Option pricing with generalized continuous time random walk models
Abstract
The pricing of options is one of the key problems in mathematical finance. In recent
years, pricing models that are based on the continuous time random walk (CTRW), an
anomalous diffusive random walk model widely used in physics, have been introduced.
In this thesis, we investigate the pricing of European call options with CTRW and generalized
CTRW models within the Black-Scholes framework. Here, the non-Markovian
character of the underlying pricing model is manifest in Black-Scholes PDEs with fractional
time derivatives containing memory terms. The inclusion of non-zero interest
rates leads to a distinction between different types of \forward" and \backward" options,
which are easily mapped onto each other in the standard Markovian framework,
but exhibit significant dfferences in the non-Markovian case. The backward-type options
require us in particular to include the multi-point statistics of the non-Markovian pricing
model. Using a representation of the CTRW in terms of a subordination (time change)
of a normal diffusive process with an inverse L evy-stable process, analytical results can
be obtained. The extension of the formalism to arbitrary waiting time distributions and
general payoff functions is discussed. The pricing of path-dependent Asian options leads
to further distinctions between different variants of the subordination. We obtain analytical
results that relate the option price to the solution of generalized Feynman-Kac
equations containing non-local time derivatives such as the fractional substantial derivative.
Results for L evy-stable and tempered L evy-stable subordinators, power options,
arithmetic and geometric Asian options are presented.
Authors
Li, ChaoCollections
- Theses [4248]