Asian Options and the Effect of a Non-gaussian Stochastic Volatility Model

In modern asset price models, stochastic volatility plays a crucial role in order to explain several stylized facts of returns. Recently, [4] introduced a class of stochastic volatility models (the so called BNS SV model) based on superposition of Ornstein-Uhlenbeck processes driven by subordinators. The BNS SV model forms a flexible class, where one can easily explain heavy-tails and skewness in returns and the typical time-dependency structures seen in asset return data. In this paper the effect of stochastic volatility on Asian options is studied. This is done by simulation studies of comparable models, one with and one without stochastic volatility. Introduction Lévy processes are popular models for stock price behavior since they allow to take into account jump risk and reproduce the implied volatility smile. Barndorff-Nielsen and Shephard [4] introduced a class of stochastic volatility models (BNS SV model) based on superposition of Ornstein-Uhlenbeck processes driven by subordinators (Lévy processes with only positive jumps and no drift). The distribution of these subordinators will be chosen such that the log-returns of asset prices will be distributed approximately normal inverse Gaussian (NIG) in stationarity. This family of distributions has proven to fit the semi-heavy tails observed in financial time series of various kinds extremely well (see Rydberg [20], or Eberlein and Keller [10]). In the comparison of the BNS SV model, we will use an alternative model, an NIG Lévy process model (LP model), which has NIG distributed log-returns of asset prices, with the same parameters as in the BNS SV case. Unlike the BNS SV model, this model doesn’t have time-dependency of asset return data. Both models are described and the effect on pricing Asian options with the two different models will be studied. This is done by a case study with calibrated parameters on stock data of the AmsterdamStock-Exchange-Index (AEX). We chosed Asian options because they are path dependent options. The time-dependency of the asset return data in the BNS SV model leads to a difference in pricing. Unlike the Black-Scholes model, closed option pricing formulae are in general not available in exponential Lévy models and one must use either deterministic numerical methods (see e.g. Carr and Madan [9] for the LP model and Benth and Groth [8] for the BNS SV model) or Monte Carlo methods. In this paper we will restrict ourselves to Monte Carlo methods. As described in Benth, Groth and Kettler [7] an efficient way of simulating a NIG Lévy process is by a quasi-Monte Carlo method. We will use a simpler Monte-Carlo method, which needs bigger sample size to reduce the error. Simulating from the BNS SV model involves simulating of an Inverse Gaussian Ornstein-Uhlenbeck (IG-OU) process. The oldest algorithm of simulating a IG-OU process is described in Barndorff-Nielsen and Shephard [4]. This is a quiet bothersome algorithm, since it includes a numerical inversion of the Lévy measure of the Background driving Lévy process (BDLP). Therefore it has a large processing time, hence we will not deal with this algorithm. The most popular algorithm is a random series representation by Rosinski [19] . The special case of the IG-OU process is described in Barndorff-Nielsen and Shephard [6]. Recently Zhang & Zhang [24] introduced an exact simulation method of an IG-OU process, using the rejection method. We will compare the efficiency and reliability of these last two algorithms. 1. The models To price derivative securities, it is crucial to have a good model of the probability distribution of the underlying asset. The most famous continuous-time model is the celebrated Black-Scholes model *The author would like to expres her gratitude to Fred Espen Benth, who provided usefull comments which leaded to an improved version of this paper. 1 (also called geometric Brownian motion), which uses the Normal distribution to fit the log-returns of the underlying asset. However the log-returns of most financial assets do not follow a Normal law. They are skewed and have actual kurtosis higher than that of the Normal distribution. Hence other more flexible distributions are needed. Moreover to model the behavior through time we need more flexible stochastic processes. Lévy processes have proven to be good candidates, since they preserve the property of having stationary and independent increments and they are able to represent skewness and excess kurtosis. In this section we will describe the Lévy process models we use. We also give a method how to fit the models on historical data. 1.1. Distributions. The inverse Gaussian distribution IG(δ, γ) is a distribution on R+ given in terms of its density, fIG(x; δ, γ) = δ √ 2π exp(δγ)x− 3 2 exp { − 2 (δ2x−1 + γx) } , x > 0 where the parameters δ and γ satisfy δ > 0 and γ ≥ 0. The IG distribution is infinitely divisible, self-decomposable. The associated Lévy process is a jump process of finite variation. The normal inverse Gaussian (NIG) distribution has values on R, it is defined by its density function, fNIG(x;α, β, δ, μ) = α π exp ( δ √ α2 − β2 + β(x− μ) ) K1(δα√1 + (x−μ δ )2) √ 1 + ( x−μ δ )2 where K1 is the modified Bessel function of the third kind and index 1. Moreover the parameters are such that μ ∈ R, δ ∈ R+ and 0 ≤ β < |α|. The NIG distribution is infinitely divisible and there exists a NIG Lévy process. The NIG distribution can be written as a mean-variance mixture of a normal distribution with an IG(δ, √ α2 − β2) distribution (see Barndorff-Nielsen [1]). More specifically, if we take σ ∼ IG(δ, √ α2 − β2) independently distributed of ∼ N(0, 1), then x = μ + βσ + σ is distributed NIG(α, β, δ, μ). See e.g. Barndorff-Nielsen [2] for an extensive description of the above distributions. In this text we will work on a probability space (Ω,F ,P) equipped with a filtration {Ft}t∈[0,T ] satisfying the usual conditions, with T <∞ being the time horizon. 1.2. Exponential Lévy process model. As model without stochastic volatility we will use a fitted NIG distribution to the log-returns. i.e. d logS(t) = dX(t), (1) where S is an arbitrary stock-price and X is an NIG Lévy process. This model is flexible to model with, but a drawback is that the returns are assumed independently. In literature this model has also been referred to as exponential Lévy process model, since the solution is of the form, S(t) = S(0)e. 1.3. Barndorf-Nielsen and Shephard stochastic volatility model. In practice there is evidence for volatility clusters. There seems to be a succession of periods with high return variance and with low variance. This means that large price variations are more likely to be followed by large price variations. This observation motivates the introduction of a model where the volatility itself is stochastic. The Barndorf-Nielsen and Shephard stochastic volatility (BNS SV) model is an extension of the BlackScholes model, where the volatility follows an Ornstein-Uhlenbeck (OU) process driven by a subordinator. In the Black-Scholes model the asset price process {S(t)}t≥0 is given as a solution to the SDE, d logS(t) = { μ+ βσ } dt+ σdB(t) (2) where an unusual drift is chosen and B(t) is a standard Brownian motion. The BNS SV model allowes σ to be stochastic. More precisely σ(t) is an OU process or a superposition of OU processes i.e.