亞式選擇權是一種路徑相依的契約，它的報酬是依賴於某一期間資產價格的平均。亞式選擇權定價是複雜的，且在一般情況下不存在封閉解。亞式選擇權定價的困難是因為對數常態分佈的算術平均不是對數常態分佈。在文獻中已提出幾種逼近解的分析方法，其中包含偏微分方程式、拉普拉斯變換、傅立葉變換和解析近似法。在本文中，我們提出新的解析近似公式，算術平均的亞式選擇權定價之高階泰勒近似法，藉由此方法得到封閉解的公式。本文最後比較一階及二階泰勒近似法。

　　Asian options are path dependent derivatives whose payoff depends on some form of averaging prices of the underlying asset. The valuation of Asian options is always complicated and no closed form solution exists, in general. The difficulties come from the fact that the distribution of the arithmetic average is no longer log normal. Several analytic approaches have been pro-posed in the literature, including, among others, partial differential equations, Laplace and Fourier transform, and analytic approximations. In this thesis we derive new analytic approximate formulas for the pricing of Asian option with arithmetic averages via higher order Taylor approximations. The resulting formulas are in closed form. Comparisons with first and quadratic orders are included.

Contents
Abstract i
1 Introduction 1
2 Prices of Asian Options 4
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . 4
2.1.2 Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Arithmetic and Geometric Asian Options . . . . . . . . . . . . . . . . 14
2.2.1 Arithmetic Average . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Geometric Average . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Arithmetic Averages via Taylor Series 22
3.1 First-Order Approximations . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Quadratic Approximations . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Third-Order Approximations . . . . . . . . . . . . . . . . . . . . . . . 32
4 Numeric Results 40
5 Conclusion 43
Reference 44

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