Efficient Option Pricing with Importance Sampling

利用重點抽樣的有效率選擇權訂價

陳醇潔(Chun-Chieh Chen) ； 傅承德(Cheng-Der Fuh) ； 鄧惠文(Huei-Wen Teng)

《中國統計學報》 51卷3期 (2013/09) Pp. 253-273

摘要

隨著金融商品迅速發展，如何正確且有效率的計算選擇權價格是一個具有挑戰性的問題。我們已知選擇權價格是選擇權連結商品的收益函數（payoff functions）的期望值，其中機率密度函數是風險中立測度。奇異選擇權（exotic options）或複雜的選擇權通常沒有封閉解，就算在布萊克-肖爾斯（Black-Scholes）假設下也是如此，因此需要使用數值方法。其中，蒙地卡羅近似是一個合適的方法，且對於複雜的收益函數也很容易做調整。雖然蒙地卡羅估計量通常是不偏的，但卻有較大的變異數。為了解決這個問題，我們提出了一個重點抽樣的方法，用指數平移測度來極小化蒙地卡羅估計量的變異數。接著我們利用這個方法計算數位選擇權（digital options）和歐式選擇權價格作為例子。

關鍵字

財務選擇權定價 ； 變異數縮減 ； 蒙地卡羅模擬 ； 重點抽樣

並列摘要

With the rapid development of financial instruments, pricing options correctly and efficiently is a challenging task. It is known that an option price is an integral, where the integrand is a product of the payoff function of an option and a probability density function under the risk-neutral probability measure. Closed-form 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 flexible and easy to be adjusted for complicated payoff functions. Although Monte Carlo estimators are usually unbiased, they suffer from large variances. To tackle this problem, we first propose an importance sampling procedure, to which it is an exponential tilting measure minimizing the variance of Monte Carlo estimators. Next we apply our method to calculate the option prices, such as digital and European options.

並列關鍵字

financial options pricing ； variance reduction ； Monte Carlo simulation ； importance sampling

參考文獻

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