| Summary: | In this thesis we consider the calibration of models based on Lévy processes to option
prices observed in some market. This means that we choose the parameters of the option
pricing models such that the prices calculated using the models correspond as closely as
possible to these option prices. We demonstrate the ability of relatively simple Lévy option
pricing models to nearly perfectly replicate option prices observed in nancial markets.
We speci cally consider calibrating option pricing models to barrier option prices and
we demonstrate that the option prices obtained under one model can be very accurately
replicated using another. Various types of calibration are considered in the thesis.
We calibrate a wide range of Lévy option pricing models to option price data. We con-
sider exponential Lévy models under which the log-return process of the stock is assumed
to follow a Lévy process. We also consider linear Lévy models; under these models the
stock price itself follows a Lévy process. Further, we consider time changed models. Under
these models time does not pass at a constant rate, but follows some non-decreasing Lévy
process. We model the passage of time using the lognormal, Pareto and gamma processes.
In the context of time changed models we consider linear as well as exponential models.
The normal inverse Gaussian (N IG) model plays an important role in the thesis.
The numerical problems associated with the N IG distribution are explored and we
propose ways of circumventing these problems. Parameter estimation for this distribution
is discussed in detail.
Changes of measure play a central role in option pricing. We discuss two well-known
changes of measure; the Esscher transform and the mean correcting martingale measure.
We also propose a generalisation of the latter and we consider the use of the resulting
measure in the calculation of arbitrage free option prices under exponential Lévy models. === PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015
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