A Pricing Process with Stochastic Volatility Controlled by a Semi-Markov Process

This paper is devoted to the investigation of the geometrical Brownian motion as a price process where the drift and volatility are controlled by a semi-Markov process. Conditions of risk-neutral measure are given as well as a formula for the risk-neutral price for European options. The discrete ver...

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Bibliographic Details
Published in:Communications in statistics. Theory and methods 2004-01, Vol.33 (3), p.591-608
Main Authors: Silvestrov, Dmitrii, Stenberg, Fredrik
Format: Article
Language:English
Subjects:
Applications
Controlling semi-Markov process
Exact sciences and technology
Inference from stochastic processes
time series analysis
Insurance, economics, finance
Mathematics
Monte Carlo algorithm
Pricing process
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)
Statistics
Stochastic volatility
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Summary:This paper is devoted to the investigation of the geometrical Brownian motion as a price process where the drift and volatility are controlled by a semi-Markov process. Conditions of risk-neutral measure are given as well as a formula for the risk-neutral price for European options. The discrete version, the binomial model controlled by a semi-Markov chain, is examined and limit theorems describing the transition from discrete time binomial to continuous time model are given. A system of partial differential equations for distribution functions of average volatility is given. Related Monte Carlo algorithms are described.
ISSN:0361-0926
1532-415X
1532-415X
DOI:10.1081/STA-120028686