# 臺灣博碩士論文加值系統

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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, closedformformulas 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 12 Our method - univariate case 42.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Upperbound-Minimization Importance Sampling . . . . . . . . . . . 52.2.2 Variance-Minimization Importance Sampling . . . . . . . . . . . . . 62.3 Univariate option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Digital options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 European options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Our method - multivariate case 113.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Rainbow options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.1 Maximum options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.2 Maximum digital options . . . . . . . . . . . . . . . . . . . . . . . . 123.3.3 Spread options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Numerical result 154.1 Univariate cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Rainbow options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Conclusion & Future work 245.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Appendix 26Reference 34
 Boyle, P., M. Broadie, and P. Glasserman (1997). Monte carlo methods for securitypricing. Journal of Economic Dynamics and Control 21, 1267–1321.Boyle, P. P. (1977). Options: A monte carlo approach. Journal of Financial Economics 4,323–338.Dupuis, P. and H. Wang (2004). Importance sampling, large deviations, and differentialgames. Stochastics and Stochastics Reports 76 (6), 481–508.Dupuis, P. and H. Wang (2005). Dynamic importance sampling for uniformly recurrentmarkov chains. The Annals of Applied Probability 15 (1), 1–38.Fu, M. C., D. B. Madan, and T. Wang (1999). Pricing continuous asian options a comparisonof monte carlo and laplace trandform inversion methods. Journal of ComputationalFinance 2 (2), 49–74.Glasserman, P., P. Heidelberger, and P. Shahabuddin (1999, April). Asymptotically optimalimportance sampling and stratification for pricing path-dependent options. Math-ematical Finance 9, 117–152.Glasserman, P. and Y. Wang (1997). Counterexamples in importance sampling for largedeviations probabilities. The Annals of Statistics 7 (3), 731–746.Hull, J. and A. White (1988). The use of the control variate technique in option pricing.Journal of Financial and Quantitative Analysis 23, 237–251.Lyuu, Y.-D. and H.-W. Teng (2011). Unbiased and efficient greeks of financial options.Finance and Stochastics, 141–181.Pellizzari, P. (2001). Efficient monte carlo pricing of European options using mean valuecontrol variates. In Proceedings of Decisions in Economics and Finance.Ross, S. M. (2006). Simulation. ELSEVIER.Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential test.The Annals of Statistics 4 (4), 673–684.Su, Y. and M. C. Fu (2000). Importance sampling in derivative securities pricing. InProceedings of the 2000 Winter Simulation Conference.Vazquez-Abad, F. J. and D. Dufresne (1998). Accelerated simulation for pricing asianoptions. In Proceedings of the 1998 Winter Simulation Conference.
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 1 以最佳重點性抽樣對資產訂價模型校準 2 降低變異數之蒙地卡羅模擬法：運用於台指選擇權評價之比較 3 評估極值相依組合信用風險之有效演算法 4 準蒙地卡羅法用於選擇權定價 5 Portfolio Risk Management with Entropy Based Importance Sampling 6 以常態逆高斯分配評估一籃子信用違約交換之演算法 7 隨機網路可靠度之估計 — 一簡易遞迴的重點抽樣和分層抽樣程序 8 偏常態因子信用組合下之效率估計值模擬 9 CalibrationofAsset-PricingModelsbyOptimalImportanceSampling 10 An Interpretation of the Metropolized Independent Sampling 11 提升模擬實驗之準確度

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