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研究生:陳仁維
研究生(外文):Ren - wei Chen
論文名稱:考慮隨機波動下評價一籃子選擇權
論文名稱(外文):Analytic Formulae for Basket Options with Stochastic Volatility
指導教授:林忠機林忠機引用關係
指導教授(外文):Chung-Gee Lin
學位類別:碩士
校院名稱:東吳大學
系所名稱:商用數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2007
畢業學年度:95
語文別:英文
論文頁數:52
中文關鍵詞:一籃子選擇權模擬法隨機波動泰勒級數展開式
外文關鍵詞:basket optionsimulationstochastic volatilityTaylor series expansion.
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因為對數常態變數之加權平均不再是對數常態,因此評價一籃子選擇權是困難的. 然而, 儘管一籃子選擇權在實務上已經相當普及, 但是對於考慮隨機波動下之一籃子選擇權之評價與避險在學術上並無太多研究. 本文是使用泰勒展示去推導考慮隨機波動下一籃子選擇權之解析近似解公式. 數值結果顯示, 當我們使用在大樣本下蒙地卡羅模擬法當作基準值, 發現本文所推導之解析近似解公式具有一定之準確度, 也比蒙地卡羅模擬法來得有效率.
The valuation of basket options is complex since the weighted average of lognormal random variables is no longer lognormal. Moreover, as the stochastic volatility inherent in the financial asset prices are extensively observed, however, few academic works pay attention to the pricing and hedging of basket options with stochastic volatility, despite the popularity of basket options in practical application. This paper derives the analytic solution for pricing and hedging basket option with stochastic volatility by integrating Taylor series expansions. Numerical experiments show that our analytic solution performs very well and is computationally efficient as benchmarked with large sample Monte Carlo simulations. Our analytic solution can also be treated as a practical tool for pricing and hedging stochastic volatility basket options, since it is easy to implement and is more desirable as the computing speed is an important consideration.
Contents
1. Introduction 1
2. Literature Review 2
3. The Model 5
3.1 Analytic Approximation Formulae with Reciprocal Gamma Distribution 5
3.2 Analytic Approximation Formulae with Lognormal Distribution 8
3.3 The Stochastic Volatility Problem 9
3.4 Analytic Approximation Basket Option with Stochastic Volatility 12
3.5 Hedge Ratios 16
4. Numerical Analyses 17
4.1 Two-Asset European Basket Call Options with Stochastic Volatility 18
4.2 Three-Asset European Basket Call Options with Stochastic Volatility 20
4.3 Four-Asset European Basket Call Options with Stochastic Volatility 21
4.4 Five-Asset European Basket Call Options with Stochastic Volatility 23
4.5 Hedge Ratios 24
5. Conclusions 25
References 27
References
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