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研究生:陳醇潔
研究生(外文):Chun-chieh Chen
論文名稱:利用重點抽樣的有效率選擇權訂價
論文名稱(外文):Efficient option pricing with importance sampling
指導教授:鄧惠文鄧惠文引用關係
指導教授(外文):Huei-wen Teng
學位類別:碩士
校院名稱:國立中央大學
系所名稱:統計研究所
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
語文別:英文
論文頁數:43
中文關鍵詞:彩虹選擇權選擇權定價蒙地卡羅重點抽樣變異數縮減
外文關鍵詞:rainbow optionvariance reductionimportance samplingMonte Carlooption pricing
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隨著金融商品迅速發展,在學術界和業界中,如何正確且有效率的對選擇權做定價依然是一個重要的問題。奇異選擇權或複雜的選擇權通常沒有封閉解,就算在布萊克-肖爾斯假設下也是如此,因此需要使用數值方法。蒙地卡羅近似是一個簡單且合適的方法,但蒙地卡羅估計量通常會有較大的變異數。為了解決這個問題,我們提出了一個重點抽樣的方法,可以找到一個指數平移測度來極小化蒙地卡羅估計量的變異數。我們應用此方法在幾個選擇權上做定價和計算希臘字母。
Along with the rapid development of financial instruments, pricing options correctly and efficiently remains a critical issue both in industry and in academy. However, closedform
formulas for exotic or complicated options price rarely exist even under the standard Black-Scholes assumptions, and consequently additional numerical techniques are required. Among them, Monte Carlo approaches are invaluable tools and are easy to implement, but Monte Carlo estimators usually suffer from large variances. To tackle this problem, we propose an importance sampling procedure with an exponentially tilted measure to minimize the variance of Monte Carlo estimators. We apply our method to calculate both the price and the Greek letters for several popular options, such as spread and maximum options.
1 Introduction 1
2 Our method - univariate case 4
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Upperbound-Minimization Importance Sampling . . . . . . . . . . . 5
2.2.2 Variance-Minimization Importance Sampling . . . . . . . . . . . . . 6
2.3 Univariate option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Digital options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.2 European options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Our method - multivariate case 11
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Rainbow options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.1 Maximum options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.2 Maximum digital options . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.3 Spread options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Numerical result 15
4.1 Univariate cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Rainbow options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Conclusion & Future work 24
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Appendix 26
Reference 34
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