Viscosity Solutions and American Option Pricing in a Stochastic Volatility Model of the Ornstein-Uhlenbeck Type

In this paper, we study the valuation of American type derivatives in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We characterize the value of such derivatives as the unique viscosity solution of an integral-partial differential equation when the payoff function satisfies a Lipschitz condition.


Introduction
In their seminal paper, Barndorff-Nielsen and Shephard [4] introduced a model that has been shown to describe particularly well financial assets for which logreturns have heavy tail distributions and display long range dependence. In this model, the volatility of the asset is described by an Ornstein-Uhlenbeck type process with a pure jump Lévy process acting as the background driving process. An empirical study was made in [4] and showed from exchange rate data that suitable distributions for the Lévy process are the so-called generalized inverse gaussian distributions from which well understood examples are the normal inverse gaussian (studied in [3]) and the gamma distribution.
The BNS model has been studied from different points of view. Benth et al. [5] considered the problem of optimal portfolio selection. Nicolato and Vernados [10] have studied European option pricing and described the set of equivalent martingale measures under this model. To evaluate these types of options, the authors propose the transform-based method and a simple Monte Carlo method.
In this paper, we consider the pricing of American options with the use of integral-partial differential equations (IPDE). Although our technique can be simplified and used for European options and certain path dependent options such as barrier options (see [6] for a definition and examples), we will mainly concentrate on American type derivatives which have not been studied for this model. The main difficulty in this case is the lack of Lipschitz continuity of some of the coefficients of the IPDE.
The connection between viscosity solutions of IPDE's and Lévy processes has been studied in the literature by various authors. Pham [11] considered a general stopping time problem of a controlled jump diffusion processes. However, his results do not apply here because the Lipschitz condition on the coefficients is not satisfied in our current setting. Cont and Voltchkova [6] studied barrier options and Barles et al. [2] established the connection between viscosity solutions and backward stochastic differential equations. In these papers, the stock price considered is modelled by a stochastic differential equation with jumps driven by a Lévy process. The main difference between the BNS model and these models is the presence of stochastic volatility. However, we will see that the lack of smoothness of the solution to our IPDE will also lead us to consider the notion of viscosity solutions as presented in [7].
The rest of the paper is organized as follows. In Section 2, we present the model and recall the results of Nicolato and Vernados [10] regarding the set of equivalent martingale measures. Section 3 is devoted to the continuity of the value function. In Section 4, we prove that the value function is the viscosity solution of the associated IPDE and the uniqueness of the solution is presented in Section 5.

Lévy Processes and the BNS Model
Let T > 0. We consider the stochastic volatility model of Barndorff-Nielsen and Shephard [4] for the price process of an asset, denoted by S = {S t } 0≤t≤T and defined on a filtered probability space (Ω, F , {F t } 0≤t≤T , P). We thus assume that the log-return X t = log(S t ) of the asset satisfies the following stochastic differential equation: in which µ, β ∈ R, λ > 0 and ρ ≤ 0. B = {B t } 0≤t≤T is a Brownian motion and Z = {Z t } 0≤t≤T is the background driving Lévy process (BDLP) under the physical measure P. In this model, Z has no gaussian component and the increments are positive. Z and B are assumed to be independent, and F = {F t } 0≤t≤T is the usual filtration generated by the pair (B, Z). The positivity of the jumps of Z insure that the process V is always positive. We denote by W the Lévy measure of Z.
Suppose Q is a probabilty measure equivalent to P under which S is a martingale. We are interested in American-type derivatives of the form in which h is the payoff function and T T is the set of all stopping times with values less or equal to T . Since {X t } 0≤t≤T and {V t } 0≤t≤T are Markov processes, U t can be written as a function of (x, v, t), say in which (X x,v t ) t≥0 is the process X for which X 0 = x and V 0 = v. We also denote by (V v t ) t≥0 the process V starting at V 0 = v at t = 0.

Equivalent Martingale Measures.
We start by summarizing the results of Nicolato and Vernados [10] concerning the set of equivalent martingale measures. In order to do so, we define the set and M ′ as the set of all equivalent martingale measures Q such that Z is still a Lévy process under Q independent of B, possibly with a different marginal distribution.
As in [10], we impose the following conditions on the process Z: (C1) The process Z is given by the characteristic triplet (0, 0, W ) so that the cumulant transform is given by for values of θ for which this expression is defined.
For z > 0 and n ≥ 1, we have 0 < z n ≤ n! b θ n 0 (e b θ0z − 1), so that µ n := ∞ 0 z n W (dz) < ∞. Furthermore, Assumption (C2) is a sufficient condition for the process Z to have finite moments of all orders.
The following theorem was proved in [10].
)ds and Z λt are respectively a Brownian motion and a Lévy process under Q. w y (x) = y(x)w(x) is the Lévy density of Z 1 under Q and κ y (θ) is the cumulant function.
In the remaining part of this paper, all expectations will be with respect to a chosen EMM Q, unless specified otherwise, and W and B will denote the associated Lévy measure and the Brownian motion associated to Q.
Let O = R × R + × [0, T ) and assume for a moment that u is Lipschitz in (x, v) and that is u is differentiable with respect to v and t, and twice differentiable with respect to x. We can then apply Itô's formula to U to find and V t is the Q-martingale given by if it can be shown that e −rt U t is also a martingale we can then expect u to satisfy the following integral-partial differential equation (IPDE) and this IPDE can be written as It is clear also that the function satisfies Condition 2.3 is in fact very restrictive and most of the time not satisfied. Despite this problem, we will see that u can still be regarded as a solution of this equation in a weaker sense.

Continuity of the Value Function
Recall the definition of the value of an American option with payoff h: In the rest of this paper, we will assume that h is positive and satisfies the Lipschitz condition, in other words ∃K > 0 such that For instance, the payoff function for an American put with strikeX > 0 is h(x) = max(X − exp(x), 0) and satisfies this condition.
Our goal is to show that the function u satisfies the IPDE (2.5) in some weak sense. In order to give meaning to this IPDE for a function u that doesn't satisfy basic differentiability conditions, we introduce the idea of viscosity solutions following Crandall and Lions [8]. Let W be the set of functions f : O → R that satisfy The function u is a viscosity solution if it is both subsolution and supersolution.
An important property of viscosity solutions is the continuity of the function. It is the content of the following proposition. Proof. In this proof, we will assume for simplicity that r = 0. The generalization to r > 0 is straightforward. Throughout, C is a positive constant that can change from line to line.
We start by showing the continuity of u with respect to (x, v), uniformly in t. We have the following representation of the volatility process: We define the integrated variance process started with V 0 = v by By Equation (2.2), we find that so that we have the following representation of the integrated variance process We also have the following identities: Using the Lipschitz condition on h, we obtain Then, Letting G = σ({Z s } 0≤s≤T ), the σ-field generated by the BDLP Z up to the maturity T , we find that is a G ∨ F t -submartingale and Doob's theorem applies. In other words, Also, And thus we proved the continuity of u in (x, v) uniformly in t since In particular u ∈ W because of the following inequality The next step of the proof is to show as |t − t ′ | → 0. This is easily obtained by first observing that As for the process V , and we need to show that E |Z λt ′ − Z λt | → 0 when |t ′ − t| → 0. We mentioned earlier that condition (C2) implies that the moments of Z t are finite for all orders. Thus Z is uniformly integrable. Since Z is also continuous in probability, it is continuous in L 1 and the conclusion follows.
Let's now show continuity with respect to time. Let 0 ≤ t ≤ t ′ ≤ T . Take τ ∈ T T −t and define τ ′ = τ ∧ (T − t ′ ). Then, . From this inequality, we readily find that which converges to zero as |t − t ′ | → 0. Global continuity follows from the following inequality and the fact that the first bound is independent of t ′ .

Viscosity Solutions
This section is devoted to the viscosity solution property of the value function u. In order to prove that u is a viscosity solution of (2.5), we need the following dynamic programming principle. It is a consequence of the martingale property of the Snell envelope stopped before its optimal stopping time and it is the key property needed in the proof of the subsolution property.
Hence we find that the process (e −rs h(X s )) 0≤s≤T is of Class D and we can apply the results of [9] to get the result.
The proof of the solution property of u makes use of the following lemma.
we can take a subsequence if necessary and find that |u(X s. with n → ∞. Taking the limit, we find Since u is continuous with respect to t, we find that η < e −rs u(x, v, t + s) − e −rs h(x) for s small enough. Then, for s small enough, for some constants δ 2 > 0 and δ 3 > 0. By the continuity in probability of the processes X et V , we know that this expression goes to zero when s → 0.
We can now show that u is a viscosity solution. Proof. We already know that u is continuous and in W. Let's start by showing that u is a supersolution of (2.5). Let (x, v, t) ∈ O and ψ satisfy the conditions given in the above definition of supersolutions. By definition, for any ∆t > 0, λds).
is also a martingale, we have the following inequality in other words, dividing by ∆t and taking the limit as ∆t → 0 Since, by definition, u(x, v, t) ≥ h(x), u satisfies Equation (2.5). To prove that u is a viscosity subsolution of (2.5), let (x, v, t) ∈ O and ψ satisfy the conditions of the above definition for subsolutions. If u(x, v, t) = h(x), the inequality (3.3) is satisfied. Otherwise, let We know from Lemma 4.1 that for any ∆t > 0. Knowing that Q(τ ǫ < ∆t) → 0 when ∆t → 0 by Lemma 4.2, dividing the preceding inequality by ∆t and taking the limit to 0, we get the desired result by Lebesgue's dominated convergence theorem.

Comparison Principles and Uniqueness of the Solution
In this section, we prove a comparison result from which we obtain the uniqueness of the solution of the IPDE. In proving comparison results for viscosity solutions, the notion of parabolic superjet and subjet as defined in Crandall et al. [8] is particularly useful. Setting y = (x, v), we define the parabolic superjet and its closure by (p n , q n , A n ) such that (p n , q n , A n ) ∈ J 2,+ u(y n , t n ) and (y n , t n ) → (y, t) .
The subjet and its closure are then defined similarly by We then have the following lemma which is essentially proved in [2] (lemma 3.3).
Pham [11]  Theorem 5.2. Let ǫ ≥ 0, and u 1 be a subsolution and u 2 a supersolution of (2.5) on O δ such that Proof. An IPDE of the form ∂ψ(x,ϑ,t) ∂t + L[ψ](x, ϑ, t) − rψ(x, ϑ, t) = 0 for (x, ϑ, t) ∈ O and ψ(x, ϑ, T ) = h(x) was shown to have a unique solution in [2] when the coefficients of L satisfy some given Lipschitz conditions. In fact when (x, ϑ, t) and 2δ |ϑ ′ − ϑ| and so the operator L satisfies the assumptions made in [2]. The extension of the uniqueness result to our current setting is straightforward and we only give the main details.
Since u 1 and u 2 are in W, the function w − φ attains its maximum (y * 1 , y * 2 , t * , s * ) (which depends on ǫ, α) in O δ × O δ . By a classical argument in the theory of viscosity solutions we can show that 1 when ǫ, α → 0. Applying Theorem 8.3 of Crandall et al. [7] to the functions w and φ, we find matrices Y 1 , Y 2 such that with a = 1 α (t * − s * ) and b = 1 ǫ (y * 1 − y * 2 ) and for 0 < ξ < 1 the inequalities are satisfied. Write these two expressions as max(A, B) ≥ 0 and max(C, D) ≤ 0.
Sending β to zero we get u 1 ≤ u 2 + ǫe r(T −t) on R × (δ, ∞) × [t 1 , T ]. As done in [2], we can repeat this argument as many times as needed to get A solution of (2.5)-(2.6) is said to be minimal if it is less or equal to any other solution of (2.5)-(2.6).
Proof. Let δ > 0 and define in which τ δ = inf{s ≥ 0 : V v s ≤ δ}. Then u δ is a viscosity solution of (2.5) on O δ with boundary conditions The proof of this statement is essentially the same as the proof for the viscosity solution property of u. The main difference is that the maturity T is replaced by τ δ . Note that V δ ′ s > δ for δ ′ > δe λT , hence u δ (x, v, t) = u(x, v, t) for all x ∈ R, t < T and v > δe λT .
Following Pham [11], we denote by U C x,v (O) the set of functions defined on O uniformly continuous in (x, v), uniformly in t. We have already shown that the function u satisfies |u(x ′ , v ′ , t) − u(x, v, t)| ≤ C |x ′ − x| + |v ′ − v| + |v ′ − v| .
Hence, u ∈ U C x,v (O). Using the two previous theorems, we can show the uniqueness in U C x,v (O).
Theorem 5.4. u is the unique viscosity solution of (2.5)-(2.6) in U C x,v (O).