Computing the Implied in Stochastic Volatility Models

HENRI BERESTYCKI École des Hautes Études en Sciences Sociales JÉRÔME BUSCA CNRS and Université Paris Dauphine AND IGOR FLORENT HSBC CCF

1 Introduction and Main Results The Black-Scholes model [6, 23] has gained wide recognition on financial mar- kets. One of its shortcomings, however, is that it is inconsistent with most observed prices. Although the model can still be used very efficiently, it has been pro- posed to relax its assumptions, and, for instance, to consider that the volatility of the underlying asset S is no longer a constant but rather a . There are two well-known approaches to achieve this goal. In the first class of models, the volatility is assumed to depend on the variables t (time) and S, giving rise to the so-called models. The second one, conceptually more ambitious, considers that the volatility has a stochastic component of its own. In the latter, the number of factors is increased by the amount of stochastic factors entering the volatility modeling. Both models are of practical interest. In these contexts, it is relevant to express the resulting prices in terms of implied volatilities. Given a price, the Black-Scholes is determined, for each given product (that is for each given strike and expiry date defining, say, the ) as the unique value of the volatility parameter for which the Black- Scholes pricing formula agrees with that given price. Actually, it is common prac- tice on trading floors to quote and to observe prices in this way. A great advantage of having prices expressed in such dimensionless units is to provide easy compari- son between products with different characteristics. In principle, the implied volatility can be inferred from computed options prices by inverting the Black-Scholes formula. It is more convenient, however, to directly analyze the implied volatility. Indeed, this approach allows us to shed light on qualitative properties that would otherwise be more difficult to establish. In partic- ular, we derive here several asymptotic formulae that are of practical interest, for example, in the calibration problem. The latter—an inverse problem that consists

Communications on Pure and Applied Mathematics, Vol. LVII, 1352–1373 (2004) c 2004 Wiley Periodicals, Inc. STOCHASTIC VOLATILITY MODELS 1353 in determining model parameters from the observation of market instruments—is typically computationally intensive. This paper addresses precisely these questions for stochastic volatility models. In an earlier paper, we had carried out a similar program in the framework of lo- cal volatility models (see [4, 5]). There, the asymptotic behavior near expiry was given explicitly by an ordinary differential equation. The situation here is more involved. We introduce a new method to determine the implied volatility in the framework of stochastic volatility models for European call or put options. It is described in Section 1.2, where we show that the implied volatility is obtained by solving a quasi-linear parabolic partial differential equation, the initial condition of which is given by the solution of an eikonal (first-order) Hamilton-Jacobi equation. We establish a uniqueness result for this equation that is new and of independent interest. The proof takes up Section 3 and involves the auxiliary notion of “effective volatility.” Its definition and some useful results about the effective volatility are given in Section 2. This notion is due to Derman and Kani [12]. Using a result of Varadhan and a large-deviation approach, the behavior of the effective volatility near expiry has been obtained by Avellaneda et al. [2, 3] in the context of basket options, and we apply the same methodology here. This is described in Section 2. We give original proofs of all these various results related to the effective volatility introducing a PDE type approach to this topic. These take up Sections 4 and 5. Furthermore, we establish here a new characterization of the Varadhan geodesic distance that is involved in the limiting theorem near expiry as a viscosity solution to a Hamilton-Jacobi equation. Lastly, in Section 6 we show how our general approach applies to a variety of specific popular stochastic volatility models. There we derive closed-form approx- imate formulae. Finally, numerical computations illustrate the high-order accuracy that is achieved through the approximation with only one- or two-term expansions.

1.1 Stochastic Volatility Models

We assume that the volatility of the underlying asset St is given as a function − = ( 1,..., n−1) of n 1 stochastic factors yt yt yt that follow diffusion processes. Specifically, we consider the following S.D.E. dSt = rdt + σ(St , yt , t)dWt (1.1) St dyt = θ(yt , t)dt + ν(yt , t)dZt ≡ 0 = ( 1,..., n−1) where Wt Zt , Zt Zt Zt , are standard Wiener processes. We define  = (ρ )  i , j =ρ θ = the correlation matrix ij 0≤i, j≤n−1 by Zt Zt ijdt. In (1.1), t (θ 1,...,θn−1) ν( , ) = (ν ) t t are drift coefficients and yt t ij 1≤i, j≤n−1 is a diffusion matrix. Precise regularity and growth conditions on these coefficients will be stated below. We assume that St ∈ (0, +∞) and that yt belongs to a domain O ⊂ Rn−1 a.s. In applications the domain O will typically be the whole space or a 1354 H. BERESTYCKI, J. BUSCA, AND I. FLORENT half-space. This framework includes such popular stochastic volatility models as the and the log-normal volatility model. We assume for simplicity in the sequel that O = Rn−1 (an example with O a half-space is discussed later on in Section 6). As is classical, we will assume that the fair value of any option is the expec- tation of its discounted payoff at maturity T under the probability for which (1.1) holds, i.e., −r(T −t) (1.2) C(S, y, t; K, T ) = E e (ST − K )+ | Ft , where Ft is the natural filtration. A discussion of this property as well as the choice of the probability measure are outside the scope of this paper. Equivalently, from the Feynman-Kac relation, C can be obtained as the solution of the linear parabolic partial differential equation in n space dimensions C + LC = 0 (1.3) t C(S, yt , t = T ; K, T ) = (S − K )+ in the domain {S > 0, y ∈ Rn−1}, where the pricing operator L is defined by

1 ∂2ϕ Lϕ = σ 2(S, y, t)S2 2 ∂ S2 ∂2ϕ + σ(S, y, t)S ρ ν , (y, t) 0 j k j ∂ S∂y 1≤ j,k≤n−1 k (1.4) 1 ∂2ϕ + ρ ν (y, t)ν (y, t) 2 ij ik jl ∂y ∂y 1≤i, j,k,l≤n−1 k l ∂ϕ ∂ϕ + rS + θ − rϕ. ∂ S i ∂y 1≤i≤n−1 i

The implied volatility function (S, y, t; K, T ) is uniquely defined by the re- lation

(1.5) C(S, y, t; K, T ) = CBS(S, t; K, T ; (S, y, t; K, T )) , where CBS(S, t; K, T ; ) is the price of call options in the Black-Scholes model [6] for a given volatility (constant) parameter >0. Recall that it is given by

BS −r(T −t) (1.6) C (S, t; K, T ; ) = SN(d1) − Ke N(d2) with ln(Ser(T −t)/K ) 1 √ √ d1 = √ +  T − t , d2 = d1 −  T − t .  T − t 2 STOCHASTIC VOLATILITY MODELS 1355

1.2 Main Results Henceforth we use the following notation corresponding to indices being shifted by one digit. The correlation matrix is now  = (ωij) with ωij = ρi−1, j−1, 1 ≤ i, j ≤ n, the diffusion matrix is given by M = (mij) with m11 = σ , m1k = mk1 = 0ifk = 1, mij = νi−1, j−1 if 2 ≤ i, j ≤ n. The drift terms are de- 1 2 = − σ ( , , ) ( ,τ)= θ − ( , ) = ,..., fined by q1 rS 2 S y t , qi x i 1 y t , i 2 n. For later pur- ˜ ˜ poses we also introduce a modified drift coefficient θ1 = 0 and θi = qi + ω1i σνi−1 for i = 2,...,n. Let us introduce the reduced variables S τ = T − t , x = ln + r(T − t), x = y − for i ≥ 2. 1 K i i 1

From now on, the “space” variables are x = (x1,...,xn). With a slight abuse, we keep the notation σ(x,τ) = σ(S, y, t) and likewise for other functions. Last, we introduce a normalized price C(S, y, t; K, T ) u(x,τ)≡ u(x,τ; K, T ) = erτ . K Throughout the paper, we shall make the following technical assumptions on the diffusion coefficients: q ∈ Cα,α/2 (H) MMT ∈ Cα,α/2  C(1 +|x|)−2|ξ|2 ≤MMT(x,τ)ξ,ξ≤C(1 +|x|)2|ξ|2 for all x,ξ ∈ Rn, τ ∈ (0, T ). Here Cα,α/2 denotes the space of functions having uniformly bounded partial Hölder differential quotients with exponent α (respec- tively, α/2) in the space (respectively, time) variables; see [15]. A straightforward computation shows that the pricing PDE in the reduced vari- ables is 1 T 2 τ =  + ( ,τ)· u 2 Tr M M D u q x Du (1.7) x u(x, 0) = (e 1 − 1)+, the equation being satisfied in Rn × (0, T ). Note that the matrix M = M(x,τ) depends on the variables x and τ. Throughout the paper, D denotes the gradient vector (∂/∂x1,...,∂/∂xn), D2 the Hessian matrix , MT the transpose of the matrix M, and Tr the trace. The following classical result asserts the solvability of the pricing equation.

PROPOSITION 1.1 Under assumption (H) there is a unique classical solution u ∈ 2,α,α/2(Rn × ( , )) ∩ (Rn ×[ , )) Cloc 0 T C 0 T that satisfies the arbitrage bounds x x (1.8) (e 1 − 1)+ < u(x,τ)

Our main result here is to derive a degenerate quasi-linear parabolic equation satisfied by the implied volatility, together with its asymptotic behavior as τ → 0.

THEOREM 1.2 Under assumption (H) the implied volatility function is the unique ( ,τ) ∈ 2,1,∞ (Rn × ( , )) solution x Wloc 0 T of the following well-posed nonlinear degenerate parabolic initial value problem 2 2 (τ )τ = H(x,τ,,D) + τL(x,τ,,D ) (1.9) (x, 0) = 0(x). Here the operators in the equation are given by (1.10) H(x,τ,,D) = x x 1 Tr MMT2 D 1 ⊗ D 1 − τ 22 D ⊗ D + τθ˜ · D   4 and (1.11) L(x,τ,,D2) =  Tr(MMT D2) n 0 0 0 in R . The initial condition is determined by  = x1/ψ (x, y) where ψ is the unique solution of the eikonal equation  H 0(x, Dψ0) ≡ Tr[MMT(x, 0)Dψ0 ⊗ Dψ0]=1 ψ0 = { = } (1.12)  0 on x1 0  0 ψ (x)>0 for x1 > 0. Furthermore, this function can be represented as (1.13) ψ0(x) = inf G(x,ξ), {ξ1=0} ∈ 1,∞(RN ×RN ) where G Wloc is the unique viscosity solution of the Hamilton-Jacobi equation   0 N H (x, Dx G) = 1 in R \{ξ} (1.14) G(x = ξ,ξ) = 0  lim|x|→∞ G(x,ξ)=+∞ for all ξ ∈ RN . In formulae (1.10)–(1.12) and throughout the paper, we use the notation A ⊗ B to denote the matrix (Ai Bj )1≤i, j≤n for any two vectors A and B. Clearly, Theorem 1.2 provides a way to directly compute the implied volatility relating it to the parameters of the model (the coefficients of the SDE (1.1)). More- over, it is a key to establishing some delicate qualitative properties. Obviously, Theorem 1.2 yields an approximation formula near expiry for the implied volatil- ity. It is worth emphasizing the fact that to get this approximation, one has to solve a Hamilton-Jacobi equation. But furthermore, equation (1.9) can be used to sys- tematically expand  in powers of τ, 0 being the leading-order term. Indeed, the STOCHASTIC VOLATILITY MODELS 1357 higher-order terms are obtained by solving transport equations. This can be done explicitly in several cases; see Section 6 for an example. These computations are quite useful in practice. We will see in Section 6 that, in practical cases, they yield remarkably accurate approximations. We wish to emphasize that the existence and uniqueness results for problem (1.12) in the theorem above are new and rather delicate. Indeed, the problem is set in the whole space with an “interior Dirichlet condition” on a hyperplane. (Thus, in practice, the problem is set in two half-spaces with a boundary condition on the limiting hyperplane.) There is a vast literature devoted to the Dirichlet problem for Hamilton-Jacobi equations in bounded domains (see, in particular, [10, 19, 22]), but comparatively little is known for unbounded domains, apart from the work of O. Alvarez [1]. There an existence and uniqueness result for Hamilton-Jacobi equations of type (1.12) is established but under the assumption that the Hamil- tonian H 0(x, p) has linear growth in x as |x|→∞. Note that, here, in view of assumption (H), the Hamiltonian has quadratic growth in x. We therefore require an extension of the result of Alvarez with some new ingredients. Incidentally, this raises the question of the range of validity of such a uniqueness result in a more general setting. In other words, how much can the result of Alvarez be extended to allow for more general growth conditions?

2 Effective Volatility and Limiting Behavior near Expiry In the proof of Theorem 1.2, we require the auxiliary notion of effective volatil- ity, which we now describe. It has been known for some time that from the point of view of valuing all European options at t = t0, it is equivalent to solving the n-dimensional PDE (1.3) for all values of K, T > 0 or to solve a one-dimensional PDE with a well-chosen diffusion coefficient. The latter is called effective local volatility, following the terminology in [12]. This result is due to Derman and Kani [12]. The precise statement is the following:1

THEOREM 2.1 [12] Let C = C(S0, y0, t0; K, T ) be the solution of (1.3), i.e., the family of all call option prices at t = t0 for an initial spot S0 and initial value of the = ( 1,..., n−1) volatility factors y0 y0 y0 . The function C is also the unique solution of the parabolic problem in the variables (K, T ) C = 1 σ 2(S , y , t ; K, T )K 2C − rKC (2.1) T 2 0 0 0 KK K C(S0, y0, t0; K, T = t0) = (S0 − K )+, where the effective local volatility σ ∈ C([0, +∞)2) is given by π( , , ; , , )σ 2( , , ) 2 S0 y0 t0 K y T K y T dy (2.2) σ (S0, y0, t0; K, T ) = , π(S0, y0, t0; K, y, T )dy

1 The authors are thankful to M. Avellaneda for pointing out the importance of the result in The- orem 2.1. 1358 H. BERESTYCKI, J. BUSCA, AND I. FLORENT and π(S, y, t; S, y, t) is the Green function of the pricing problem, i.e., the solu- tion of πt + Lπ = 0 (2.3)     π(S, y, t = t ; S , y , t ) = δ(S,y)=(S,y), where L is defined in (1.4), defined for S, S > 0,y, y ∈ Rn−1,t< t.

Let us note that, with this definition of the Green function, the solution C of (1.3) is given by      (2.4) C(S, y, t; K, T ) = π(S, y, t; S , y , T )(S − K )+ dS dy .

This can be seen as a generalization of a well-known formula regarding lo- cal volatility of B. Dupire [14] for the following reason: Consider the case when σ is independent of y: σ(S, y, t) ≡ σ(S, t)—the so-called local or determinis- tic volatility model; we find that (2.2) reduces to σ(S0, y0, t0; K, T ) ≡ σ(K, T ), that is, Dupire’s result. It is also related to the papers of Derman and Kani [11], Dupire [14], and the formula of Breeden and Litzenberger [7]. For a review of those results, see [21]. The latter states that state price densities are recovered from options prices by differentiating twice with respect to the strike variable. It reads ∂2C (2.5) π(S, y, t; S, y, t) = C(S, y, t; S, t). ∂ K 2 That equation (2.1) defines a well-posed problem, however, is far from obvious. It relies on the fact that the effective volatility remains locally bounded as T → t0; this is very much linked to the assumption that the stochastic volatility factors follow a diffusion process. Indeed, this property may be violated for other kinds of stochastic processes. One of the difficulties here in proving that the effective volatility remains locally bounded as T → t0 is that the volatility of the diffusion process is not necessarily bounded. For these reasons we give here, in Section 4, a new and complete proof of Theorem 2.1 relying on PDE methods. Note that in view of expression (2.2), the equivalent volatility σ can be thought of as the conditional expectation 2 S0,y0,t0 2 (2.6) σ (S0, y0, t0; K, T ) = E σ (K, yT , T ) | ST = K . This formula results from a formal application of Ito’s formula on the terminal payoff of the call, (ST − K )+ (see [12]). In using the results of Theorem 2.1, an essential role is played by the limiting behavior close to expiry. This is the object of the following result, in which we identify the limit of the equivalent local volatility for short time to maturity: STOCHASTIC VOLATILITY MODELS 1359

THEOREM 2.2 When T −t0 → 0 the effective volatility converges locally uniformly to σ(S, y, t = T ; K, T ), which satisfies the limiting first-order problem 0 Tr(MMT Dd ⊗ Dσ) = 0 ∀S > 0, ∀y ∈ Rn−1, (2.7) σ(S = K, y, T ; K, T ) = σ(K, y, T ) ∀y ∈ Rn−1, where d = d(S, y; K ) is Varadhan’s signed geodesic distance to the hyperplane {S = K, y ∈ Rn−1} associated to the elliptic operator L defined in [24]. This result is analogous to that in [2, 3], and we apply the same methods. How- ever, it is stated here in the framework of stochastic volatility models. In Section 5 below, we give an original and rigorous proof of the result with PDE arguments. We will make use of another characterization of the geodesic distance. In the next statement, we derive a way of actually computing the function d.

THEOREM 2.3 The signed geodesic distance d defined above is the unique viscosity solution of Tr(MMT Dd ⊗ Dd) = 1 ∀S > 0, ∀y ∈ Rn−1, (2.8) d(S = K, y; K ) = 0 ∀y ∈ Rn−1.

Note that d is nothing else than ψ0 defined in (1.12) in the variables (S, y). This result yields a simple means of computing σ at t0 = T by applying the method of rays. Indeed, given K > 0, one can solve the system of ODEs (S˙(τ), y˙(τ)) = T MM ∇d(S(τ), y(τ)), (S(0), y(0)) = (S0, y0) where ∇d is smooth, that is, at least in a neighborhood2 of the hyperplane {S = K, y ∈ Rn−1}. It is easily seen that ∗ (2.9) σ(S0, y0, T ; K, T ) = σ(K, y(τ ), T ), where τ ∗ is the first time S(τ) hits the hyperplane {S = K, y ∈ Rn−1}.An interpretation of (2.9) is that the averaging that takes place in (2.2) is replaced in the asymptotic regime t0 → T by an evaluation of the stochastic volatility at a point y = y(τ ∗) that is determined by the geodesic distance d. Intuitively, the higher the ∗ “volatility of the volatility factors” is, the farther y(τ ) will deviate from y0, thus creating a more pronounced smile. An important consequence of Theorem 2.1 is that the implied volatility remains bounded and bounded away from 0 as T − t0 → 0—again an essential feature of diffusion models, which will be used in the proof of Theorem 1.2. We state this property in the following proposition:

PROPOSITION 2.4 Given τ0 > 0, there exists a continuous function ,  = (R,τ0), such that −1 (2.10) 0 <(R,τ0) ≤ (x,τ)≤ (R,τ0) n for all x ∈ BR ={x ∈ R :|x| < R}, τ ∈ (0,τ0).

2 Actually, this ODE is solvable in the classical sense up to the cut locus of (K, y0). 1360 H. BERESTYCKI, J. BUSCA, AND I. FLORENT

3 Determining the Implied Volatility The proof of Theorem 1.2 will take up all this section. It is divided into several parts. 3.1 Derivation of the PDE for the Implied Volatility We first show that the nonlinear PDE in (1.9) follows from the pricing equation (1.3) in a straightforward way. Let us denote by U the solution to (1.7) corresponding to the Black-Scholes model with unit volatility, that is, MMT ≡ e ⊗ e . Then U solves 1 1 1 τ = ( − ), τ > , ∈ R, U 2 Ux1x1 Ux1 0 x1 (3.1) x U(x, 0) = (e 1 − 1)+. The explicit solution to (3.1) is readily seen to be √ √ x1 1 x1 1 (3.2) U(x,τ)≡ U(x ,τ)= ex1 N √ + τ − N √ − τ , 1 τ 2 τ 2 where y 1 − y2 (3.3) N(y) = √ e 2 dy. 2π −∞ Note that this is nothing else but the classical Black-Scholes formula [6] written in reduced variables. Next, owing to the obvious scaling properties of (3.1), we observe that the implied volatility (1.5) is equivalently defined by 2 (3.4) u(x,τ)= U(x1,(x1,τ) τ). Next, we make use of the following relations, which hold for the solution of (3.1): 2 Ux1τ 1 x1 Uττ 1 x1 1 (3.5) (x1, s) = − , (x1, s) =− + − . Uτ 2 s Uτ 2s 2s2 8 With these, a direct computation allows us to derive the nonlinear PDE in (1.9) from the pricing equation (1.7). 3.2 Construction of the Limiting Solution We are going to construct a solution of the eikonal equation (1.12). It is clearly 2 equivalent to solving first H(x, 0,,D) =  and then setting ψ = x1/.For this, we first localize the problem in a ball BR and solve the localized problem by the vanishing viscosity method [22]. Then we prove uniqueness for the limiting problem and let R →∞. A difficulty in the first step is the lack of coercivity of the Hamiltonian on account of the D(x1/) term in (1.10). This makes it difficult to construct a solution by the usual Perron method. However, since x1 (3.6)  = δ1,i − x1(ln )i ,  i STOCHASTIC VOLATILITY MODELS 1361 we are naturally led to the natural change of variable w = ln . For a fixed R > 0 we now show that the localized problem in the new variables can actually be solved. ε LEMMA 3.1 Let >0, >0. There exists a unique classical solution  ∈ 2,α( ) ∩ ( ) Cloc BR C BR of H(x, 0,ε, Dε) + ε(ln ε) − (ε)2 = 0 (3.7) ε =  |∂ BR (where H is defined in (1.10)) that satisfies the bound (3.8) 0 < min , inf σ(x, 0) ≤ ε(x, 0) ≤ max , sup σ(x, 0) x∈B R x∈BR uniformly in ε>0. When ε → 0, for  and R fixed, the function ε converges locally uniformly 0 in BR to the unique viscosity solution  of (3.9) H(x, 0,0, D0) = (0)2 n 0 0 in R . The function ψ = x1/ is the solution of (1.12).

ε ε PROOF OF LEMMA 3.1: Setting w =−ln  , equation (3.7) can be rewritten as T ε ε ε −2wε  ( , ) (δ , + w )(δ , + w ) − εw − = . (3.10) Tr M M x 0 1 i x1 i 1 j x1 j i, j e 0 This equation is clearly uniformly elliptic and coercive and hence enjoys a com- T 2 parison principle. Now, noting that Tr(MM (x, 0)e1 ⊗ e1) = σ (x, 0),wesee that the constants − ln min , inf σ(x, 0) and − ln max(, sup σ(x, 0)) x∈B R x∈BR are respectively sub- and supersolutions of (3.10). Perron’s method [9] thus applies in a standard way to give the existence of a viscosity solution satisfying the bounds (3.8), uniqueness being a trivial consequence of the comparison principle. The C2,α regularity of solutions follows from (H) and from the general theory of Hölder regularity for elliptic equations [16]. 

The uniqueness property for the limiting equation (3.9) is fairly unusual given that there is no apparent boundary condition. As a matter of fact, (3.9) is degenerate on the hyperplane {x1 = 0}, which turns out to imply a boundary condition there that is automatically satisfied by any solution.

PROPOSITION 3.2 There exists a unique viscosity solution of (2.8); it is given by the representation formula (1.13). 1362 H. BERESTYCKI, J. BUSCA, AND I. FLORENT

PROOF: The proof has three steps. Step 1: Construction of the fundamental solution. Let us fix ξ ∈ RN . By the results of Kružkov [20], it is classical to construct (using the vanishing viscosity 0 method) a viscosity solution G(x,ξ)of H (x, Dx G) = 1 satisfying G(ξ, ξ) = 0 and G > 0for|x − ξ| > 0. If one defines the radial envelopes of MMT(x, 0) by A(r) = sup MMT(x, 0), A(r) = inf MMT(x, 0), | |= |x|=r x r together with the solutions of the radial problems  ( ) , ≡ ( ) 2 = ,  ( ) , = , ( ) = ( ) = , A r DG DG A r Gr 1 A r DG DG 1 G 0 G 0 0 it is classical (by applying a comparison principle when constructing G) that G ≤ G ≤ G. In view of assumption (H), one gets the behavior at infinity −1( + ( | − ξ|) ) ≤ ( ,ξ)≤ ( +| − ξ|2), = , , (3.11) C0 1 ln x + G x C0 1 x i 1 2 so that in particular (3.12) lim G(x,ξ)=∞. |x|→∞ Step 2: Uniqueness of the fundamental solution. The solution we have just constructed is unique. The proof of this fact here follows the approach of Alvarez [1]. Recall that [1] is restricted to linear growth of the Hamiltonian, while the growth here is quadratic. Therefore some new estimates are needed. 1,∞ Let us consider two solutions G1 and G2 in W of (1.14). Since the first-order equation is satisfied a.e., each Gi satisfies (3.11) and (3.12). Let us now perturb ˜ −1 G1 into a strict supersolution by introducing G1 =  (G1) with (z) = z for 6MC0 all z ∈[0, M], (z) = 1/(3C0) ln(ze /(2M)) in z ∈ (2M, +∞), and  increasing and C1 in (0, +∞). This is a supersolution of the eikonal equation in (1.14) since we clearly have T ˜ ˜ 1 Tr MM (x, 0)DG1 ⊗ DG1 = ≥ 1 . 2 ˜  (G1) ˜ A key observation is now that G1 ≥ G2 for |x|1. Indeed, in view of (3.12) one ˜ 3 has G1(x,ξ)≥ C|x| ≥ G2(x,ξ)for |x|≥R = R(M, C0,ξ). We can then apply the classical comparison principle [9] in the bounded domain BR \{ξ} to infer that ˜ G1 ≥ G2 there. Sending now M to ∞, one gets G1 ≥ G2 everywhere. Step 3: Representation formula. It is classical that, given any set S, the function 0 ζ = infξ∈S G(x,ξ) is a viscosity solution of H (x, Dx ) = 1 that vanishes on S.   Choosing S = ∂ωR, ωR ={x = (x1, x ) | x1 > 0, |x |≤R}, and noting that ψ ≥ 0in{x1 ≥ 0}, we can apply the classical comparison principle in ωR to ψ ≥ ( ,ξ) ω > ( ,ξ) →∞ infer that infξ∈∂ωR G x in R for all R 0. Since G x when |x − ξ|→∞, we have in the limit R →∞ ψ( ) ≥ ( ,ξ) { > } . (3.13) x inf G x in x1 0 {ξ=(ξ1,ξ )|ξ1=0} STOCHASTIC VOLATILITY MODELS 1363

To get the reverse inequality, we use a similar argument to that in step 2 which we ϕ ( ) =   ( ,ξ) only briefly sketch. We define n x inf{ξ=(ξ1,ξ )|ξ1=0,|ξ |≤n} G x and modify it at infinity into a strict supersolution with the help of  defined in step 2. Since 2 1/2 ψ satisfies (1.12), by (H) we have |Dx ψ|≤C(1 +|x − ξ| ) , so that ψ can grow at most quadratically at ∞. We infer from here that ψ ≤ ϕn for |x|≥R =  R(n). Applying the classical comparison principle in ωR \{ξ = (ξ1,ξ ) | ξ1 = 0,  |ξ |≤n}, one gets ψ ≤ ϕn there. Sending n to ∞ yields the reverse inequality in (3.13). 

3.3 A Comparison Principle

LEMMA 3.3 Suppose  ∈ C(BR × (0,τ0)) (respectively,  ∈ C(BR × (0,τ0))) satisfies

2 2 (3.14) (τ )τ ≤ H(x,τ,D) + τL(x,τ,D )

(respectively, , ≥) in BR × (0,τ0) in the viscosity sense, together with the lateral comparison

(3.15)  ≤  on ∂ BR × (0,τ0) and the growth condition at τ = 0

2 2 (3.16) lim τ (x,τ)= lim τ (x,τ)= 0 ∀x ∈ BR. τ→0 τ→0 Then

(3.17) (x,τ)≤ (x,τ) ∀x ∈ BR, ∀τ ∈ (0,τ0).

PROOF: By a change of function this boils down to applying a classical maxi- mum principle for solutions to linear parabolic PDEs. Setting u(x,τ) = 2 U(x,2(x,τ))and u(x,τ)= U(x,  (x,τ)), differentiating, and using relations (3.5) show that u is a subsolution (respectively, u is a supersolution) of (1.7), and u ≤ u on the lateral boundary as well as at initial time. It should be noted that x (3.16) has been used to ensure that u(x, 0) = u(x, 0) = u(x, 0) = (e 1 − 1)+. We now apply the comparison principle (Proposition 3.3) to infer that u(x,τ)≤ u(x,τ)≤ u(x,τ)everywhere, that is,

U(x,(x,τ)2τ) ≤ U(x,(x,τ)2τ) ≤ U(x, (x,τ)2τ).

Since U is increasing as a function of its time argument, this in turn gives

(3.18) (x,τ)≤ (x,τ)≤ (x,τ) for all x ∈ BR, τ ∈ (0, T ).  1364 H. BERESTYCKI, J. BUSCA, AND I. FLORENT

3.4 Sub- and Supersolutions We are going to define sub- and supersolutions of the time-dependent equation (1.9) from a relaxed version of problem (3.7). For this we introduce for δ,ε > 0 ε,δ the solution  (respectively, ε,δ)of ε,δ ε,δ ε,δ ε,δ H(x, 0,  , D ) + ε(ln  ) − ( )2 =−δ (3.19) ε,δ  = (R, 1) + 1 ≡ (R) |∂ BR (respectively, ε,δ, δ →−δ, (R, 1) + 1 → ((R, 1) + 1)−1), where (R, 1) is defined in Proposition 2.4. Ultimately we intend to treat (1.9) for small times as a perturbation of (3.7). The difficulty that arises in this task is the lack of a uniform estimate on the Hessian of the solutions of (3.19), which will eventually be needed to control the time- dependent terms in (1.9). A standard way to get around this is to regularize by sup and inf convolution [9], i.e., define for η>0 the function ε,δ,η ε,δ 1 (3.20)  (x) = inf  (ξ) + |ξ − x|2 , ξ∈BR 2η and, respectively, ε,δ,η( ) = ε,δ(ξ) − 1 |ξ − |2 . (3.21) x sup η x ξ∈BR 2 For the reader’s convenience we list some of the well-known properties of sup and inf convolution that we shall use below. We refer to [9] for the definition of the semijets J 2,± as well as the notion of pointwise twice differentiability.

LEMMA 3.4 The following properties hold for all ε, δ, η>0: ε,δ,η (i) The functions  and ε,δ,η are locally Lipschitz and a.e. twice differ- entiable. ε,δ,η ε,δ ε,δ,η ε,δ (ii)  →  as η → 0(respectively,  ,  ) uniformly in BR. ε,δ,η ε,δ,η (iii)  (respectively, ε,δ,η) is semiconcave with D2 ≤ (1/η)I (re- spectively, D2ε,δ,η ≥−(1/η)I ) a.e. ε,δ,η 2,+ ε,δ,η 2,− (iv) If (p, N) ∈ J ( (x0)) (respectively, J ,  ) for x0 ∈ BR, then ε,δ,η 2,+ ε,δ 2,− (p, N) ∈ J ( (x0 + ηp)) (respectively, J ,  ,x0 − ηp) and ε,δ,η( ) + 1 η| |2 = ε,δ( + η ), (3.22) x0 2 p x0 p ε,δ,η (respectively,  , −η). ε,δ,η ε,δ ε,δ,η |  | ∞ ≤ (|  | ∞ , 1 ( ))  ) (v) D L (BR ) min D L (BR ) η C R (respectively, . For the proof of this lemma, see [9] for (i) through (iv) and [8] for (v). We claim that these functions are super- and subsolutions of the time-dependent equation (1.9) for sufficiently small times. More precisely, we have the following result: STOCHASTIC VOLATILITY MODELS 1365

LEMMA 3.5 For all R > 0 and δ>0, there exist η0 = η0(R,δ) > 0 such that for all 0 <η<η0(R,δ), all 0 <ε<ε0(R,δ,η), and 0 <τ<τ0(R,δ,η), the ε,δ,η function  (respectively, ε,δ,η) is a supersolution (respectively, subsolution) of the equation in (1.9) in the domain BR × (0,τ0). Let us first make clear how the last result (1.12) in Theorem 1.2 follows from this lemma. Applying the comparison principle (Lemma 3.3) we get that ε,δ,η (3.23) ε,δ,η(x) ≤ (x,τ)≤  (x) in BR × (0,τ0). Now this implies ε,δ,η (3.24) ε,δ,η(x) ≤ lim inf (x,τ)≤ lim sup (x,τ)≤  (x). τ→ 0 τ→0 We now send first ε → 0 and observe that by the arguments in the proof of ε,δ Lemma 3.1, it is easy to see that  (respectively, ε,δ) converges locally uni- δ δ formly in BR to  (respectively,  ) solution of δ δ δ (3.25) H(x, 0,  , D ) − ( )2 =−δ (respectively, δ, δ →−δ). Now, using (3.20)–(3.21) we see that when ε → 0 ε,δ,η δ,η one has  →  (respectively, ), the inf (respectively, sup) convolution of δ the above-defined  (δ). Using the uniqueness result in Section 3.2 we infer δ that  and δ converge as δ → 0to0 defined in (1.12). We finally deduce that (x,τ)converges as τ → 0tothe0 solution of the eikonal equation (1.12). This will eventually complete the proof of Theorem 1.2. To conclude this section we now prove Lemma 3.5.

PROOF: We observe that for all ϕ>0 | ( ,τ,ϕ, ϕ) − ( , ,ϕ, ϕ)|≤ (|ϕ| +| ϕ| )τ H x D H x 0 D C W 1,∞ ln W 1,∞ and, by using (iii) in Lemma 3.4, that ε,δ,η ε,δ,η ( ) τL( ,τ, , 2 ) ≤ τ R x D η (respectively, τL(x,τ,ε,δ,η, D2ε,δ,η) ≥−τ(R)−1/η), together with ε( ε,δ,η) ≤ ε , |ε,δ,η| 1 (3.26) ln C R W 1,∞ η ε( ε,δ,η) ≥−ε ( , |ε,δ,η| )/η (respectively, ln C R W 1,∞ ). Finally, let us note that ε,δ |D |L∞ ≤ C(R) (respectively, ) [22]. |ε,δ,η| ≤ ( , η)/η Using (3.26) and Lemma 3.4(v) leads to the estimates W 1,∞ C R |ε,δ,η| ≤ ( )/η ε δ ε,δ,η and W 1,∞ C R uniformly in and . This implies that satisfies ε,δ,η ε,δ,η ε,δ,η ε,δ,η ε,δ,η τ( )2 ≥ ( ,τ, ,  ) + τL( ,τ, , 2 ) τ H x D x D + δ − εC(R,η)− τC(R,η)− C(R)η 1366 H. BERESTYCKI, J. BUSCA, AND I. FLORENT

(respectively, ε,δ,η, ≤), these last terms accounting for the shift in the space variable due to the sup convolution; see Lemma 3.4(iv). For suitable ε, τ, and η, the remainder term is positive (respectively, negative); hence the statement in Lemma 3.5. 

4 Effective Volatility: Existence and Uniqueness

PROOF: Let us introduce the generalized Dupire differential operator 1 (4.1) L = σ 2(S , y , t ; K, T )K 2∂2 − rK∂ . 2 0 0 0 KK K In order to establish (2.1), we intend to show by applying the maximum prin- ciple that the function w = (∂T − L)C vanishes identically for the choice of dif- fusion coefficient (2.2). For this we compute the Lie bracket (defined by [A, B]= AB − BA) of the pricing operator (1.4) with (4.1); we find 1 ∂2C (4.2) ∂ − L,∂ + L (C) = K 2(∂ + L) σ 2 . T t 2 t ∂ K 2 Differentiating the pricing equation (1.4) twice with respect to K and using the π definition of the Green function leads to 2 ∂ C   (4.3) (S, y , t; K, T ) = dy π(S, y , t; S , y, T )δ = dS ∂ 2 0 0 S K K

(4.4) = π(S, y0, t; K, y, T )dy. Hence by (4.2) if one sets σ as (4.5) π(S, y, t; K, y, T )dy σ 2(S, y, t; K, T ) = dS π(S, y, t; S, y, T )φ(S, y; K, T )dy for any function distribution φ such that (4.5) is integrable, we see that ∂2C (4.6) (∂ + L) σ 2 = 0 , t ∂ K 2 so that the function w = (∂T − L)C satisfies the backward parabolic equation

(4.7) (∂t + L)w = 0 . 2  In a final step, we show that the choice φ ≡ δS=K σ (K, y , T ) ensures the terminal condition wt=T = 0. This allows us to apply the maximum principle to (4.7) to infer that w ≡ 0, which establishes (2.1) and (2.2). We write the pricing equation (1.3) in the distribution sense at t = T to find that

1 2 2 (4.8) C + σ (S, y, T )S δ = + rS1I > − r(S − K )+ = 0 . t 2 S K S K STOCHASTIC VOLATILITY MODELS 1367

Since C(S, y,ξ; K,ξ) = (S − K )+ for all ξ>0, we find that Ct (S, y, t = T ; K, T ) =−CT (S, y, t = T ; K, T ), so that we can rewrite (4.8) as 1 (4.9) −C + σ 2(K, y, T )K 2C − rKC = 0 T 2 KK K in the distribution sense at t = T . Now, (4.5) evaluated at t = T yields (in the distribution sense) 2 (4.10) δS=K σ (S, y, t = T ; K, T ) = φ(S, y; K, T ). 2 Hence φ(S, y; K, T ) = δ = σ (K, y, T ) is the only choice for which S K (4.11) w|t=T ≡ ∂T − L C|t=T = 0 is satisfied. In view of (4.7) and (4.11), the maximum principle implies w ≡ 0. 

5 Asymptotic Behavior of the Effective Volatility The derivation of the asymptotic behavior rests on a formula of Varadhan that is essentially a large-deviation result. We first show that the function σ 2 converges to a limit as t0 → T and then prove that this limit satisfies the first-order Hamilton- Jacobi equation (2.7). Varadhan’s formula [24], which we are going to use, states that 2 (5.1) lim (T − t0) ln(π(S0, y0, t0; K, y, T )) =−d (S0, y0; K, y). t0→T Here d is the geodesic distance defined by 2 T ˙ ˙ (5.2) d (S0, y0; K, y) = inf MM (X(s), 0)X(s), X(s)ds , where the infimum is computed over all smooth paths (X(s))s∈[0,1] valued in n−1 (0, +∞) × R such that X(0) = (S0, y0) and X(1) = (K, y). We now note that −1 (5.3) π(S0, y0, t0; K, y, T )dy π(S0, y0, t0; K, y, T ) can be viewed as a family of positive functions of y0 indexed by τ = T − t0 with unit L1 mass. Applying standard stationary-phase arguments, one sees clearly from D(Rn−1) µ (5.1) that this family of kernels converges in to a measure S0,y0,K with unit mass supported in the set S ={ ∈ Rn−1 : ( , ; , ) = ( , ; )} , (5.4) S0,y0,K y d S0 y0 K y d S0 y0 K where d is the geodesic distance to the hyperplane {S = K }, i.e.,

(5.5) d(S0, y0; K ) = inf d(S0, y0; K, y). y∈Rn−1 This argument shows that σ 2( , , ; , ) = σ 2( , , )µ ( ). (5.6) lim S0 y0 t0 K T K y T S0,y0,K dy t0→T 1368 H. BERESTYCKI, J. BUSCA, AND I. FLORENT

2 Henceforth this limit will be denoted by σ (S0, y0, T ; K, T ). Note that for S0 sufficiently close to K , the cut locus of d is not reached, which implies that the S µ set S0,y0,K is a singleton, and, in turn, that S0,y0,K is a Dirac mass at that point.

PROOF: We now intend to establish that σ(S0, y0, T ; K, T ) satisfies the first- order Hamilton-Jacobi equation (2.7). For this we expand (4.6) to find (5.7) (∂ + L)(σ 2) +MMT D(ln π),˜ D(σ 2)=0 , t where π˜ = CKK = π(S0, y0, t0; K, y, T )dy. We now multiply (5.7) by a test φ ≡ φ( , ) ∈ ∞(( , +∞) × Rn−1) ∈ ( , ) > function S y C0 0 , integrate it in t t0 T , S 0, y ∈ Rn−1, and integrate by parts in the space variables the first two terms. The time term gives rise to boundary values that cancel out when t0 → T thanks to (5.6). Integrating by parts in the space variables, the term involving L disappears as well due to the local boundedness of σ . Finally, the bracket term can be rewritten using the identity T (5.8) ln π˜ dt = t 0 1 (T − t0) ds ln π(S0, y0, t0 + s(T − t0); K, y, T )dy . 0 Thanks to (5.1) this term clearly establishes (2.7) in the distribution sense; now by (2.2) σ is Lipschitz, so that (2.7) is actually satisfied a.e. in (0, +∞) × Rn−1. 

PROOF OF PROPOSITION 2.4: It follows from the above proof that the effec- tive local volatility σ is locally bounded in (S0, y0, K, T ) uniformly in t0 < T .We shall now prove that the result in Proposition 2.4 follows from here. For this we observe that since the prices of call options satisfy (2.1), we have reduced the prob- lem, for each (S0, y0, t0), to a pricing equation in one space dimension. Indeed, this was the main point of introducing the notion of effective volatility. This allows us to use the setting of one-dimensional local volatility models studied in [4, 5]. Set- ting σ(˜ x,τ) = σ(S0, y0, t; K, T ) with x = (x1, y0), x1 = ln(S0/K ) + r(T − t0), and τ = T −t0 as usual, we know that the implied volatility (x,τ)is the solution of the one-dimensional nonlinear parabolic equation 1 2 2 x1 2 1 2 2 2 (τ )τ =  ( ) + τx x − τ   (5.9) σ˜ 2(x,τ)  x1 1 1 4 x1 ( , ) = ( 1 x dξ )−1 x 0 x 0 σ(ξ,˜ 0) that one may term an effective nonlinear PDE on the implied volatility. In [5] we established that (5.9) satisfies a comparison principle (i.e., all sub- and superso- lutions must be ordered). Consequently, all we need do is construct a sub- and a supersolution of (5.9) in order to prove (2.10). Let us recall (see [5]) that on account of the degenerate feature of (5.9), it is not necessary to check that the sub and supersolutions are ordered at τ = 0; indeed, this is automatically satisfied since the initial condition simply reflects the limiting equation at τ = 0. STOCHASTIC VOLATILITY MODELS 1369

We set, for x ∈ (−1, 1), (x , x) = (1 + x2) with 1 1 1  = max 2supσ(x,τ),1 , |x|≤1 0<τ<1 1 and set (x) = 2|x1| in |x1| > 1. Note that this is a C function. For |x1| > 1 the right-hand side in (5.9) is negative while the left-hand side is positive. In the set |x1|≤1, one computes x 1 1 − x2 (5.10) 1 = 1 ,   ( + 2)2 x1 1 x1  / ≤  whereas x1x1 2. From here we clearly see that is a supersolution for 0 <τ<1/(22). A 2 As for the subsolution we set, for A > 1,  (x) = λ(A)(1 − (x1/A) ) for x1 ∈ 1 A (−A, A) with λ(A) = min(1, inf |x|≤A σ(x,τ)), and set  = 0in{|x1|≥A}. 2 0<τ<1 To correctly interpret (5.9) when  vanishes, one should keep in mind that (5.9) is but the rephrasing of the pricing equation in terms of the implied volatility; thus setting  ≡ 0 in an interval corresponds to considering a price function identically x equal to the payoff function (e − 1)+ there. A direct computation shows that the payoff function is a subsolution of the pricing equation [5], whatever σ˜ is. Hence 2  appear as the maximum of 0, which is a subsolution, and λ(A)(1 − (x1/A) ). It thus remains to show that the latter function is a subsolution in {|x1| < A}.For this we note that (1 − x A / A)2 ≥ 1, | A A |≤2λ(A)2/A2 ≤ 2, | A A |≤ x1 x1x1 x1 2λ(A)/A ≤ 2, so that the left-hand side in (5.9) is no greater than the right-hand <τ< 1 >  A side for 0 8 . This shows that, for each A 0, the function is a ( ) =  A( ) ( , ) ≥ subsolution of (5.9), and thus so is x supA>1 x . Note that x1 x 1 |˜|<| | σ(˜ ˜, ) ∈ R 4 inf x x1 x 0 for all x1 . Applying the comparison principle easily yields the local boundedness of the implied volatility function (2.10). 

6 Examples and Further Remarks In these models the space dimension is n = 2 (one stochastic volatility factor). For convenience we adopt the notation x = x1 = ln(S0/K )+r(T −t0), τ = T −t0, and y = x2. 6.1 A Log-Normal, Mean-Reverting Stochastic Volatility Model In this model the dynamics of the underlying asset is given by dS = rS dt + y S σ(S )dW1 (6.1) t t t t t t = (θ − ) + κ 2 dyt a yt dt yt dWt  1, 2=ρ θ κ with dWt dWt dt, where a, , and are nonnegative constants. We simply rephrase our main result for this model. The signed geodesic distance d is the 1370 H. BERESTYCKI, J. BUSCA, AND I. FLORENT unique solution of  σ( )2 2 2 + ρκ 2σ( ) + κ2 2 2 =  x y dx 2 y x dx dy y dy 1 (6.2) d(x = 0, y) = 0  d(x, y)>0forx > 0 with the usual notational abuse on σ . It turns out that d is given by the explicit expression 1 κz − ρ + 1 − 2ρκz + κ2z2 (6.3) d(x, y) = ln , κ 1 − ρ =ˆ/ ˆ = x ζ/σ(ζ) where z x y, x 0 d . From Theorem 1.2 one readily gets x (6.4) 0(x, y) ≡ (x, y,τ = 0) = . d(x, y) As for the equivalent local volatility σ , we know that it is the unique solution of (2.7). It can be computed explicitly by noting that, using (5.9), one has 1 (6.5) σ(x, y, 0) = = y 1 − 2ρκz + κ2z2 . dx (x, y) 6.2 The Heston Model In this model the driving process is given by √ dS = rS dt + y S dW1 (6.6) t t t t √ t = (θ − ) + κ 2 dyt a yt dt yt dWt where a, θ, and κ are nonnegative constants. As was mentioned earlier, all of our results, which are stated for a process yt valued in R, extend in a straightforward way to the case yt ∈ (0, +∞). Hence the signed geodesic distance d is the unique solution of   2 + ρκ + κ2 2 = ydx 2 ydx dy ydy 1 (6.7) d(x = 0, y) = 0  d(x, y)>0forx > 0. Once d is (numerically) computed, using Theorem 1.2 and 2.2 one easily infers the value of the asymptotic implied volatility 0(x, y) ≡ (x, y,τ = 0) = x/d(x, y) and the local volatility σ(x, y, 0) = 1/dx (x, y). 6.3 Further Results The main purpose of Theorem 1.2 was twofold. First, we determined the limiting value of the implied volatility function  in the limit τ → 0; second, we showed that  solves the nonlinear evolution problem (1.9). Following this methodology, it becomes clear that one can get further terms in a Taylor expansion in powers of τ. Indeed, this task is much simpler than determining the zero-order term, since any singularity is now removed from the problem. For this reason we STOCHASTIC VOLATILITY MODELS 1371

FIGURE 6.1. Implied , formula (6.4) vs. PDE solution for model (6.1) with r = 0, S0 = 4%, T = 2 years, σ ≡ 1, y0 = 0.2, a = 0, κ = 0.3, ρ =−0.2. will not undertake this task in full generality; rather, we give an example in the case (6.1) with a = ρ = 0. If one makes the first-order time to maturity expansion (6.8) (x, y,τ)= 0(x, y)(1 + τ1(x, y) + O(τ 2)) , it is not difficult to see that 1 must satisfy the transport equation 1 0 1 0 (6.9) 21 + y2σ 2(x)dd 1 + κ2 y2 dd 1 = σ 2(x)y2 xx + κ2 y2 yy , x x y y 2 0 2 0 where d is defined in (6.3). This equation happens to be explicitly solvable; we find 1 x 1 1 (6.10) 1(x, y) =− ln √ , d2 d(x, y) y σ(0)σ (x) (1 + κ2z2)1/4 where the notation is defined in Section 6.1 above. Remark. Following a completely different and interesting approach based on for- mal asymptotics and educated guesses, Hagan et al. [17] have independently ad- dressed in the question of asymptotic expansions on implied volatilities in stochas- tic volatility models. That paper is in the same spirit as an earlier work [18] on local volatility models. They derive in particular two-term expansions analogous to (6.8). It should be noted, however, that formula (A65) in [17, p. 101] appears to be somewhat different from ours.

6.4 Numerical Examples We examine the accuracy of the asymptotic formula in (1.9) by comparing it to benchmark prices computed by solving the pricing problem (1.7) on a refined finite difference grid. We address the example of the model described in Section 6.1 with 1372 H. BERESTYCKI, J. BUSCA, AND I. FLORENT

FIGURE 6.2. Implied volatility smile, formulae (6.4) and (6.10) vs. PDE solution for model (6.1) with r = 0, S0 = 4%, T = 2 years, σ ≡ 1, y0 = 0.2, a = 0, κ = 0.5, ρ = 0. a negative asset/volatility correlation that leads to a characteristic “skew” phenom- enon (Figure 6.1) and with a zero correlation that leads to a symmetric smile in log variables (Figure 6.2). Moreover, we illustrate the gain in accuracy provided by the two-term expansion (6.8) in Figure 6.2. We observe a satisfactory agreement between the asymptotic formula and the numerically computed smile. Acknowledgments. This paper was written as part of a research program at HSBC CCF Derivatives Research. The authors thank M. Avellaneda for many useful discussions about this paper. The authors also thank J. M. Roquejoffre for helpful comments about Proposition 3.2 and O. Alvarez for reference [1].

Bibliography [1] Alvarez, O. Bounded-from-below viscosity solutions of Hamilton-Jacobi equations. Differen- tial Integral Equations 10 (1997), no. 3, 419–436. [2] Avellaneda, M.; Boyer-Olson, D.; Busca, J.; Friz, P. Reconstruction of volatility: Pricing index options using the steepest-descent approximation. Risk (2002), October, 87–91. [3] Avellaneda, M.; Boyer-Olson, D.; Busca, J.; Friz, P. Application of large deviation methods to the pricing of index options in finance. C. R. Math. Acad. Sci. Paris 336 (2003), no. 3, 263–266. [4] Berestycki, H.; Busca, J.; Florent, I. An inverse parabolic problem arising in finance. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 12, 965–969. [5] Berestycki, H.; Busca, J.; Florent, I. Asymptotics and calibration of local volatility models. Quant. Finance 2 (2002), no. 1, 61–69. [6] Black, F.; Scholes, M. The pricing of option and corporate liabilities. J. Political Economy 81 (1973), no. 3, 637–654. [7] Breeden, D. T.; Litzenberger, R. H. Prices of state contingent claims implicit in option prices. J. Business 51 (1978), no. 4, 621–651. [8] Cabré, X. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equa- tions. Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 331–359. STOCHASTIC VOLATILITY MODELS 1373

[9] Crandall, M.G.; Ishii, H.; Lions, P.-L. User’s guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc. 27 (1992), no. 1, 1–67. [10] Crandall, M. G.; Lions, P.-L. Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Illinois J. Math. 31 (1987), no. 4, 665–688. [11] Derman, E.; Kani, I. Riding on a smile. Risk 7 (1994), no. 2, 32–39. [12] Derman, E.; Kani, I. Stochastic implied trees: arbitrage pricing with stochastic term and strike structure of volatility. Internat. J. Theoret. Appl. Finance 1 (1998), no. 1, 61–110. [13] Derman, E.; Kani, I.; Chriss, N. Implied trinomial trees of the volatility smile. J. Derivatives 3 (1996), no. 4, 7–22. [14] Dupire, B. Pricing with a smile. Risk 7 (1994), no. 1, 18–20. [15] Friedman, A. Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, N.J., 1964. [16] Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order. 2nd ed. Springer, Berlin–Heidelberg, 1998. [17] Hagan, P. S.; Kumar, D.; Lesniewski, A. S.; Woodward, D. E. Managing smile risk. Wilmott 18 (2002), no. 11, 84–108. [18] Hagan, P.; Woodward, D. E. Equivalent black volatilities. Appl. Math. Finance 6 (1999), 147–157. [19] Ishii, H. A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type. Proc. Amer. Math. Soc. 100 (1987), no. 2, 247–251. [20] Kružkov, S. N. Generalized solutions of Hamilton-Jacobi equations of eikonal type. I. State- ment of the problems; existence, uniqueness and stability theorems; certain properties of the solutions. Mat. Sb. (N.S.) 98(140) (1975), no. 3(11), 450–493. [21] Lee, R. W. Implied and local volatilities under stochastic volatility. Internat. J. Theoret. Appl. Finance 4 (2001), no. 1, 45–89. [22] Lions, P.-L. Generalized solutions of Hamilton-Jacobi equations. Research Notes in Mathemat- ics, 69. Pitman, Boston–London, 1982. [23] Merton, R. C. Theory of rational pricing. Bell. J. Economics 4 (1973), no. 1, 141–183. [24] Varadhan, S. R. S. On the behavior of the fundamental solution of the heat equation with vari- able coefficients. Comm. Pure Appl. Math. 20 (1967), 431–455.

HENRI BERESTYCKI JÉRÔME BUSCA École des Hautes Études CNRS and Université Paris Dauphine en Sciences Sociales CEREMADE C.A.M.S. Place Maréchal de Lattre de Tassigny 54 Boulevard Raspail 75775 Paris Cedex 16 75270 Paris Cedex 06 FRANCE FRANCE E-mail: jerome.busca@ E-mail: [email protected] normalesup.org

IGOR FLORENT HSBC CCF 103 Avenue des Champs-Elysées 75419 Paris Cedex 08 FRANCE E-mail: [email protected] Received August 2003.