华东师范大学：《金融工程》英文版 Chapter 12 Options on Stock Indices, Currencies, and Futures

12.1 Options on Stock Indices Currencies and futures Chapter 12 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.1 Options on Stock Indices, Currencies, and Futures Chapter 12

12.2 European Options on Stocks Paying continuous dividends We get the same probability distribution for the stock price at time T in each of the following cases 1. The stock starts at price So and provides a continuous dividend yield g The stock starts at price Soe-qn and provides no dividend yield Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.2 European Options on Stocks Paying Continuous Dividends We get the same probability distribution for the stock price at time T in each of the following cases 1. The stock starts at price S0 and provides a continuous dividend yield = q 2. The stock starts at price S0e -qT and provides no dividend yield

12.3 European Options on Stocks Paving continuous dividends (continued) We can value European options by reducing the stock price to Soe q/ and then behaving as though there is no dividend Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.3 European Options on Stocks Paying Continuous Dividends (continued) We can value European options by reducing the stock price to S0 e –q T and then behaving as though there is NO dividend

12.4 Extension of Chapter 7 Results (Equations 12.1 to 12.3) Lower Bound for calls Xe Lower Bound for puts p≥e"-So oe 9? Put Call Parity C+Xe=p+se9? Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.4 Extension of Chapter 7 Results (Equations 12.1 to 12.3) 0 qT rT c S e Xe − −  − Lower Bound for calls: Lower Bound for puts 0 rT qT p Xe S e − −  − Put Call Parity 0 rT qT c Xe p S e − − + = +

12.5 Extension of Chapter 11 Results(equations 12.4 and 12.5 soe g n(di-xe n(d2) where d, ln(S0/X)+(r-q+a/2)7 √T n(S/X)+(r-q-a-/2) √T Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.5 Extension of Chapter 11 Results (Equations 12.4 and 12.5) 0 1 2 2 0 1 0 1 0 2 ( ) ( ) ( ) ( ) 2 ln( / ) ( / 2) where 2 ln( / ) ( / 2) q r r T T T T q c S N d Xe N d p Xe N d S N d S X r T d T S X r T d e e q q T     − − − − = − = − − − + + = + − − − =

12.6 The binomial model f 0 (7 0 p)f fer!Ipfi+(l-plfdI p Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.6 The Binomial Model S0u ƒu S0d ƒd S0 ƒ f=e-rT[pfu +(1-p)fd ] p=?

12.7 TThe Binomial model (continued In a risk-neutral world the stock price grows at r-g rather than at r when there is a dividend yield at rate q The probability, p, of an up movement must therefore satisfy pOut(I-p)Sod=spe(r-g)r so that q)7 d Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.7 The Binomial Model (continued) • In a risk-neutral world the stock price grows at r-q rather than at r when there is a dividend yield at rate q • The probability, p, of an up movement must therefore satisfy pS0u+(1-p)S0d=S0 e (r-q)T so that ( ) r T q e d p u d − − = −

12.8 Index options Option contracts are on 100X the index The most popular underlying indices are the s&P 100(American)OEX the s&P 500(European) SPX the Major Market Index(XMD Contracts are settled in cash Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.8 Index Options • Option contracts are on 100× the index • The most popular underlying indices are the S&P 100 (American) OEX the S&P 500 (European) SPX the Major Market Index (XMI) • Contracts are settled in cash

12.9 Index Option Exampl oe Consider a call option on the oEX index with a strike price of 560 Suppose 1 contract is exercised When the index level is 580 What is the payoff? (Ans=2000) Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.9 Index Option Example • Consider a call option on the OEX index with a strike price of 560 • Suppose 1 contract is exercised when the index level is 580 • What is the payoff? (Ans=2000)

12.10 Valuing European Index Options We can use the formula for an option on a stock paying a continuous dividend yield Set So current index level Set g average dividend yield expected during the life of the option Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 12.10 Valuing European Index Options We can use the formula for an option on a stock paying a continuous dividend yield • Set S0 = current index level • Set q = average dividend yield expected during the life of the option