Computing the Implied Volatility 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 option 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 stochastic process. 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 local volatility 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 implied volatility is determined, for each given product (that is for each given strike and expiry date defining, say, the call option) 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 Heston model 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.