本研究在有限時間軸的條件下利用動態規劃方法, 建構美式選擇的定價模型。假設股價服從幾何布朗運動, 無風險利率及波動率為常數。選擇權的價值計算主要利用逐步線性內插法及動態規劃反向程序逐步計算之。同時導入有最佳止步時間反自由邊界問題概念並計算自由邊界, 最後數值例顯示所有結果都符合一般性之結果。

We propose a dynamic programming (DP) approach for pricing American options over a finite time horizon. We model uncertainty in stock price that follows geometric Brownian motion (GBM) and let interest rate and volatility be fixed. A procedure based on dynamic programming combined with piecewise linear interpolation approximation is developed to price the value of options. And we introduce the free boundary problem into our model. Numerical experiments illustrate the relation between value of option and volatility.

Abstract iii
1 Introduction 1
2 Preliminaries 3
2.1 Definition of Terminology . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Option Pricing by Dynamic Programming Algorithm 11
3.1 The Discretization of a Geometric Brownian Motion . . . . . . . . . . 11
3.2 Optimal Stopping Problem . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Free Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 DP Problem and Approximation Procedure . . . . . . . . . . . . . . 17
4 Numerical Experiments 22
4.1 Accuracy and Convergence . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Volatility and Free Boundary . . . . . . . . . . . . . . . . . . . . . . 25
5 Summary and Future Work 27
Appendix 30

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