Two Option Pricing Models for Leptokurtic Risk-Neutral Asset Return Distributions and Their Comparison with Series Expansion Approaches

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研究生：	牛玉琉 Siti - Maghfirotul Ulyah
論文名稱：	Two Option Pricing Models for Leptokurtic Risk-Neutral Asset Return Distributions and Their Comparison with Series Expansion Approaches Two Option Pricing Models for Leptokurtic Risk-Neutral Asset Return Distributions and Their Comparison with Series Expansion Approaches
指導教授：	繆維中 Wei-Chung Miao 林昌碩 Chang-Shuo Lin
口試委員:	劉代洋 Day-Yang Liu 謝劍平 C.-P. Shieh
學位類別：	碩士 Master
系所名稱：	管理學院 - 財務金融研究所 Graduate Institute of Finance
論文出版年：	2016
畢業學年度：	104
語文別：	英文
論文頁數：	86
中文關鍵詞：	leptokurtic 、multimodality 、jump-diffusion 、normalmixture 、seriesapproximation
外文關鍵詞：	leptokurtic, multimodality, jump-diffusion, normal mixture, series approximation
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The characteristics of asset return distribution in the market are far from normal distribution, especially for exchange rate return. It usually leptokurtic, heavy-tailed, and sometimes having more than one mode (bimodal). Skewness and kurtosis are the common measures of leptokurtic and asymmetry issues in the asset return. Jump is also considered as a source of leptokurtic feature. The purpose of this study is to give some evidences to show that series approximation is not proper in the option pricing. Another purpose is to have an option pricing model that can account the extreme characteristics of the asset return distribution such as very high kurtosis and multimodality issues. Two constructed models are the Kou's jump-diffusion model with one jump (KJD1) that will be applied to quanto options, and a model based on the standardized normal mixture (SNM) distribution. This study concludes that those two models can account for arbitrarily high kurtosis and very skewed return distribution. Besides, the second model can account for bimodality. This study also presents that the series approximation models should not be applied to the return with high kurtosis (>7). For in-the-money (ITM) options, the pricing errors are smaller than that for out-of-the-money (OTM) and at-the-money (ATM) options. With very large kurtosis, the price from the series approximations may become negative for ATM options, which are not trustable.

Abstract i
Acknowledgements ii
Contents iii
List of Figures vi
List of Tables viii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Kou's Jump-diffusion Model with One Jump (KJD1) . . . . . . . . . . . 3
1.3 The Option Pricing Model Based on the Standardized Mixture Normal
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 The Organization of this Thesis . . . . . . . . . . . . . . . . . . . . . . . 6
2 The Kou’s Jump-diffusion Model with One Jump 7
2.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Constructed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Return Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Probability Density Function . . . . . . . . . . . . . . . . . . . . 13
2.3.2 The Main Statistics of the Return . . . . . . . . . . . . . . . . . 15
2.3.3 Call Option Price . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Numerical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 The Option Prices of the KJD1 Model with Symmetric Jump Size
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 The Option Prices of the KJD1 Model with Asymmetric Jump
Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 The Comparison with KJD Model . . . . . . . . . . . . . . . . . . . . . 25
2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 The Option Pricing Model Based on the Standardized Normal Mixture Distribution 28
3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 The Normal Mixture Distribution . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 The Main Statistics of The Normal Mixture Distribution . . . . . 30
3.2.2 The Standardized Normal Mixture (SNM) Distribution . . . . . . 31
3.2.3 The Modality of Normal Mixture Distribution . . . . . . . . . . . 32
3.2.4 The Symmetric Case of the SNM Distribution . . . . . . . . . . . 33
3.2.5 The Asymmetric Case of the SNM Distribution . . . . . . . . . . 35
3.3 Return Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 The Main Statistics of the Return . . . . . . . . . . . . . . . . . 37
3.3.2 The Condition of Unimodality . . . . . . . . . . . . . . . . . . . 38
3.3.3 The Unimodal-Symmetric Return . . . . . . . . . . . . . . . . . 38
3.3.4 The Unimodal-Asymmetric Return . . . . . . . . . . . . . . . . 42
3.3.5 The Bimodal Return Distribution . . . . . . . . . . . . . . . . . 45
3.3.6 Call Option Price . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Numerical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 The Comparison with Black-Scholes Model . . . . . . . . . . . . . . . . 52
3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Conclusion 56
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A The Series Expansions 59
A.1 The pdf of Gram-Charlier and Edgeworth Series Expansions . . . . . . . 59
A.2 Some Figures of the pdf of Gram-Charlier and Edgeworth Returns . . . . 60
B The Kou’s Jump-diffusion Model with One Jump 64
B.1 The Call Price Functions in KJD1 Call Option Price Formula . . . . . . 64
C Numerical Works in Chapter 3 66
C.1 The Unimodal-Symmetric Case . . . . . . . . . . . . . . . . . . . . . . 66
C.2 The Unimodal-Asymmetric Case . . . . . . . . . . . . . . . . . . . . . . 68
C.3 The Bimodal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
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