| Summary: | 博士 === 國立中央大學 === 數學研究所 === 94 === This thesis consists of seven chapters. Its organization is stated as below.
In chapter 1, we study the locally risk-minimizing hedging strategy for asset models
with jumps. For this model, we give the minimal martingale measure under which we
give a PDE satisfied by the price of the option and we then construct a hedging strategy
for the contingent claim in the locally risk-minimizing sense.
Chapter 2 is devoted to finding the distribution of a squared Bessel process run for
an exponentially distributed time and applying this result to find the price of a zero
coupon bond at time zero when the pricing model involves a squared Bessel interest
process and there is one jump.
In chapter 3, we derive some properties of variants of squared Bessel processes
known as CIR processes in the finance literature, as they were introduced by Cox,
Ingersoll and Ross for the modelling of interest rates. By using the time-space transformation
from a Bessel-squared process to a CIR process, we are able to easily derive
the transition probability density function of a square-root process and compute explicitly
the resolvent density of CIR processes. As a consequence we can derive the
density of CIR processes sampled at an independent exponential time. Finally, we
exploit martingale methods to derive expressions for the Laplace transform of the first
hitting time of a point by a square-root process.
In chapter 4, we are concerned with the computation of Asian options when the
underlying asset has a jump. In the Black and Scholes model, Geman and Yor give
a closed-form formula for the price of an Asian option at a random exponential distributed
maturity (it then “suffices” to invert the Laplace transform to have the price
at a fixed time). The aim of this chapter is to obtain such a formula in a model (which
seems more realistic) of Black and Scholes with a jump at a random time, which ex-
tends the well-known case of the continuous Black and Scholes model. Furthermore, we
treat the multi-jump case. We present an integro-differential equation whose solution
leads to the time zero price of an Asian option with multi-jump, which is based upon
an identity in law between the exponential functional under study and the value at a
fixed time of a Markovian process.
Chapter 5 is concerned with the computation of Asian options on the underlying
asset driven by a combined geometric Brownian motion and a geometric compound
Poisson process. We present an integro-differential equation whose solution leads to
the risk-neutral price at time t of Asian options.
In chapter 6, we use Geman-Yor’s contingent pricing formula to obtain a hedging
strategy replicating Asian options.
Finally, chapter 7 is concerned with pricing and hedging catastrophe options. We
derive a hedging strategy minimising the risk at maturity if this date t belongs to the
loss period and replicating the option if this date t belongs to the development period.
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