Recent years have seen an increased level of interest in pricing equity options under a stochastic volatility model such as the Heston model. Often, simulating a Heston model is difficult, as a standard finite difference scheme may lead to significant bias in the simulation result. Reducing the bias to an acceptable level is not only challenging but computationally demanding. In this paper we address this issue by providing an alternative simulation strategy -- one that systematically decreases the bias in the simulation. Additionally, our methodology is adaptive and achieves the reduction in bias with "near" minimum computational effort. We illustrate this feature with a numerical example.
Under the standard Black-Scholes framework, the asset price dynamics is given by the lognormal model dS t S t = µdt + σdW t (1.1) where the drift parameter µ and the volatility σ are considered constant (or at best piecewise constant). The popularity of this model lies in its convenience and simplicity; however, these features come at a price. The standard Black-Scholes model is not able to capture the volatility smile observed in the trading market. The Heston model assumes that the variance V = σ 2 is itself a stochastic process -more specifically a square-root diffusion model of the CIR (Cox-Ingersoll-Ross) type (see [11], [9]). It further allows the variance, V , to be correlated with the stock price S, thereby capturing the volatility smile. Furthermore, the Heston model provides a closed-form solution for pricing European Options, allowing one to fit the model to observed option prices. The Heston model is given by the coupled SDE (stochastic differential equations):
where the variance V is modelled by a square-root diffusion process with parameters κ which is the speed at which the process mean reverts to the long term variance θ, and σ V , the volatility of the variance. We denote by ρ the instantaneous correlation between the two noise processes: d W S , W V = ρ dt.
By Cholesky factorization, one can rewrite equations (1.2)-(1.3) as:
(2) t
(1.4)
(1.5)
with independent Brownian motions dW
(1) t and dW
(2)
t . There is a wide range of literature on the simulation of this model. Some of these are based on Euler discretisation; others on the improved finite difference approximations such as higher-order Milstein schemes and Predictor-Corrector methods; and still others based on distributionally exact ( [5]) or approximate ( [3]) simulation methods. In our study we focus mainly on the Exact simulation method as proposed by Broadie and Kaya [5] but we will avoid the numerical inversion of the Laplace Transform which seems to be the most time consuming part of their simulation technique. Doing so, will mean introducing bias in the simulation. We will address this problem by providing an adaptive strategy that systematically controls the bias in the simulation. Furthermore, this is achieved with minimal computational overhead. In most cases, our method should prove faster than that of [5], as there is no costlier inversion to perform.
Here is the layout of the rest of the paper. In Section 2, we recall the general algorithm (cf. Broadie and Kaya [5]) for simulating the Heston model, to emphasise the basic difficulty: the simulation of the integral of the variance process. In Section 3 we focus on the dynamics of the variance process and study its properties by transforming it into a canonical process. We also describe methods for simulating the latter process. In Section 4 we describe a method for simulating bridges corresponding to the latter process, and their use in simulating the integral of the variance process. In Section 5 we relate the integral of the variance process to a weighted integral of the canonical process, and we also calculate some moments of the latter for the canonical bridge process. In Sections 6, 7, we explore adaptive computation of the integral. In Section 8, we describe the results of some numerical experiments on the accuracy and efficiency of our adaptive algorithm. Finally, we provide concluding remarks and future research ideas in Section 9. Lengthy or highly technical proofs are given in Appendices A-C.
All the details of the simulation algorithm for the equity price governed by the stochastic differential equation in (1.4), are given in [5]. We reproduce them here for the sake of completeness and to emphasise the role played by the integral of the variance process, V . Integrating (1.4) between two dates of interest (e.g., coupon/reset dates), t j-1 and t j , we obtain
V s dW (2) s where ∆t j = t j -t j-1 . Similarly, integrating (1.5) we obtain
V s dW (1) s .
(2.1)
In the next section we will see that it is not complicated to simulate V j in a manner that is not based on (2.1). We can easily use (2.1) to obtain:
where ∆V j = V j -V j-1 . The only component left is
s ; but since V s is independent of the Brownian increments dW
Using this result we have:
V s dW (1) s
where, given V s , 0 ≤ s ≤ t j , Z j ∼ N (0, 1) and is conditionally independent of W (1) . Based on (2.2) and (2.3), it is clear that simulation of the Heston model is straightforward except for the simulation of the integral tj tj-1 V s ds. In [5], it is done in an unbiased manner by using the distribution of this integral. The distribution is not available in closed form and is computationally intensive to compute. Our approach is to use a random quadrature with an adaptive control of bias.
Our first task is to be able to simulate equation (1.5) exactly. In order to do so, we cite the following result from [12]: Then for all real β = 0,
where Q d x is the distribution of the d-d
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