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Option is one of successful examples of innovation practice in financial markets and it contributes a lot to the development of financial theory. British option is a new class of early-exercise option that provide its holder with an inherent protection mechanism to realize the `best prediction' assuming the real drift of the stock price equals the contract drift. British option can provide its holder a protection mechanism against unfavorable prices change and bring high returns from favorable prices change. British option addresses liquidity and return completely endogenously. After the introduction of some famous option pricing models, such as the Black-Scholes option pricing model, extensions of Black-Scholes option pricing model and the unified Black-Scholes model for strong path dependent options, this paper focuses on the British Asian option and the British Russian option. The arbitrage-free price, rational exercise boundary and nonlinear integral equation of Financial Mathematics the path dependent British option are all discussed in this paper.选择权是一种创新实践中成功的例子，在金融市场和金融理论的发展作出了巨大贡献。英国的选择是一类新的早期行使购股权，其持有人具有内在的保护机制来实现'最佳预测假设的股票价格真正的漂移等于合同漂移。英国选项可以其持有人提供一个保护机制，防止不利的价格变化，从有利的价格变化带来高回报。英国的方案解决流动性和内源性完全恢复。出台后，一些著名的期权定价模型，如Black-Scholes期权定价模型，Black-Scholes期权定价模型的扩展和统一的强烈的路径依赖期权的Black-Scholes模型，本文侧重于英国亚洲选项，英国俄罗斯选项。所有本文中讨论的无套利价格，合理的行使边界和非线性积分方程的路径依赖英国选项。

ontents

Chapter1: Introduction 1
1.1 Background information 1
1.2 Main contents 2
Chapter2: Literature review 4
2.1 Early study on option pricing theory 5
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Chapter3: Some basic facts about option 10
3.1 Option definition 10
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3.5 The functions of options 15
Chapter 4: Option pricing models 16
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4.3 Unified Black-Scholes model for strong path dependent options 22
Chapter5: The British Asian option 24
5.1 Definition of British Asian option 24
5.2 Arbitrage-free price of British Asian option 25
5.3 Other British Asian options 28
Chapter6: The British Russian option 30
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6.3 Other British look back options 35
Chapter7: Conclusions and future research 36
7.1 Conclusions 36
7.2 Future research 36
References 38

Chapter1: Introduction

1.1 Background information

With the rapid development of financial markets and various financial products, more and more investors are involved in derivatives trading. The advantages of low cost, flexible pricing methods and powerful leverage effect make the financial derivatives popular in meeting the demands of different risk preferences and effectively hedge investors’ position. In mature financial markets such as European financial market and U.S. financial market, the financial derivatives are trading frequently and the value they created is growing exponentially. Of which, the contribution of option products is significant. Option is an important financial derivative. In many developed countries, the options market is an important component of the securities market. Stock options, stock index options, currency options and currency options all have great performance.在成熟的金融市场，如欧洲金融市场和美国金融市场，金融衍生品交易频繁，他们创造的价值成倍增长。其中，期权产品的贡献是显着的。期权是一种重要的金融衍生工具。在许多发达国家，期权市场是证券市场的重要组成部分。所有股票期权，股票指数期权，外汇期