Longtime large deviations for the multiasset Wishart stochastic volatility model and option pricing Aurélien Alfonsi^{1}^{1}1Université ParisEst, Cermics (ENPC), INRIA, F77455 MarnelaVallée, France.
David Krief ^{2}^{2}2LPSM, Université Paris Diderot, Paris, France
Peter Tankov ^{3}^{3}3ENSAE ParisTech, Palaiseau, France
In this paper, we prove a large deviations principle for the class of multidimensional affine stochastic volatility models considered in (Gourieroux, C. and Sufana, R., J. Bus. Econ. Stat., 28(3), 2010), where the volatility matrix is modelled by a Wishart process. This class extends the very popular Heston model to the multivariate setting, thus allowing to model the joint behaviour of a basket of stocks or several interest rates. We then use the large deviation principle to obtain an asymptotic approximation for the implied volatility of basket options and to develop an asymptotically optimal importance sampling algorithm, to reduce the number of simulations when using MonteCarlo methods to price derivatives.
Key words: Large deviations, Wishart process, Importance sampling, Basket options, Implied volatility
MSC2010: 60F10, 91G20, 91G60
1. Introduction
The Heston stochastic volatility model [Heston, 1993] is one of the most popular models in quantitative finance for the evolution of a single asset price. The Wishart stochastic volatility model is its natural extension to a basket of assets, since it coincides with the Heston model in dimension and preserves the affine structure. This model, proposed in [Gourieroux and Sufana, 2010], assumes that under the riskneutral probability, the vector of asset prices is modelled as an Itô process
(1.1) 
where the volatility matrix follows the Wishart process with dynamics
(1.2) 
where and are independent standard dimensional and dimensional Brownian motions, and is the diagonal matrix whose diagonal elements are given by the vector .
The matrix process (1.2) has been introduced by [Bru, 1991] to model the perturbation of experimental biological data. As shown by [Bru, 1991] and [Cuchiero et al., 2011] in a more general framework, for (resp. ), the SDE (1.2) has a unique strong (resp. weak) solution. Furthermore, since is positive semidefinite [Bru, 1991, Prop. 4], Wishart processes turn out to be very suitable processes to model covariance matrices. This, and the affine property of the Wishart process, led several authors to use them in stochastic volatility models for a single asset, such as [Da Fonseca et al., 2008] and [Benabid et al., 2008] and in the Wishart stochastic volatility model for multiple assets (1.1)–(1.2). Subsequently, this model has been extended by [Da Fonseca et al., 2007] to include a constant correlation between and in a way to preserve the affine structure.
By using the affine property, the Laplace transform of the model (1.1)–(1.2) is computed as follows [Da Fonseca et al., 2007].
(1.3) 
where and satisfy the matrix Riccati equations
with initial conditions , and . Since the Riccati equations can be solved explicitly, the Laplace transform can be expressed explicitly in terms of matrix exponentials and inverses.
The goal of the present paper is to prove a large deviations principle the Wishart stochastic volatility model (1.1)–(1.2) in the largetime asymptotic regime. Since the Laplace transform of the logprice vector in the Wishart model is known explicitly, a natural path towards a large deviations principle is via GärtnerEllis theorem. However, despite the explicit form of the Laplace transform, it is not easy to calculate its longtime asymptotics and to check the assumptions of the theorem because of the multidimensional setting. In this paper we therefore focus on a (large enough) subclass of the model (1.1)–(1.2) which enables us to obtain a simpler formula for the limiting Laplace transform and then prove a large deviations principle.
Beyond its theoretical interest, knowing that a given model satisfies a large deviations principle, and knowing the explicit form of the rate function, enables one to develop a number of important applications. One can mention e.g., efficient importance sampling methods for Monte Carlo option pricing; asymptotic formulas for option prices and implied volatilities in various asymptotic regimes, approximate evaluation of risk measures, simulation of rare events and others. We refer the reader to [Pham, 2007] for a review of various applications of large deviations methods in finance. In this paper we develop applications to variance reduction of Monte Carlo methods and to the asymtotic computation of implied volatilities far from maturity.
Our variance reduction method follows previous works of [Guasoni and Robertson, 2008], [Robertson, 2010] and [Genin and Tankov, 2016] and uses Varadhan’s lemma of large deviations theory to approximate the optimal measure change in the importance sampling algorithm. Note that since the Laplace tranform is known explicitly, Fourier inversion methods can be used, as explained in [Da Fonseca et al., 2007]. However, these methods are much less competitive than in dimension since they require to approximate an integral on . When, for complexity reasons, Fourier methods are not an option, the use of a large number of MonteCarlo simulations is necessary. [Ahdida and Alfonsi, 2013] present an exact simulation method for Wishart processes and a second order scheme for the Gourieroux and Sufana model (1.1)–(1.2). Thus, it is possible to sample efficiently such processes, and it is relevant to develop variance reduction techniques to reduce computational costs.
The approximation of implied volatility far from maturity extends earlier results on the Heston model and the onedimensional affine stochastic volatility models [Forde and Jacquier, 2011, Jacquier et al., 2013] to the multidimensional setting of Wishart model. Once again, this approach is more relevant in the multidimensional setting, since in onedimensional affine models the implied volatility may be quickly computed by Fourier inversion.
In this paper, we denote the set of real squared matrices, the set of symmetric matrices and , (resp. ), the sets of symmetric an nonnegative (resp.) positive definite. For a Borel set , we denote by the closure of and by the interior of .
The paper is structured as follows. In Section 2, we describe the model, make certain assumptions on the parameters and give some properties of the model. In Section 3, we prove that the asset logprice vector satisfies large deviations principle when maturity goes to infinity. In Section 4, we calculate the asymptotic put basket implied volatility, following the approach of [Jacquier et al., 2013]. In Section 5, we develop the variance reduction method using Varadhan’s lemma. Finally, in Section 6, we test numerically the results of Sections 4 and 5.
2. The Wishart stochastic volatility model
In this section we introduce the subclass of the Wishart stochastic volatility models, in which we are interested in the present paper, and compute the Laplace transform of the log stock price process.
Let be a dimensional vector stochastic process with dynamics
(2.1) 
where , , is dimensional standard Brownian motion and the stochastic volatility matrix is a Wishart process with dynamics
(2.2) 
with , invertible, and is a matrix standard Brownian motion independent of . Note again that [Bru, 1991, Prop. 4]. Let us also assume that is such that .
Remark 2.1.
Remark 2.2.
In dimension one, the model defined by eqs. (2.1) and (2.2) corresponds to the famous Heston model [Heston, 1993] and being negative definite yields the mean reversion property of the stochastic volatility process.
Defining the logprice , a simple application of Itō’s lemma gives
(2.3) 
We are interested in the Laplace transform of . In order to calculate it, we first cite the following proposition.
Proposition 2.3.
[Alfonsi et al., 2016, Prop. 5.1.]. Let , , and with dynamics (2.2). Let be such that
If , then we have for
with
If besides, , then
and
The following proposition provides and explicit formula for the Laplace transform of the log stock price in the model (2.1)–(2.2).
Proposition 2.4.
Let be the function defined by
(2.4) 
Let , be the set defined by
Then, for all , the Laplace transform of is
where
Proof.
By conditioning on the trajectory of , we have
where
Let . Then and
Therefore, by Proposition 2.3,
(2.5) 
where
Since , we can write , where is diagonal, is orthonormal and .
Replacing by the latter expression finishes the proof.∎
Remark 2.5.
Note that, when , is not invertible. The notation is therefore abusive and is to be interpreted as the finite limit
Remark 2.6.
The set is bounded. Indeed, let , with and . Then, letting , we have
It follows that is contained, e.g., in the set with
3. Longtime large deviations for the Wishart volatility model
In this section, we prove that the Wishart stochastic volatility model satisfies a large deviation principle when time tends to infinity.
3.1. Reminder of large deviations theory
Let us recall some standard definitions and results of large deviations theory. For a wider overview of large deviations theory, we refer the reader to [Dembo and Zeitouni, 1998]. We consider a family of random variables on a measurable space , where is a topological space.
Definition 3.1 (Rate function).
A rate function is a lower semicontinuous mapping . A good rate function is a rate function such that, for every , is compact.
Definition 3.2 (Large deviation principle).
satisfies a large deviation principle with rate function if, for every , denoting and the interior and the closure of ,
Definition 3.3.
Let be a convex function with domain . is called essentially smooth if is differentiable on and for every , .
The following theorem is the celebrated GärtnerEllis theorem of the large deviations theory. [Dembo and Zeitouni, 1998] give a version of this theorem for a family of random variables parameterized by an integer number (see paragraph 2.3 in their book), but the version for families parameterized by a real number is easily deduced from the abstract GärtnerEllis theorem given in paragraph 4.5.3.
Theorem 3.4 (GärtnerEllis).
Let be a family of random vectors in . Assume that for each ,
(3.1) 
exists as an extended real number. Assume also that 0 belongs to the interior of . Denoting
the FenchelLegendre transform of , the following hold.

For any closed set ,

For any open set ,
where is the set of exposed points of , whose exposing hyperplane belongs to the interior of .

If is an essentially smooth, lower semicontinuous function, then satisfies a large deviations principle with good rate function .
Remark 3.5.
The function of (3.1) is a convex function. Indeed, let and . A direct application of Hölder’s inequality yields
Applying the logarithm then proves that and therefore are convex.
Theorem 3.6 (Varadhan’s Lemma, extension of [Guasoni and Robertson, 2008]).
Let be a metric space with its Borel field. Let be a family of valued random variables that satisfies a large deviations principle with rate function . If is a continuous function which satisfies
for some , then, for any ,
where denotes the law of
3.2. Longtime behaviour of the Laplace transform of the logprice
Let and define the transformation , which corresponds to the longtime behaviour of . We are interested in the function
We first give the following lemma.
Lemma 3.7.
Let such that est invertible for all . Then, is bounded for all sufficiently large .
Proof.
Since is invertible, for all ,
where . Now, the fact that est invertible for means that the eigenvalues of satisfy or for all . This implies for some , and since the adjugate matrix of has coefficients of order , we get that is bounded for . Therefore, is bounded, and as well, whenever is sufficiently large. ∎
We now characterise the asymptotic behaviour of the Laplace transform of .
Proposition 3.8.
Define
(3.2) 
For every ,
Proof.
Let . By Proposition 2.4,
(3.3)  
Write , where is diagonal, is orthonormal and let . Then
Let and be square matrices with and . We then have
and
Therefore,
(3.4)  
and
where the invertibility of is guaranteed for every by the existence of the Laplace transform. Since and , and is therefore invertible. Hence
and
But
where is bounded by Lemma 3.7. Therefore,
and as . Using (3.4), we find
We have , since the latter determinant is a nonzero polynomial of (for the determinant is clearly positive). Thus, by passing to the limit, . Furthermore, since , is bounded. Therefore,
Finally, passing to the limit in (3.3) finishes the proof. ∎
The next proposition proves the essential smoothness of .
Proposition 3.9.
The function defined in (3.2) is essentially smooth.
Proof.
The function defined in (3.2) is a lower semicontinuous proper convex function with domain . Furthermore, since for every , , is of class on . Only remains to prove that when goes to the boundary of . Let . By Proposition 3.8
Then for every ,
where satisfies
Multiplying this equation by and using the cyclic property of the trace, we get
and therefore
(3.5) 
where
We write with diagonal and denote , which is invertible since is orthonormal and . Then