Stochastic Volatility Models with Jumps - Computational Methods for Quantitative Finance - page 242

Fig. 15.2 Convergence rate
of the one-factor ( top )and
two-factor ( bottom ) Bates
model on sparse tensor
product space

15.5 Further Reading

A stochastic volatility model with jumps comparable to the BNS model is the so-
called COGARCH(1 , 1) process introduced by Klüppelberg et al. [105]. This model
is a continuous time version of the popular GARCH model in discrete time and
states that the asset log-price X is given by d X t = σ t d L t , where L is a Lévy pro-

cess and the jumps L of this Lévy process are also used to define the volatility
process σ . The model is generalized to the COGARCH( p,q ) process by Brockwell
et al. [30].
A large class of stochastic volatility models is described by Carr et al. [37] where
the stock price process S is given as the ordinary or the stochastic exponential of a
stochastic volatility process Z t =

L Y t , which is obtained by subordinating a Lévy
process L to Y (which, for example, is given by the CIR model).