應用局部微分積分法求解選擇權評價問題

本文主要闡述以局部微分積分數值方法求解選擇權評價問題之過程以及結果。在Black, Scholes以及Merton於1973年提出之模式帶來之重大貢獻後，為了使其假設更接近真實金融市場或是更廣泛應用選擇權理論於特定商品的需求，眾多改良自Black-Scholes模式的選擇權評價模式隨之被提出。一般而言，這些模式依照其針對標的資產價格之動態過程假設之數學形式大致可分為兩種。第一種模式僅假設標的資產價格之動態過程滿足布朗運動, 可歸類為「擴散模式」；另一種則假設標的資產價格之動態過程包括布朗運動以及隨機跳躍，可歸類為「跳躍─擴散模式」。本研究主要目標則為提供一無論於「擴散模式」或「跳躍─擴散模式」下進行選擇權評價時皆能有效運行之數值模式。為求有效提升本問題數值運算效益，故本研究採用針對收益函數之一階偏微分導數不連續位置進行加密佈點之網格，並使用適用於該類網格之局部微分積分法進行數值計算。針對「跳躍─擴散模式」，由於其統御方程式較「擴散模式」多一非局部積分項，本研究亦提供一能在非均勻佈點網格上有效求解該積分之數值流程。數值案例皆為典型且常見之選擇權評價問題，包括歐式、美式、回顧型以及障礙型選擇權。經過與參考資料比較結果後，發現本研究建議之方法流程於「擴散模式」及「跳躍─擴散模式」皆擁有良好穩定性以及表現，並且適用於各類典型常見之選擇權評價問題，故可得此方法可成功應用於選擇權評價問題之結論。

關鍵字

局部微分積分法； Black-Scholes 模式；跳躍─擴散模式；歐式選擇權；美式選擇權；回顧選擇權；障礙型選擇權

並列摘要

This study demonstrates the numerical procedure of solving option-pricing problems by the local differential quadrature (LDQ) method. After the remarkable contribution of Black, Scholes and Merton in 1973, many option-pricing models are developed for relaxing the restrictions of the Black-Scholes (BS) model or extending the theory to much wider applications. In general, these models can be divided into two kinds based on the assumption of the dynamic process for the underlying assets. Ones are diffusion models in which the dynamics of the underlying-asset price follow the Brownian process; the others are jump-diffusion models with the random jumps considered in the dynamics. Our work aims to develop a numerical process for solving both the diffusion and jump-diffusion models efficiently. Because of the non-differentiability of the payoff functions, non-uniformly distributed nodes are applied, and thus the LDQ method is used due to its advantage for arbitrary grid nodes. For the jump-diffusion models, we also provide an efficient scheme for computing the non-local integral of the governing equations on the non-uniform nodal girds. Numerical experiments include typical option-pricing problems, such as European, American, lookback, binary and barrier options. According to the comparison with the reference data, our results show that the proposed method has robustness and good performance for both diffusion and jump-diffusion type option-pricing models. In brief, one can conclude that it is a successful application.

並列關鍵字

local differential quadrature ； the Black-Scholes model ； jump-diffusion model ； European option ； American option ； lookback option ； barrier option

參考文獻

[1] Ahn, D. H.; Gao, B.; Figlewski, S. (1999): Pricing Discrete Barrier Options with the Adaptive Mesh Model. The Journal of Derivatives, Vol. 6, No. 4, pp. 33–44.

[2] Ait-Sahalia, Y.; Lo, A. W. (1998): Nonparametric Estimation of State-price Densities Implicit in Financial Asset Prices. The Journal of Finance, Vol. 53, Iss. 2, pp. 499–547.

[3] Albert, M.; Fink, J.; Fink, K. (2008): Adaptive Mesh Modeling and Barrier Option Pricing Under a Jump-Diffusion Process. Journal of Financial Research, Vol. 31, No. 4, pp. 381-408.

[4] Almendral, A.; Oosterlee, C. W. (2005): Numerical Valuation of Options with Jumps in the Underlying. Applied Numerical Mathematics, Vol. 53, pp. 1-18.

[5] Almendral, A.; Oosterlee, C. W. (2007): On American Options Under the Variance Gamma Process. Applied Mathematical Finance, Vol. 14, No. 2, pp. 131-152.

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應用局部微分積分法求解選擇權評價問題

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