本論文主要在探討一些較實際的衍生性金融商品的評價與避險策略。本文共分為七章，其主要結果分述如下： 在第一章中，我們主要探討具風險的標的在模型 $S_t=S_0+int_0^tS_s(mu ds+sigma dB_s)+sum_{j=1}^{N_t}S_{tau_j^-}U_j.$ 下的評價與避險，因為雜訊過程有跳躍，所以該金融市場為不完備市場，因為其具有許多個等價的鞅測度使得其貼現標的為鞅，所以對此市場的選擇權，一般而言，不可能做完美的避險。因此，我們利用Föllmer和Schweizer發展的理論建構我們模型的最小鞅測度，它將使我們的風險最小化。在此最小鞅測度下，我們 建立了一偏微分方程式，其解為市場的選擇權價，並且在局部最小風險意義下， 我們建立了一避險策略。 在第二章中，我們得到貝氏平方過程(BESQ)在獨立指數分配時間的機率密度函數，利用此結果，我們導出當評價模型為具有一個跳躍的貝氏平方利率過程時零息債券在時間零時的價格。 在第三章中，我們建立了CIR過程一些有用的性質。Pitman和Yor [PY] 建立了BESQ過程與 CIR過程的時間-空間變換關係，利用此結果，我們導出 CIR過程的機率推移密度函數。另外，藉由融合超幾何函數，我們建立了CIR過程與Kummer微分方程式解的關係，更進一步，利用此關係我們得到CIR過程在獨立指數分配時間的機率密度函數。最後，從財務金融的觀點來看，有關 barrier option，我們有興趣研究CIR過程的撞擊時間。藉由鞅性質的方法，我們得到CIR過程的撞擊時間的拉普拉斯變換。 在第四章，我們考慮有跳躍的亞洲選擇權在時間零的評價。Geman 和Yor [GYa1] 曾經對連續的亞洲選擇權在時間零時於隨機指數分配的到期時間給予一封閉解，且經拉普拉斯逆變換可得到期時間為固定時的亞洲選擇權在時間零的評價，受其啟示，首先我們推廣其結果到具有一個跳躍的亞洲選擇權，它似乎更符合實際金融市場的情況。另外，我們建立了一積分-微分方程式與多重跳躍的亞洲選擇權在時間零的評價關係。 在第五章，我們利用 change-of-numéraire 方法，我們建立了一積分-微分方程式與多重跳躍的亞洲選擇權在任意時間t時的評價關係。 在第六章中，我們主要探討亞洲選擇權的避險策略。令 $q:=frac{sigma^2}{4S_t}(K(T-t_0)-int_{t_0}^tS_udu).$ q雖為隨機但在時間t卻為已知，由觀察q在時間t的值，我們對亞洲選擇權做不同的避險投資組合。 在第七章中，我們考慮劇變選擇權的避險。我們採用Geman和Yor [GYa3]的模型。若時間在虧損區間內，我們建立一風險最低的投資組合。另一方面，若時間在發展區間內，因為有關劇變的訊息已知，我們建立一複製劇變選擇權的避險策略。 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.
