臺灣博碩士論文加值系統

這篇論文的主要目的，是將跳躍擴散過程加入標的資產報酬的波動方程式中，以評價複合選擇權。為了改善Black and Scholes (1973) 建立的模型假設，造成選擇權具有波動度微笑的現象，即資產報酬的分配呈現出高峰與左右兩端厚尾的特性，在此一跳躍擴散過程中，我們假設跳躍的頻率是服從複合卜瓦松過程，跳躍的幅度是服從由Kou (2002)所提出的雙指數分配。而數值分析中，加入雙指數跳躍擴散過程的標資產報酬分配，的確可以改善波動度微笑的現象; 我們也分析了在各三種不同歐式買權和複合買權的模型下加入雙指數跳躍擴散過程對選擇權價值的影響，當跳躍次數愈多，具有跳躍擴散的選擇權價格會愈大; 相反地當具有跳躍擴散的選擇權模型不存在跳躍的情況下，它們的選擇權價值會各自回復到B-S (1973) 或是 Geske (1979) 的價格。除此之外，我們的模型(DEJDCC)比Gukhal (2004) 和 Geske (1979) 一般化，更能描述波動度微笑的現象，在財務上的應用也因此更為廣泛。

This paper introduces the jump-diffusion process into pricing compound options and derives the related valuation formulas. We assume that the dynamic of the underlying asset return process consists of a drift component, a continuous Wiener process and discontinuous jump-diffusion processes which have jump times that follow the compound Poisson process and the logarithm of jump size follows the double exponential distribution proposed by Kou (2002). Numerical results indicate that the advantage of combining the double exponential distribution and normal distribution is that it can capture the phenomena of both the asymmetric leptokurtic features and the volatility smile. In addition, in order to examine the effect of the jumps, we compare three European option call models and three compound option models with and without jumps, and we observe that the higher the jump frequency we set, the greater the option values we obtain. The numerical results also show that the European call option and compound option models with jumps can reduce to those models without jumps when the jump frequency is set to zero. Furthermore, the compound call option under the double exponential jump diffusion model which we derived is more generalized than Gukhal (2004) and Geske (1979), and thus has wider application.

Contents

Chapter 1　Introduction 1
1.1 Introduction 1
1.2 Motivations and Contributions 4
1.3 Organization of this Thesis 5
Chapter 2　Literature Review 6
2.1 Compound Options 6
2.2 Amending the Deficiencies of Black and Scholes 11
2.2.1 Jump-Diffusion Models 12
2.2.2 Stochastic Volatility Models 14
2.2.3 Local Volatility Models 15
Chapter 3　 Compound Option Valuation Model 17
3.1　Assumptions for the Model 17
3.2　The Model 20
3.3 Compound and European Call Options 32
Chapter 4　Numerical Results 35
4.1 Advantage of Double Exponential Distribution in Option Pricing 35
4.2　The Effect of the Jumps on Value of European Call Options 41
Chapter 5　Conclusion 58
References 60
Appendix A 64
Appendix B 69
Appendix C 72
Appendix D 75
List of Tables

Table 4.1 Comparing the European Call Option Values when the Models are with and without Jumps 44
Table 4.2 Comparing the Compound Call Option Values when the Model is with and without Jumps and 51
Table 4.3 Comparing the Compound Call Option Values when the Model is with and without Jumps and 52
Table 4.4 Comparing the Compound Call Option Values when the Model is with and without Jumps and 53

List of Figures
Figure 3.1 Lifetimes of the Compound Option with Double Exponential Jumps 21
Figure 3.2 The Relation between Two Kinds of Option Values: European Call Options and Compound Call Options 33
Figure 4.1 The Density of the Sum of the Double Exponential and Normal Random Variable 38
Figure 4.2 Comparing the Density of the Sum of the Double Exponential with the Normal Random Variable when is Set as Positive and Negative 39
Figure 4.3 Comparing the European Call Option Models when changes from 2 to 4 46
Figure 4.4 Comparing the Kou and Black and Scholes Models when changes from 2 to 10 48
Figure 4.5 Comparing the Compound Call Option Models when changes from 2 to 4 56

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