Black-Scholes 歐式選擇權估價模型可以用很多方法推導，在這篇論文中，我們呈現包括偏微分方程、風險中立估價、平賭測度、變換度量在內的幾種技巧

The Black-Scholes European option pricing formula can be derived in several ways. In this dissertation we present several methods that can be used to derive
this formula, including partial differential equation method, the risk-neutral pricing method, the martingale measure method, and the change of numeraire technique

1 Introduction 1
2 Model Setup 2
3 No-arbitrage and Completeness 5
4 Black-Scholes Equation 7
5 Derivation of Black-Scholes Formula via Heat Equation 9
6 Martingale Measures and Numeraires 15
6.1 EMM as numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.2 Underling Risky Asset as numeraire . . . . . . . . . . . . . . . . . . . . . 19
7 Risk-Neutral Valuation 21
8 Martingale Valuation 22
9 Derivation of Black-Scholes Formula via Change of Numeraire 22
References 26

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measure and option pricing, J.Applied Probability 32 (1995), 443-458
[4] A.Esswe, "Pricing in (In)Complete Markets," Lecture Notes in Economics and Math-
ematical Systems 537, Springer, 2004
[5] J.Andreasen, B.Jensen, and R.poulsen, Eight valuation in ‾nancial mathematics:The
Black-Scholes formula as an Example, Math.Scientist 23(1998),18-40.