亞式選擇權是一種路徑相關的選擇權，它的報酬是依賴於一段時間股價的平均。亞式選擇權的本質可以是歐式或美式的。股價平均的時段可以是整個選擇權的有效時間或是從選擇權開始的某個時間點至選擇權的執行時間。股價的平均可分為算術平均和幾何平均。
在本文中，我們計算出向前時置亞式選擇權定價的一個解析近似方法，其中分別有固定執行價幾何平均的亞式選擇權以及變動執行價算術平均亞式選擇權。在此定價公式中，我們利用線性近似的方式去求出近似的計價公式。而且我們所得到的對於浮動執行價向前時置亞式買權的近似計價公式與參考文獻[1]中的計價公式有些差異，我們相信我們是正確的，他們是錯誤的。

An Asian option is a path-dependent option whose payoff depends on the average of the underlying asset price over a certain time interval. It can be European or American. The time interval can be the entire interval of the option's life from the initiation to the expiration, or beginning from some time later than the initiation until the option's expiration. The average can be arithmetic or geometric.
This paper derives a closed-form solution for the valuation of European Geometric average fixed strike
Asian call option and a closed-form solution for the valuation of a forward-starting Asian call option with arithmetic average floating strike.
The valuation formula is obtained by relying upon a slight linear approximation. And the valuation formula of Asian call option with arithmetic average floating strike we have derived is different from that of L. Bouaziz, E. Briys and M. Crouhy (1994). We believe that our argument here is correct, and theirs is wrong.

論文審定書i
誌謝ii
摘要iii
Abstract iv
1 Introduction 1
1.1 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 European Arithmetic Average Fixed Strike Asian Call . . . . 3
1.1.2 European Geometric Average Fixed Strike Asian Call . . . . . 3
2 The Black-Scholes Pricing Theory 8
2.1 Change of Probability-Measures Brownian Motion . . . . . . . . . . . 8
2.2 The Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 The Representation of Brownian Motion . . . . . . . . . . . . . . . . 13
2.4 The Risk-Neutral Pricing Formula . . . . . . . . . . . . . . . . . . . . 13
3 The Forward-Starting Asian Options 17
3.1 Geometric Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 Fixed Strike . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Arithmetic Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Time Window Prior to T − T0 . . . . . . . . . . . . . . . . . . 22
3.2.2 Time Window Posterior to T − T0 . . . . . . . . . . . . . . . . 25
Reference 30

[1] L. Bouaziz, E. Briys and M. Crouhy, The pricing of forward-starting asian
options, Journal of Banking & Finance 18 (1994), 823-839.
[2] Bernt Oksendal, Stochastic Differential Equations, 6ed (2003), springer.
[3] Steven E. Shreve, Stochastic Calculus for Finance I The Binomial Asset Pricing
Model, (2004), springer.
[4] Steven E. Shreve, Stochastic Calculus for Finance II Continuous-Time Models,
(2004), springer