## Asian options and stochastic volatility

##### Summary

In modern asset price models, stochastic volatility plays a crucial role in order to explain several stylized facts of
returns. Recently, [3] introduced a class of stochastic volatility models (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 thesis 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
Levy 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 [3] introduces a class of stochastic
volatility models (BNS SV model) based on superposition of Ornstein-Uhlenbeck processes driven by
subordinators (Levy 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 [17], or Eberlein and
Keller [9]).
In the comparison of the BNS SV model, we will use an alternative model NIG Levy 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 stochastic volatility and time-dependency
of asset return data. Both models are described and provided with theoretical background. Moreover
difference in pricing Asian option with the two different models will be studied.
Unlike the Black-Scholes model, closed option pricing formulae are not available in exponential Levy
models and one must use either deterministic numerical methods (see Carr [8] for the LP model and
Benth [7] for the BNS SV model) or Monte Carlo methods. In this thesis we will restrict ourselves to
Monte Carlo methods.
As described in Benth [6] the best way of simulating a NIG Levy process is by a quasi-Monte Carlo
method. We will use a simpler Monte-Carlo method, which needs bigger samplesize 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 [3].
This is a quiet bothersome algorithm, since it includes a numerical inversion of the Levy measure of the
Background driving Levy process (BDLP). Therefore it has a large processing time, hence we will not
deal with this algorithm.
The most popular algorithm is a series representation by Rosinski [16] . The special case of the IGOU
process is described in Barndorff [5]. Recently Zhang & Zhang [21] introduced an exact simulation
method of an IG-OU process, using the rejection method. We will compare these last two algorithms.