Black-Scholes 公式是廣泛被使用的選擇權定價模型，可以由目前的標的物價、選擇權履約價、到期日、波動利率及無風險利率根據公式計算出合理的歐式選擇權價格，而在模型下歐式買權的價格對於標的物價格為一遞增凸函數，假設標的物價格的變動過程為廣義幾何布朗運動 (generalized geometric Brownian motion)，標的物價格的波動愈大，則選擇權愈有價值，所以選擇權的價格與標的物價格呈現正向變動的關係。若標的物價的波動利率σ(t) 變動過程介於 σ_1 ≤ σ(t) ≤ σ_2 之間，σ_1 及 σ_2 皆為常數，使得 0 ≤ σ_1 ≤ σ_2。
將 C_i(t, S_t) 表示為在時間 t 時，對於不同常數波動利率σ_i (i = 1,2) 買權的價格，我們將推導出在定價模型之下，買權的初始價格也會介於不同波動利率買權初始價格的區間 [C_1(0, S_0),C_2(0, S_0)] 中。

The Black-Scholes formula was the widely-used model for option pricing, this formula can be use to calculate the price of option by using current underlying asset prices, strike price, expiration time, volatility and interest rates. The European call option price from the model is a convex and increasing with respect to the initial underlying asset price. Assume underlying asset prices follow a generalized geometric Brownian motion, it is true that option prices increasing with respect to the constant interest rate and volatility, so that the volatility can be a very important factor in pricing option, if the volatility process σ(t) is constant (with σ(t) =σ  for any t ) satisfying σ_1 ≤ σ(t) ≤ σ_2 for some constants σ_1 and σ_2 such that 0 ≤ σ_1 ≤ σ_2. Let C_i(t, S_t) be the price of the call at time t corresponding to the constant volatility σ_i (i = 1,2), we will derive that the price of call option at time 0 in the model with varying volatility belongs to the interval [C_1(0, S_0),C_2(0, S_0)].

致謝i
摘要ii
Abstract iii
1 Introduction 1
2 The Generalized Black-Scholes Model 6
3 The Black-Scholes Formula 9
4 Monotonicity of Option Prices Relative to Volatility 14
References 23

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[3] Shreve, Steven E., (2004), Stochastic Calculus for Finance II: Continuous-Time
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[4] Michael Meyer, (2001), Continuous Stochastic Calculus with Applications to
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[5] Lawrence C. Evans, An Introduction to Stochastic Dierential Equations.
Lecture notes (Department of Mathematics,UCBerkeley), available from
http://math.berkeley.edu/evans/.
[6] Follmer, H. and Schied, A., (2004), Stochastic Finance. An Introduction in
Discrete Time, Walter de Gruyter, Berlin.
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