Stochastic Volatility Models - Computational Methods for Quantitative Finance - page 105

Chapter 9
Stochastic Volatility Models

In Sect. 4.5, we considered local volatility models as an extension of the Black-
Scholes model. These models replace the constant volatility by a deterministic
volatility function, i.e. the volatility is a deterministic function of s and t .In stochas-
tic volatility (SV) models, the volatility is modeled as a function of at least one addi-
tional stochastic process Y 1 ,...,Y n v , n v ≥

1. Such models can explain some of the
empirical properties of asset returns, such as volatility clustering and the leverage
effect. These models can also account for long term smiles and skews.
We focus on SV models which are pure diffusion models, i.e. the processes
Y 1 ,...,Y n v used to model the volatility are diffusions. We will also assume that
there is only one risky underlying S . The flexibility offered by SV models comes
at the cost of an increase in dimension. In SV models, the price S is no longer
Markovian, since the price process is determined not only by its value but also by
the level of volatility. To regain a Markov process, one must consider the (n v +

1 ) -

dimensional process (S, Y 1 ,...,Y n v ) . This results in pricing PDEs in n v +

1 space
dimensions: each additional source of randomness Y i gives an additional dimension
in the pricing equation. Hence, we can use the discretization techniques developed
for the pricing of multi-asset options also for the pricing of options in SV models.

9.1 Market Models

A large class of diffusion SV models can described as follows. Consider the asset
price process S following the SDE

σ t S t d W t ,

d S t = μS t d t + (9.1)

where σ ={ σ t : t ≥ 0 } is called the volatility process. A widely used model for σ is

ξ(Y t ,...,Y n t ), σ t =

(9.2)

(Y 1 ,...,Y n v ) , n v

n v

where ξ : R → R ≥ 0 is some function and Y := ≥ 1, is

n v -valued diffusion process. The dynamics of the vector process Z

an R :=