A Switching Black-Scholes Model and Option Pricing
Date
2003
Authors
Webb, Melanie Ann
Editors
Advisors
Hoek, John van der
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Thesis
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Abstract
Derivative pricing, and in particular the pricing of options, is an important area of current research in financial
mathematics. Experts debate on the best method of pricing and the most
appropriate model of a price process to use.
In this thesis, a ``Switching Black-Scholes'' model of a price
process is proposed. This model is based on
the standard geometric Brownian motion (or Black-Scholes) model of a
price process.
However, the drift and volatility parameters are
permitted to vary between a finite number of possible values at
known times, according to the state of a hidden Markov chain.
This type of model has been found to
replicate the Black-Scholes implied volatility smiles observed in
the market, and produce option prices which are closer to market
values than those obtained from the traditional Black-Scholes
formula.
As the Markov chain incorporates a second source of uncertainty into
the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no
unique option pricing methodology exists. In this thesis, we apply
the methods of mean-variance hedging, Esscher transforms and minimum
entropy in order to price options on assets which evolve according to
the Switching Black-Scholes model. C programs to compute these prices
are given, and some particular numerical examples are examined.
Finally, filtering techniques and reference probability methods are
applied to find estimates of the model parameters and state of the
hidden Markov chain.
School/Discipline
Applied Mathematics
Dissertation Note
Thesis (Ph.D.)--Applied Mathematics, 2003.