Derivatives_8_Binomial Option Pricing Model (1).pptx - Lecture Note Eight Binomial Option Pricing FINA0301/2322 Derivatives Faculty of Business and

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FINA0301/2322 Derivatives Faculty of Business and Economics University of Hong Kong Dr. Huiyan Qiu Lecture Note Eight: Binomial Option Pricing 8-1

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Outline How the binomial model works and how to use it One-period and multiple period binomial tree Call options and put options European options and American options Options on other assets: currencies, futures contracts Appendix: Pricing options using real probability Reading: Chapter 10, 11 8-2
Introduction to Binomial Option Pricing Binomial option pricing enables us to determine the price of an option , given the characteristics of the stock or other underlying asset To perform synthetic replication A simple yet powerful approach Assumption: the price of the underlying asset follows a binomial distribution — that is, the asset price in each period can move only up or down by a specified amount. 8-3

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A One-Period Binomial Tree Consider a European call option on the stock of XYZ, with a \$40 strike and 1 year to expiration XYZ does not pay dividends, and its current price is \$41 The continuously compounded risk-free interest rate is 8% The following figure depicts possible stock prices over 1 year, i.e., a binomial tree \$41 \$60 \$30 S = 41 K = 40 r = 8% δ = 0 T = 1 8-4
Option Payof Stock Price in 1 Year \$30 \$60 Option Payof \$0 \$20 \$41 \$60 \$30 C \$20 \$0 8-5

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Computing the Option Price Let’s look at the payof on the following portfolio: Buy 2/3 shares of XYZ Borrow \$18.462 at the risk-free rate (8%) Then the portfolio payof is Stock Price in 1 Year \$30 \$60 2/3 purchased shares \$20 \$40 Repay the loan of \$18.462 – \$20 – \$20 Portfolio Payof 0 \$20 8-6
Computing the Option Price (cont’d) The payof on the portfolio is the same as the payof on the option. What is the price of the 40-strike call option? The current price (or cost) of the portfolio is 2/3 × 41 – 18.462 = \$8.871 If the option price is higher  sell the option, buy 2/3 of a share of XYZ, borrow \$18.462  positive time-0 cash flow, zero final payof If the option price is lower  also arbitrage 8-7

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Computing the Option Price (cont’d) No arbitrage  the price of 40-strike call option must be \$8.871, the cost of the replicating portfolio. The payof to a call is replicated (duplicated) by borrowing to buy shares. The replicating portfolio forms a synthetic call option The value 2/3 is the delta ( ) of the option: the number of shares that replicates the option payof 8-8
The Binomial Solution The length of one period is h and stock price in each period goes up or down by a specified amount. Let uS 0 denotes the stock price when the price goes up, and dS 0 denotes the stock price when the price goes down. What is C 0 , the value of the option? S 0 uS 0 dS 0 C 0 C u C d 8-9

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The Binomial Solution (cont’d) Suppose that the stock has a continuous dividend yield of , which is reinvested in the stock, and the risk-free interest rate is r .

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