(y)2 − σ2)

(

∂2V0

∂x2− ∂V0

∂x

)

= 0,

13

which implies that

V2(t, x, y) = −1

2L−1

0 (σ0(y)2 − σ2)

(

∂2V0

∂x2− ∂V0

∂x

)

.

We may therefore write V2 as

V2(t, x, y) = −1

2(φ(y) + c(t, x))

(

∂2V0

∂x2− ∂V0

∂x

)

, (17)

where φ(y) is a solution of the Poisson equation

L0φ = σ0(y)2− < σ20 > .

Moving on to the next scale (√

ε), the relevant equation is

L2V1 + L1V2 + L0V3 +1

8(V0,xx − V0,x)(V1,xx − V1,x) + ρν

√2αV2,xy(g(V0,xx, V0,x, y) − 1)

− ρν√

2αV2,y(1/g(V0,xx, V0,x, y) − 1) = 0.

Solvability for V3 implies

LBS(σ0)V1 +1

8(V0,xx − V0,x)(V1,xx − V1,x) + ρν

√2α

⟨

g(V0,xx, V0,x, y)∂2V2

∂x∂y

⟩

− ρν√

2α(µ − r)

⟨

1

g(V0,xx, V0,x, x)

∂V2

∂y

⟩

−√

2α√

1 − ρ2

⟨

γ∂V2

∂y

⟩

= 0. (18)

The averages in the last three terms of Equation (18) may be further expanded into terms withderivatives in V0, using Equation (17). Specifically, we obtain the following relations

⟨

g(V0,xx, V0,x, y) ∂2V2

∂x∂y

⟩

= 12 〈gφ′〉

(

∂3V0

∂x3 − ∂2V0

∂x2

)

⟨

1g(V0,xx,V0,x,x)

∂V2

∂y

⟩

= 12

⟨

1gφ′⟩(

∂2V0

∂x2 − ∂V0

∂x

)

⟨

γ ∂V2

∂y

⟩

= 12 〈γφ′〉

(

∂2V0

∂x2 − ∂V0

∂x

)

(19)

So our equation for the correction V1 becomes

LBS(σ0)V1 +1

8(V0,xx − V0,x)(V1,xx − V1,x) + A3

∂3V0

∂x3+ A2

∂2V0

∂x2+ A1

∂V0

∂x= 0, (20)

which is a linear equation with a source term. In particular, the coefficients A1, A2 and A3 aregiven by

A1 = ρν√

α1√2(µ − r)

⟨

1

gφ′⟩

+ ν√

α1√2

√

1 − ρ2⟨

γφ′⟩

A2 = −ρν√

α1√2

⟨

gφ′⟩− ρν√

α1√2(µ − r)

⟨

1

gφ′⟩

− ν√

α1√2

√

1 − ρ2⟨

γφ′⟩

A3 = ρν√

α1√2

⟨

gφ′⟩

14

We note that, with this method, and as it stands now, we need the correlation ρ and the variousparameters of the stationary distribution of the stochastic volatility driving factor.

4.3 Determination of the group parameters

With the approximation given in Equation (15), our parameters may be written as

A1 =√

ε

(

ρν√

α1√2(µ − r)

⟨

1

σ0φ′⟩

+ ν√

α1√2

√

1 − ρ2⟨

γφ′⟩)

A2 = −√

ε

(

ρν√

α1√2

⟨

σ0φ′⟩+ ρν

√α

1√2(µ − r)

⟨

1

σ0φ′⟩

+ ν√

α1√2

√

1 − ρ2⟨

σ0φ′⟩)

A3 =√

ερν√

α1√2

⟨

σ0φ′⟩ . (21)

These parameters, which we call our group parameters, are the same as the Ai but they are nowscaled by

√ε.

We may determine these parameters as combinations of parameters that are observable from thesmile. To do this, we outline the results in Fouque, Papanicolaou and Sircar [12]. In their work,they found that the price of an option, C may be expanded as

C(t, x; T, K) = CBS(t, x; T, K) − (T − t)

(

(2V3 − V2)∂CBS

∂x+ (V2 − 3V3)

∂2CBS

∂x2+ V3

∂3CBS

∂x3

)

,

(22)where CBS is the Black-Scholes option price and x = log(S). Here,

V2 =√

ε

(

ν√

α1√22ρ⟨

σ0φ′⟩− ρν

√α

1√2(µ − r)

⟨

1

σ0φ′⟩

− ν√

α1√2

√

1 − ρ2⟨

σ0φ′⟩)

V3 =√

ερν√

α1√2

⟨

σ0φ′⟩ (23)

We deduce, then, that our approximate Ai’s given by Equation (21 )are exactly

A1 = 2V3 − V2

A2 = V2 − 3V3

A3 = V3.

The parameters V2 and V3 may be determined from the smile. By expanding the price of the optionin powers of ε, we may write

C(t, x; T, K; I) = CBS(t, x; T, K; σ0) +√

εI1∂CBS

σ, (24)

15

where I1 is the correction to the implied volatility. By equating Equations (22) and (24), we findthat the implied volatility I is just:

I = σ0 +V3

σ20

(r +3

2σ2

0) −V2

σ0− V3

σ30

log(K/S)

T − t.

The implied volatility is therefore an affine function of the log-moneyness-to-maturity-ratio

LMMR =log(K/S)

T − t,

which means that we may fit the parameters aFPS and bFPS below to the smile curve determinedby options prices

I = aFPSLMMR + bFPS ,

where

aFPS = −V3

σ30

bFPS = σ0 +V3

σ30

(

r +3

2σ2

0

)

− V2

σ0.

This means that the scaled parameters Ai which are required to determine the correction term√εV1 are explicitly expressed as combinations of the parameters aFPS and bFPS . Moreover, given

any such parameters a and b determined from a smile curve, the parameters Ai are given by

A1 = −2aσ30 − σ0((σ0 − b) − a(r +

3

2σ2

0))

A2 = σ0((σ0 − b) − a(r +3

2σ2

0)) + 3aσ30

A3 = −aσ30 (25)

We have thus expressed the parameters needed for the correction term as simple expressions involv-ing observable quantities. This is the power of the fast mean-reversion setting. Individual estimatesof the parameters of the stochastic volatility models are not needed. Rather, group parameters likethe Ai are all that are needed to not only price options as in Fouque, Papanicolaou and Sircar [12]but also to determine minimal entropy volatility surfaces as in this paper.

4.4 Summary of our analysis

What we have achieved now is a set of equations that incorporates the objectives we wanted

1. Uncertainty over the prior in the form of stochastic volatility that is independent of thespecific form of the volatility since we only really need the group parameters, A1, A2 and A3.

16

2. A value function that does not depend on the stochastic volatility driving factor because ofthe use of fast mean-reversion.

3. Uncertainty in the observed market prices

The asymptotics helped us isolate the leading two terms of the value function that solved the HJBequation. These two terms were, indeed, independent of the stochastic volatility driving factor.The first term, moreover, solved the same equation that was solved in Avellaneda et al. [5].

5 Description of the estimation procedure

Our analysis permits a simplified approach to the estimation of the stochastic volatility surface.There are two possible approaches from here. We may estimate the parameters Ai once or we maytry to self-consistently determine the same parameters in an iteration scheme.

In the first approach, we may determine A1, A2 and A3 by following the smile-fitting proceduregiven in Section 4. We may then solve for V0 using Equation (16). With this V0 and the coefficients{Ai}, we may solve for

√εV1 for each value of λ using Equation (20). Using some gradient search

algorithm, we may then minimize (over λ) V (0, x0)−∑

i λiCi+∑

i |λi|βi, where V = V0+√

εV1 andx0 is the log of the current stock price. Finally, the appropriate derivative of V gives a volatilitysurface as described by Equation (5).

We choose to iterate this scheme. We may obtain A1, A2 and A3 from the prices as before, solvefor V0 and V1 and obtain a volatility surface. But instead of stopping there, we price the same setof options in our data set again using the newly derived volatility surface and use these prices toget adjusted estimates for A1, A2 and A3. Repeating this procedure a few times then leads to aconverged set of parameters. The fact that our procedure leads to a converged set of parameterstells us the group parameters are intrinsic to the volatility surface as a whole and that they maybe self-consistently determined. They thus have the same role in volatility surface estimation asthey do in option pricing.

We call this method the volatility surface iteration procedure. It is used in our numerical calcula-tions and is summarized succinctly as

1. Fit the implied volatility smile to an affine function of the log-moneyness-to-maturity ratio.This affine fit gives us the parameters aFPS and bFPS from which we obtain A1, A2 and A3.

2. Solve for V0 and√

εV1.

3. Obtain a volatility surface.

4. Price options again using the volatility surface in Step 3. We do this using the Black-ScholesPDE with volatility σ.

17

5. Derive the implied volatility smile from the prices obtained in Step 4. Fit this smile to thelog-moneyness-to-maturity ratio to obtain parameters a and b. Obtain new estimates of theparameters A1, A2 and A3 from the fitted a and b. .

6. Stop if adjusted parameters differ from the previous iteration by less than δ, a small parameter(which we take to be of order 10−5 in the l2 norm). Otherwise, repeat the procedure fromStep 2.

6 Numerical Methods

6.1 The highest-order equation

Supposing that the stock pays dividend of d (constant), then the function V0(x, t) satisfies theequation:

supσ

(

V0,t − rV0 + (r − d)V0,x +1

2σ2(V0,xx − V0,x) − (σ2 − σ2

0)2

)

= −∑

i

λihi(ex)δ(t − Ti)

Here, we choose σ ∈ [σmin, σmax] in order to preserve the stability of our numerical scheme. Now,using the substitution V0 = e−rtV0, yields:

supσ∈[σmin,σmax]

(

V0,t + (r − d)V0,x +1

2σ2(V0,xx − V0,x) − e−rt(σ2 − σ2

0)2

)

= −∑

i

λihi(ex)e−rtδ(t − Ti)

We discretize the interior equation (spatial h, temporal ∆t), and supposing that we have the optimalσ at time t, we have the equation (after dropping the tildes):

V0(x, t) − V0(x, t − ∆t)

∆t+

1

2σ2 V0(x + h, t) + V0(x − h, t) − 2V0(x, t)

h2

+ (r − d − 1

2σ2)

V0(x + h, t) − V0(x − h, t)

2h− e−rt(σ2 − σ2

0)2

= −∑

i

e−rtλihi(ex)δ(t − Ti)

In order to regularize the problem, we replace each option payoff e−rthi(ex)δ(t − Ti) by Pi =

C(ex, ∆t, σ)1t≤Ti<t+∆t, where C is the price of an option with payoff hi and σ is the volatility fromthe previous time step. This type of regularization was used, for example, in [14].

18

We may now write an equation for the value function at (x, t − ∆t) as:

V0(x, t − ∆t) = V0(x, t)

(

1 − ∆tσ2

h2

)

+ V0(x − h, t)

(

∆t

(

σ2

2h2− r − d − 1

2σ2

2h

))

+ V0(x + h, t)

(

∆t

(

σ2

2h2+

r − d − 12σ2

2h

))

− (σ2 − σ20)

2∆te−rt

+∑

i

λiPi

The terminal condition is taken to be V (T, x) = 0.

Finally, we specify how to get σ. We notice that we are taking the supremum in a closed interval,and we have therefore that:

σ2 =1

4ert(V0,xx − V0,x) + σ2

0 (26)

but if the right hand side in this equation is bigger than σ2max, then σ2 = σ2

max and, similarly, if itis smaller than σ2

min, then σ2 = σ2min.

As written our discretized scheme is itself an HJB equation. In order for this interpretation to bevalid, we need the condition

1 − ∆tσ2

h2≥ 0

for stability, and, hence, we take h2 ≥ σ2max∆t.

We note that in our implementation, our grid is cut off 3.5 standard deviations away from theunderlying price of the stock. Furthermore, the boundary conditions are chosen so that σ = σ0 atthe boundary.

This numerical scheme is more transparent than the trinomial tree-based scheme presented in the[5]. Once we have the volatility surface from our scheme, the derivative of V0 with respect to λi

may be readily computed by discretizing the linear PDE for V0,i = ∂V0

∂λi. This PDE is just:

LBS(σ)V0,i = −Pi (27)

with terminal condition 0, as before. These derivatives are needed for the optimization over theλi’s.

The PDE (27) tells us that V0,i = λi (which is the first-order condition for optimality in the workof [5]) is solvable as long as σmax and σmin are chosen so that the interval [σmin, σmax] contains theimplied volatility of option i.

19

6.2 The PDE for the correction term

Our correction equation may be written as:

LBS(σ)V1 +

(

A1∂3V0

∂x3+ A2

∂2V0

∂x2+ A3

∂V0

∂x

)

= 0,

where V1 =√

εV1. This is a linear PDE and a simple explicit scheme may be employed. Noticethat one needs the solution V0 from the first PDE in order to solve this PDE because we need σ.

In order to get a prior for the parameters A1, A2 and A3, we use the procedure outlined in Section4. We fit the smile (the implied volatilities) to the log-moneyness-to-maturity-ratio of the optionsand then use Equation (25) to determine the parameters. We do this at the start with quoted pricesand then at each subsequent iteration with newly calculated prices from intermediate estimationsof the volatility surface.

7 Numerical Results

The fast mean-reversion asymptotics cannot be expected to apply to options covering all expirationdates and all strikes. The stochastic volatility driving factor cannot be averaged for options thatare too close to expiration as in these situations the volatility term has not had enough time tofluctuate. Options that are far away from maturity may not be used either as other effects such asslowly varying volatility may dominate the fast mean-reversion effects. Similarly, options that aretoo far out-of-the-money or too far in-the-money should not be used.

It is for these reasons that the actual procedure we use does not correct the volatility surface allthe way up to expiry. The reason for this is that the correction is not valid all the way to expirybecause the fast mean-reversion correction does not apply to short-dated options. It is thereforereasonable to expect that this is also true of the correction to the volatility surface close to expiry.Hence, our results are derived from corrections to the volatility surface up to 35 days from expiry.

Our data set consists of options on the S&P 500 index between 3 May, 2004 and 12 May, 2004.Out of this data set, we pick call options that have strikes within 5% of the at-the-money level andthat expire on 19 June, 2004. To allow for enough distance between the strikes so that a very finemesh is not needed in the numerical procedures, we make sure that the strikes of the call optionswe choose are at least 10 apart. This means that on each day, we have between 6 and 8 optionsfrom which to build a volatility surface. We choose 3/4 of the bid/ask spread on the options to bethe uncertainty in our prices (our βs).

Table 1 shows the results of the iterative procedure for finding the Ais on 3 May, 2004. Startingfrom a prior derived from the work of Fouque, Papanicolaou and Sircar, each succeeding row of theAi represents coefficients derived from the volatility surface, until convergence in the Ai is achieved.

20

iteration A1 A2 A3

1 -0.00100 0.00072 0.000282 -0.00294 0.00267 0.000273 -0.00298 0.00271 0.000274 -0.00296 0.00269 0.000275 -0.00298 0.00270 0.00027

Table 1: Results of iterating the FPS parameters for 3 May, 2004 for options on the S&P 500 index.The first iteration has parameters that are equal to the original FPS estimates.

Date aFPS bFPS a b

05/03/2004 -0.0688 0.1525 -0.0664 0.140205/04/2004 -0.0339 0.1416 -0.0310 0.134805/05/2004 -0.0861 0.1518 -0.0870 0.126705/06/2004 -0.0666 0.1573 -0.0529 0.149105/07/2004 -0.0518 0.1588 -0.0424 0.149605/10/2004 -0.0422 0.1674 -0.0318 0.163005/11/2004 -0.0414 0.1595 -0.0177 0.150105/12/2004 -0.0727 0.1763 -0.0629 0.166005/13/2004 -0.0459 0.1641 -0.0156 0.155105/14/2004 -0.0571 0.1679 -0.0373 0.1574

Table 2: FPS parameters that are obtained from an affine fit of the implied volatility to the log-moneyness-to-maturity ratio, using the original FPS results and using our procedure. These arethe parameters that are shown to be stable in previous work, and are seen to be stable here too.

If this is done for each of the days that we correct the surface, we obtain the following series ofvalues for the parameters a and b. Figure 1 graphically shows the comparison between the two setsof coefficients.

The other part of this framework is the volatility surface obtained. We show the effects of thecorrection to the volatility surface for 3 May, 2004 in Figures 2-4. The two surfaces incorporateuncertainty in prices, while the last figure shows the difference between the corrected and uncor-rected surfaces as a percentage of the uncorrected surface for the duration of the correction. Wecan see that the stochastic volatility correction introduces a ripple into the surface (because of thethird derivative term). This ripple, though smaller in magnitude, is still present 3 days later, ascan be seen in Figure 5.

8 Sensitivity to parameters

It was shown in [5] that the value function is convex in the λi for the case where the Ai are all zero.When the Ai are not all zero, numerical computations show that the value function need not have

21

1 2 3 4 5 6 7 8 9 10−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

aFPS

ab

FPS

b

Figure 1: Comparison of parameters obtained via the FPS theory and via our new method

010

2030

4050

800

1000

1200

1400

16000.1

0.12

0.14

0.16

0.18

0.2

0.22

Trading daysS&P 500 index price

σ un

corr

ecte

d

Figure 2: Uncorrected volatility surface for 8 European call options on the S&P 500 index, struckon May 3, 2004, and expiring on June 19, 2004. The strikes of these call options were 1050, 1075,1085, 1100, 1110, 1120,1130 and 1140. σ0 = 0.16, σmin = 0.1, σmax = 0.25, r = 0.119 and thedividend rate, d was 0.017, while the underlying price was 1117.49.

22

010

2030

4050

800

1000

1200

1400

16000.1

0.15

0.2

0.25

Trading daysS&P 500 index price

σ co

rrec

ted

Figure 3: Corrected volatility surface for the same 8 European call options on the S&P 500 index

0

5

10

15

800

1000

1200

1400

1600−0.15

−0.1

−0.05

0

0.05

Trading daysS&P 500 index price

diffe

renc

e in

σ a

s fr

actio

n

Figure 4: The difference (as a fraction) between the corrected volatility surface and the uncorrectedvolatility surface, shown for the first 15 days of the correction.

23

02

46

810

800

1000

1200

1400

1600−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Trading daysS&P 500 index price

diffe

renc

e in

σ c

orre

cted

and

σ u

ncor

rect

ed

Figure 5: The difference (as a fraction) between the corrected volatility surface and the uncorrectedvolatility surface, shown for the first 10 days of the correction for the trading date 6 May, 2004.

a stationary point. Large values of Ai, in particular, can make it impossible to find a minimum.Figures 8 and 7 show what happens to the objective function (V − λiCi + βi|λi|) as we vary theparameters A1 and β1 for a typical day.

The problem of obtaining a minimum is solved in two ways in our procedure. Firstly, we onlycorrect the volatility surface in the regime where we expect the fast mean-reversion asymptotics tohold, i.e. far enough away from maturity. Secondly, we make the βi nonzero, thereby including someprice uncertainty. With everything else the same, nonzero βi forces λ to be smaller in magnitude.We can see this in Figure 8. In fact, as βi gets larger, λi goes to zero, which means that large priceuncertainty forces our volatility surface to be constant and equal to the prior volatility σ0 at allpoints. This means that for these large values, a = 0 and b = σ0.

Although we cannot expect our volatility surface to be the same for all values of β, we can seefrom Table 3 that over the range of βi that are practical (and lead to convergence in our numericalscheme), the reduced parameters a and b are fairly similar. More importantly, Table 3 tells us alot about the role that price uncertainty has on the determination of the volatility surface. Priceuncertainty turns out to be very important as a lot of price uncertainty implied that the bestvolatility surface under our objective of minimizing deviation from the prior is to use the prioritself. In this case, however, we are not really fitting the data. Rather we are taking a positionon it. With absolute confidence in the prices, we believe the market entirely and our stochasticvolatility surface estimation procedure will rarely converge. Table 3 tells us that the middle groundbetween complete market certainty and historical estimates of parameters may be reached with ourestimation procedure.

24

−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

λ1

Obj

ectiv

e F

unct

ion

Figure 6: Using data from 6 May, 2004, we plot the dependence of the objective function on λ1 (allother λ’s are zero) for three cases: i) ’-’: Uncorrected surface; ii) ’•’: FPS fitted values of Ai areused with zero βi; iii) ’�’: FPS fitted values with βi derived from the bid/ask spread.

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.5

0

0.5

1

1.5

2

2.5

3

3.5

λ1

Obj

ectiv

e F

unct

ion

Figure 7: Changing the Ai to one hundred times the original value can lead to a loss of a minimumas can be seen in curve ’×’; however, increasing the value of β to five times its original value canrestore that convexity.

25

β value a b

0 - -β4 -0.0841 0.1387β2 -0.0727 0.1467β -0.0529 0.149132β -0.0186 0.15012β 0.0002 0.1600

Table 3: Dependence of parameters a and b on β for options whose prices are given on 6 May, 2004.β here is 3/4 the bid/ask spread. The first line indicates that there was no convergence when thevolatility surface was corrected, even for the prior values of Ai.

9 Conclusion

We formulate the problem of determining the volatility surface from option prices when the under-lying price process exhibits stochastic volatility and when there is some uncertainty in the observedmarket prices of the options. Using fast mean-reversion asymptotics, we simplify the problem andrewrite our HJB equation so that the leading-order terms were no longer dependent on the un-observable stochastic volatility driving factor. Importantly, other parameters associated with thestochastic volatility model can be obtained directly from the options prices and do not need sepa-rate estimation. With this analysis in hand, we solve the HJB equation and determine the groupparameters derived from the options’ smile. Our procedure admits an iterative determination ofthese stochastic volatility parameters, which are the same parameters found earlier in Fouque, Pa-panicolaou and Sircar [12]. The iteration makes these parameters consistent with the equationsthat determine the volatility surface.

Our numerical results showed that the stochastic volatility group parameters determined throughthis iterative procedure are reasonably stable in time. The parameters are also stable with respectto the volatility surface iteration procedure. This added level of stability can only be expected toarise from a model such as ours because it reduces the complexity of the estimation problem. Thegroup parameters are thus intrinsic to the entire surface because the equations that determine thesurface involve the same parameters that are obtained from the options’ smile. More importantly,we found that uncertainty in prices significantly affect both the group parameters and the volatilitysurface itself. The bigger the uncertainty, the more our surface resembles our prior, while thesmaller the uncertainty, the closer the parameters are to the original parameters found in Fouque,Papanicolaou and Sircar [12]. Our procedure provides a natural way of interpolating between thesetwo cases.

We hope to extend our work to incorporate maturity cycles and multiple expiration dates as in[14]. It is also believed that the proofs about the validity of our approximations which are foundin [13] should extend to the present problem. We leave this to future work.

26

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[1] Y. Achdou and O. Pironneau. Volatility smile by multilevel least square. International Journal

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[2] L.B.G. Andersen and R. Brotherton-Ratcliffe. The equity option volatility smile: an implicitfinite difference approach. Journal of Computational Finance, 1:5–38, 1997.

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A Relative entropy in a stochastic volatility setting

We have the following dynamics under P0, our prior,

dSt

St= rdt + σ0(St, Yt)dWt

Yt = γ0,tdt + βdZt

where, as before, we assume the correlation between the two Brownian motions is ρ. Under P , wehave

dSt

St= rdt + σ(St, Yt)dW ∗

t

Yt = γtdt + βdZ∗t

In what follows, we assume that γ = γ0.

Instead of using a trinomial tree-based discretization, as in Avellaneda et. al. [5] we choose to usea simple Euler discretization for the Xt ≡ log(St) and Yt processes under P .

Xn+1 = Xn + (r − 1

2σ2)∆t + σ

√∆tε1

Yn+1 = Yn + γ∆t + β√

∆tε2

We may do the same for the processes under P0 with no change except for subscripts on some ofthe parameters. Here, ε1 and ε2 are each normally distributed but have correlation ρ.

The relative entropy we seek is a discrete approximation to the general form

H(P |P0) = EP

[

lndP

dP0

]

.

28

In the discrete-time case, this may be wrriten as

H(P |P0) =∑

paths

(

N−1∏

n=0

πPn

)

ln

(

∏

πPn

∏

πP0n

)

= EP

[

ln

(

∏

πPn

∏

πP0n

)]

= EP

[

∑

n

EPn

[

ln

(

πPn

πP0n

)]

]

Here, En is the conditional expectation taken with respect to the information available up to timen and πP

n and πP0

n are just the probabilities associated with one step along any given path undereach of the probability laws.

We therefore just need to work out the log-likelihood ratio to calculate the relative entropy. First,however, we calculate the associated probabilities.

πPn =

1

2πσβ√

1 − ρ2exp

−

(

(x−µ)2

σ2 − 2ρx−µσ

y−µy

β+

(y−µy)2

β2

)

2(1 − ρ2)

,

is the likelihood function for a bivariate normal distribution and, similarly,

πP0

n =1

2πσ0β√

1 − ρ2exp

−

(

(x−µ0)2

σ2

0

− 2ρx−µ0

σ0

y−µy

β+

(y−µy)2

β2

)

2(1 − ρ2)

.

Here, µ = Xn +(r− 12σ2)∆t and a similar expression holds for µ0, while µy = Yn +γ∆t. Evaluating

the log likelihood gives us the following expression

−1

2ln

σ2

σ20

+1

2(1 − ρ2)

(

(x − µ0)2

σ20

− (x − µ)2

σ2+ 2ρ

(

(x − µ)(y − µy)

σβ− (x − µ0)(y − µy)

σ0β

))

,

which we average against the joint probability density of the bivariate normal. We do this becausewe want to find

EPn

[

lnπP

n

πP0n

]

.

This yields

−1

2ln

σ2

σ20

+1

2(1 − ρ2)

(

−1 +σ2

σ20

)

+ρ2

1 − ρ2

(

1 − σ

σ0

)

.

We may expand this expression for small σ2 − σ20, which implies

EP

[

lnπP

n

πP0n

]

≈ 1

4σ40

(

(σ2 − σ20)

2 +2ρ2σ2

0(3σ0 + σ)(σ0 − σ)

1 − ρ2

)

.

29

If we suppose that ρ is small, we may omit the last term in our entropy. We therefore choose ourentropy generated per unit of time to be

η(σ) = (σ2 − σ20)

2. (28)

We note here that a similar entropy (with a ρ2 term also) may be obtained using a trinomial tree-based discretization, where care must be taken to appropriately find the joint probabilities giventhe correlation. This approach yields the same entropy as in Equation (28).

B A Derivation of the HJB equation under our new formulation

Consider the dynamics of e−rtV (St, Yt, t):

(

e−rtV (St, Yt, t))

= e−rt

(

Vt − rV +1

2σ2S2VSS + rSVS + γVy

+1

2β2Vyy + ρσβSVSy)

)

dt

+ Brownian (under P) terms

The above equation comes from applying Ito to the expression and using the dynamics under Pthat we have assumed (Equation (1)). Let

Φ(X, Y ) = supσ

Xσ2 + Y σ − η(σ).

From the definition of Φ, we know that:

Xσ2 + Y σ ≤ η(σ) + Φ(X, Y )

letting X = −12S2VSS and Y = ρβSVSy, we may therefore write (at least formally), the following

inequality

d(

e−rtV (St, Yt, t))

≤ e−rt

(

Vt − rV + rSVS + γVy +1

2β2Vyy + Φ(

1

2S2VSS , ρβSVSy) + η(σ)

)

dt

+ Brownian (under P) terms

Suppose we solve our HJB equation:

Vt − rV + rSVS + γVy +1

2β2Vyy + Φ(

1

2S2VSS , ρβSVSy) = −

∑

i

λiδ(t − Ti)hi(S).

Then the inequality above reads

d(

e−rtV (St, Yt, t))

≤ e−rtη(σ)dt −∑

i

λie−rtδ(t − Ti)hi(S)dt + Brownian terms.

30

Integrating and taking expectations under P gives:

EPt

[

e−rT V (ST , YT , T )]

− e−rtV (s, y, t) ≤ −EPt

[

∑

i

λihi(STi)e−rTi

]

+ EPt

[∫ T

t

η(σu)e−rudu

]

.

Noting that our HJB solution also satisfies V (ST , YT , T ) = 0, we can conclude, then, that

V (s, y, t) ≥ EPt

[

∑

i

λihi(STi)e−r(Ti−t)

]

− EPt

[∫ T

t

η(σ)e−r(s−t)ds

]

and equality may be achieved by setting σ = σ∗, where σ∗ is the supremum attained in the equationfor Φ(X, Y ). So, indeed the function, V , that solves the HJB equation is a value function for

supσ

EPt

[

∑

i

λihi(STi)e−r(Ti−t)

]

− EPt

[∫ T

t

η(σ)e−r(s−t)ds

]

.

31