Correlated Continuous Time Markov Chains and Derivatives Pricing
Author(s)
Dalessandro, Antonio
Type
Thesis
Abstract
In this thesis, properties and results on the continuous-time Markov chain approximation
for multivariate diffusions and Lévy processes are presented and applied to the problem
of derivative pricing. Numerical methods are formulated in order to approximate a specific
stochastic process by a corresponding continuous-time Markov chain. We produce
desired convergence results for such approximation schemes. A great emphasis is given
to the approximation of correlated multidimensional processes and to the construction of
the associated correlated continuous-time Markov chain. In fact, the correlation among
continuous-time Markov chains is the theme of this dissertation. We show how to build
a multi-dimensional continuous-time Markov chain that closely follow the dynamics of
a multivariate diffusion. We apply this result to compute the price of European options
where the underlying process dynamics is given by the Heston’s stochastic volatility
model. Furthermore, we introduce a method to create high-dimensional correlated
continuous-time Markov chains that approximate specified stochastic processes. We apply
this methodology to model the point-in-time credit rating dynamics and also to price
standardized Index collateralized debt obligations and more generally credit derivatives
portfolios.
for multivariate diffusions and Lévy processes are presented and applied to the problem
of derivative pricing. Numerical methods are formulated in order to approximate a specific
stochastic process by a corresponding continuous-time Markov chain. We produce
desired convergence results for such approximation schemes. A great emphasis is given
to the approximation of correlated multidimensional processes and to the construction of
the associated correlated continuous-time Markov chain. In fact, the correlation among
continuous-time Markov chains is the theme of this dissertation. We show how to build
a multi-dimensional continuous-time Markov chain that closely follow the dynamics of
a multivariate diffusion. We apply this result to compute the price of European options
where the underlying process dynamics is given by the Heston’s stochastic volatility
model. Furthermore, we introduce a method to create high-dimensional correlated
continuous-time Markov chains that approximate specified stochastic processes. We apply
this methodology to model the point-in-time credit rating dynamics and also to price
standardized Index collateralized debt obligations and more generally credit derivatives
portfolios.
Date Issued
2012
Date Awarded
2012-07
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Davis, Mark
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Master of Philosophy (MPhil)