S TOCHASTIC M ODELS L ECTURE 4 P ART III B ROWNIAN M OTION Nan Chen MSc Program in Financial Engineering The Chinese University of Hong Kong (Shenzhen) Nov 25, 2015

Outline 1.One-period financial market 2.No arbitrage opportunity and risk neutral probability 3.Martingale measure

4.5 M ARTINGALE M EASURE AND N O A RBITRAGE C ONDITION

Arbitrage In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more financial assets. One example from FX market: – 1 USD = 7.5 HKD – 1 USD = 150 JPY – 1 HKD = 25 JPY

Arbitrage Free and Efficient Market In an efficient market, there should not be any arbitrage opportunities because – Investors can “buy low and sell high” to lock in risk-free profit – When a massive population follows this strategy, that will close down the price difference.

A Simple Mathematical Model for Financial Market Consider a one-period model – Time: (now) and (future) – States: Physical probabilities for each state: – Stocks: – Each stock can be characterized by Price at time : Payoff at time : the investor will receive for each share of stock he has when state realizes at time

Matrix Form of the Model Formulation Price vector for the stocks: Payoff matrix:

Example I: Two-Stock Market Let there be three states and two stocks. Stock 1 is risk free and has payoff. Stock 2 is risky with payoff Then, the payoff matrix of this market is given by

Portfolios Investors can construct investment portfolios to achieve certain objectives. A portfolio in this simple market is composed of holdings of the securities. Suppose that is the holding of stock in a portfolio. Then, – Portfolio price: – Portfolio payoff At state

Example I: Two-Stock Market Suppose that one investor purchase 1 share of stock 1 and 2 shares of stock 2 in Example I. Then the payoff of his portfolio is given by

Arbitrage in this Simple Market We say an arbitrage opportunity arises in this simple market if there exists a portfolio such that – Either, and – Or, and

Example II: Arbitrage Consider two stocks with payoffs and. Their prices are One arbitrage portfolio can be constructed by

Farkas Hyperplane Separation Lemma Farkas Lemma: Let be an matrix and be an dimensional vector. There does not exist a vector such that – and – Or, and if and only if there exists a strictly positive vector such that

Fundamental Theorem in Finance Theorem: There is no arbitrage in the market if and only if there exists a strictly positive vector such that

Example III: A No-Arbitrage Market Revisit Example II. If we change the stock prices to and then there should be no arbitrage in the market. Here the corresponding

Risk Neutral Probabilities We may use to define a new probability distribution such that with

Risk Neutral Pricing Under this new probability distribution, we can represent the stock price in Example III by where 0.8 can be viewed as the discounting factor of risk free investment, and

Risk Neutral Pricing The example provides a very useful framework to evaluate financial assets: Price = Discounting factor Expected Payoff But, the expectation should not be computed in the physical probability distribution. It should be computed in the risk neutral probability distribution.

Risk Neutral Pricing and Martingale Consider stock 2. Its discounted values at time 0 and 1 constitute a simple stochastic process: – – It is very easy to see that is a martingale.