降低變異數演算法在不同選擇權評價上的應用

Variance Reduction Algorithm for Pricing Various Options

林怡廷(I-Ting Lin)

指導教授 ： 張傳章

國立中央大學/理學院/統計研究所/碩士(2008年)

摘要

本篇論文是結合Longstaff and Schwartz (2001) 提出的最小平方法與Huge and Rom-Poulsen (2004) 提出降低變異數 (variance reduction) 的技巧來評價美式選擇權。Longstaff and Schwartz (2001) 使用最小平方法估計美式選擇權的持有價值，Huge and Rom-Poulsen (2004) 則是利用最小平方法計算標的資產的價格。當標的資產須用蒙地卡羅的方式模擬時，計算選擇權的報酬會產生偏誤的現象。我們應用這個演算法分別去估計標的資產為債券的障礙選擇權 (barrier option) 與重設選擇權 (reset option) 的價格，並且以數值模擬的結果呈現出降低變異數的效果。

關鍵字

重設選擇權 ； 障礙選擇權 ； 偏誤縮減 ； 最小平方蒙地卡羅法

並列摘要

This paper develops an algorithm that combines the Longstaff and Schwartz (2001) simulation algorithm and the variance reduction technique proposed in Huge and Rom-Poulsen (2004) to simulate American-style option prices on securities such that their prices can be found by the Monte Carlo simulations. Longstaff and Schwartz (2001) used the least squares method to estimate the optimal exercise boundary of American options, Huge and Rom-Poulsen (2004) used the same method to calculate the price of underlying security. In this paper, we apply this algorithm to value various options, such as barrier option and reset option. Bias reduction is also involved in algorithm, since we know that using simulated prices of the underlying security to compute option payoff causes an upward bias in option prices. We use numerical results to show that this algorithm can provide significant improvement on efficiency and accuracy for pricing barrier bond option and reset bond option.

並列關鍵字

barrier option ； bias reduction ； least squares Monte Carlo simulation ； reset option

參考文獻

[1] Andersen, L., “A Simple Approach to the Pricing of Bermudan Swaptions in the Multi-Factor Libor Market Model”, Journal of Computational Finance, Vol. 3, No. 2, pp. 5-32, 2001.

[2] Barone-Adesi, G. and R. Whaley, “Efficient Analytic Approximation of American Option Values”, Journal of Finance, Vol. 42, pp. 301-320, 1987.

[3] Barraquand J. and D. Martineau, “Numerical Valuation of High Dimensional Multivariate American Securities”, Journal of Financial and Quantitative Analysis, Vol. 30, No. 3, pp. 383-405, 1995.

[4] Black, F. and M. Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, Vol. 81, No. 3, pp. 637-654, 1973.

[5] Boyle, P. P., “A Monte Carlo Approach”, Journal of Financial Economics, Vol. 4, pp. 323-338, 1977.