Lévy Models - Computational Methods for Quantitative Finance - page 123

Chapter 10
Lévy Models

One problem with the Black-Scholes model is that empirically observed log re-
turns of risky assets are not normally distributed, but exhibit significant skewness
and kurtosis. If large movements in the asset price occur more frequently than in
the BS-model of the same variance, the tails of the distribution of X t , t> 0, should
be “fatter” than in the Black-Scholes case. Another problem is that observed log-
returns occasionally appear to change discontinuously. Empirically, certain price
processes with no continuous component have been found to allow for a consider-
ably better fit of observed log returns than the classical BS model. Pricing derivative
contracts on such underlyings becomes more involved mathematically and also nu-
merically since partial integro-differential equations must be solved. We consider a
class of price processes which can be purely discontinuous and which contains the
Wiener process as special case, the class of Lévy processes . Lévy processes contain
most processes proposed as realistic models for log-returns.

10.1 Lévy Processes

We start by recalling essential definitions and properties of Lévy processes and refer
to [18] and [143] for a thorough introduction to Lévy processes.

Definition 10.1.1 An adapted, càdlàg stochastic process X ={ X t : t ≥ 0 } on

F P F R

such that X 0 = (Ω, , , ) with values in 0 is called a Lévy process if it has

the following properties:

(i) Independent increments: X t −

F s ,0 ≤ ∞

X s is independent of s<t< ;

(ii) Stationary increments: X t − X s has the same distribution as X t − s ,0 ≤ s<

;
(iii) Stochastically continuous: lim t → s X t =

t< ∞

X s , where the limit is taken in proba-

bility.