任课老师：戴滨林 上海交通大学博士 复旦大学博士后 副教授 硕士导师

戴滨林 E-mail：bldai@sohu.com 办公室：数学系404 办公室电话：65904529

电子课件 金融数学

电子课件 期权定价的数学模型和方法 姜礼尚 著（第二版）

Financial Mathematics Fundament of Financial Mathematics -- Option Pricing

期权定价的数学模型和方法 第一章 风险管理与金融衍生物 第二章 无套利原理 第三章 期权定价的离散模型（二叉树方法） 第一章 风险管理与金融衍生物 第二章 无套利原理 第三章 期权定价的离散模型（二叉树方法） 第四章 Brown运动和Ito公式 第五章 欧式期权定价（Black-Scholes公式） 第六章 美式期权定价与最佳实施策略 第七章 有关Black-Scholes公式的推广与应用

课程目的/Major Subjection of Course/ 学生通过学习，具备金融数学的基础知识，掌握各种金融衍生物定价的数学建模，求解方法与技巧，以及几种数值方法，如二叉树方法、有限差分方法。 本课程的重点是金融衍生物定价模型的建立与计算。特别是二叉树方法。它既是一种计算金融衍生物价格的计算格式，同时它本身也是一种离散的金融模型，并且这种模型具有明显的金融意义。因此，这是一种普遍被金融界接受的计算方法。我们将着重讲清它在金融上的无套利意义。同时对于用到的一些数学基础知识与运算技巧，例如：倒向归纳法与概率上的近似方法作充分的讲解。 课时/Periods/ 3节/周（51学时） 考试/Examination/ 闭卷：期末考试。 参考书目/Reference Books/ 《期权定价的数学模型和方法》,姜礼尚著，高等教育出版社，2007年 金融数学,蔡明超译（斯塔夫里和古德曼著），机械工业出版社，2006年 （目前，国内外金融工程专业普遍将J.Hull的专著“Option, Futures and other Derivatives”作为教材或参考书）

金融数学（规范金融数学和实证金融数学） 期权定价的数学模型和方法 ----金融数学（规范金融数学）简介 金融衍生品定价的最早起源应可追溯到1900年法国Louis Bachelier 发表了他的学位论文“Theorie de la Speculation”（投机交易理论）。在他的论文中首次利用随机游动的思想给出了股票价格运行的随机模型，提出了期权的定价问题，它被公认是现代金融学的里程碑。1964年Paul Samuelson (Nobel奖获得者)对L. Bachelier的模型进行了修正，提出了股票运行的几何Brown运动模型。基于这个模型，Fischer Black 和Myron Scholes在1973年建立了看涨期权定价公式。正是由于这个公式及由此产生了期权定价理论方面的一系列贡献，M.Scholes和 R.Merton 1997年获得Nobel经济奖。此后基于这种思想的期权定价理论在国外得到了迅速的发展。

课程包括金融数学的基础知识、无套利原理、随机过程基本知识与Brown运动、金融衍生物定价数学建模的Δ－对冲方法、数理方程的变换技巧以及差分方法与二叉数方法等。学生通过学习，具备金融数学的基础知识，掌握各种金融衍生物定价的数学建模，求解方法与技巧，以及几种数值方法，如二叉树方法、有限差分方法。本课程的重点是金融衍生物定价模型的建立与计算。特别是二叉树方法。它既是一种计算金融衍生物价格的计算格式，同时它本身也是一种离散的金融模型，并且这种模型具有明显的金融意义。因此，这是一种普遍被金融界接受的计算方法。我们将着重讲清它在金融上的无套利意义。同时对于用到的一些数学基础知识与运算技巧，例如：倒向归纳法与概率上的近似方法作充分的讲解（离散情形和连续情形）。 。

教 材： 《期权定价的数学模型和方法》,姜礼尚著，高等教育出版社，2007年. 参考书目： 金融数学,蔡明超译（斯塔夫里和古德曼著），机械工业出版社，2006年 Options, Futures and other Derivatives, Fourth Edition, Prentice- Hall,2000。 。

一、金融数学的重要性 二、金融数学的复杂性 三、金融数学的学习方法

四、具体要求 预习、复习、作业

休息片刻继续

第二章 无套利原理

Chapter 2 Arbitrage-Free Principle

Financial Market Two Kinds of Assets Risk free asset Bond Risky asset Stocks Options …. Portfolio – an investment strategy to hold different assets

Investment At time 0, invest S When t=T, Payoff = Return = For a risky asset, the return is uncertain, i.e., S is a random variable

A Portfolio a risk-free asset B n risky assets a portfolio is called a investment strategy on time t, wealth: portion of the cor. Asset

Arbitrage Opportunity Self-financing - during [0, T] no add or withdraw fund Arbitrage Opportunity - A self-financing investment, and Probability Prob

Arbitrage Free Theorem the market is arbitrage-free in time [0, T], are any 2 portfolios satisfying &

Proof of Theorem Suppose false, i.e., Denote B is a risk-free bond satisfying Construct a portfolio at

Proof of Theorem cont. r – risk free interest rate, at t=T Then From the supposition

Proof of Theorem cont. It follows Contradiction!

Corollary 2.1 Market is arbitrage free if portfolios satisfying then for any

Proof of Corollary Consider Then By Theorem, for Namely

Proof of Corollary 2.1 In the same way Then Corollary has been proved.

Option Pricing European Option Pricing Call-Put Parity for European Option American Option Pricing Early Exercise for American Option Dependence of Option Pricing on the Strike Price

Assumptions The market is arbitrage-free All transactions are free of charge The risk-free interest rate r is a constant The underlying asset pays no dividends

Notations ------ European call option price, ------ the risky asset price, ------ European call option price, ------ European put option price, ------ American call option price, ------ American put option price, K ------ the option's strike price, T ------ the option's expiration date, r ------ the risk-free interest rate.

Theorem 2.2 For European option pricing, the following valuations are true:

Proof of Theorem 2.2 consider two portfolios at t=0: lower bound of (upper leaves to ex.) consider two portfolios at t=0:

Proof of Theorem 2.2 cont. At t=T, and By Theorem 2.1 i.e.

Proof of Theorem 2.2 cont. cont. Now consider a European call option c Since and By Theorem 2.1 when t<T i.e. Together with last inequality, 2.2 proved.

Theorem 2.3 For European Option pricing, there holds call-put parity

Proof of Theorem 2.3 2 portfolios when t=0 when t=T

Proof of Theorem 2.3 cont. So that By Corollary 2.1 i.e. call-put parity holds

Theorem 2.4 For American option pricing, if the market is arbitrage-free, then

Proof of Theorem 2.4 Take American call option as example. Suppose not true, i.e., s.t At time t, take cash to buy the American call option and exercise it, i.e., to buy the stock S with cash K, then sell the stock in the stock market to receive in cash. Thus the trader gains a riskless profit instantly. But this is impossible since the market is assumed to be arbitrage-free. Therefore, must be true. can be proved similarly.

American Option v.s. European Option For an American option and a European option with the same expiration date T and the same strike price K, since the American option can be early exercised, its gaining opportunity must be >= that of the European option. Therefore

Theorem 2.5 If a stock S does not pay dividend, then i.e., the ``early exercise" term is of no use for American call option on a non-dividend-paying stock.

Proof of Theorem 2.5 By above inequalities, there holds This indicates it is unwise to early exercise this option

Theorem 2.6 If C,P are non-dividend-paying American call and put options respectively, then,

Proof of Theorem 2.6 (right side) It follows from call-put party, and Theorem 2.5, thus the right side of the inequality in Theorem 2.6 is proved.

Proof of Theorem 2.6 (left side) Construct two portfolios at time t If in [t, T], the American put option $P$ is not early exercised, then

Proof of Theorem 2.6 (left side) cont.1 Namely, when t=T

Proof of Theorem 2.6 (left side) cont.2 If the American put option P is early exercised at time , then By Theorem 2.2,2.5

Proof of Theorem 2.6 (left side) cont.3 According to the arbitrage-free principle and Theorem 2.1, there must be That is, The Theorem has been proved.

Theorem 2.7 Let be the price of a European call option with the strike price K. For with the same expiration date,

Financial Meaning of Theorem 2.7 For 2 European call options with the same expiration date, the option with strike price ,leaves its holder profit room and is therefore priced , the difference between the two options shall not exceed the difference between the strike prices.

Proof of Theorem 2.7 (left side) Leave the right side part to reader Construct two portfolios at t: when t=T:

Proof of Theorem 2.7 (left side) cont.1 Case 1

Proof of Theorem 2.7 (left side) cont.2 Case 2 So,

Proof of Theorem 2.7 (left side) cont.3 Case 3 Thus, when t=T By Theorem 2.1 & Arbitrage Free Principle, for 0<t<T,

Theorem 2.8 For two European put options with the same expiration date, if then

Theorem 2.9 European call (put) option price is a convex function of K, i.e.,

Proof of Theorem 2.9 Only prove the first one, the second one left to the reader Construct two portfolios at t=0 On the expiration date t=T, Discuss in 4 cases

Proof of Theorem 2.9 cont.1 Case 1 Case 2 Therefore

Proof of Theorem 2.9 cont.2 Case 3 Case 4

Proof of Theorem 2.9 cont.3 In all Cases, when t=T But By Arbitrage Free P. and Theorem 2.1

Theorem 2.10 European call (put) option price is a linear homogeneous function of the underlying asset price and the strike price K. i.e. for

Financial Meaning of Theorem 2.10 Consider buying European options, with each option to purchase one share of a stock on the expiration date at strike price K; Also consider buying 1 European option to purchase shares of the same stock at strike price K on the expiration date; The money spent on the options in these two cases must be equal. Proof leaves to exercise

五、小结 基本概念 集合, 区间, 邻域, 常量与变量, 绝对值. 函数的概念 函数的特性 有界性,单调性,奇偶性,周期性. 反函数

思考题

思考题解答 设 则 故

练 习 题

作业

思考与练习 1.曲线上任一点的切线与两坐标轴所围成的三角形 的面积都等于常数 ,求该曲线所满足的微分方程. 解: 由题目条件有: