2020/9/16,1,Options onStock Indices, Currencies, and Futures,Chapter 12,2020/9/16,2,European Options on StocksPaying 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 S0eq T and provides no income,2020/9/16,3,European Options on StocksPaying Continuous Dividendscontinued,We can value European options by reducing the stock price to S0eq T and then behaving as though there is no dividend,2020/9/16,4,Extension of Chapter 7 Results(Equations 12.1 to 12.3),Lower Bound for calls:,Lower Bound for puts,Put Call Parity,2020/9/16,5,Extension of Chapter 11 Results (Equations 12.4 and 12.5),2020/9/16,6,The Binomial Model(Risk-neutral world),S0u u,S0d d,S0 ,p,(1 p ),f=e-rTpfu+(1-p)fd ,2020/9/16,7,The Binomial Modelcontinued,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=S0e (r-q)T so that,2020/9/16,8,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 Contracts are settled in cash,2020/9/16,9,Index Option Example,Consider a call option on an index with a strike price of 560 Suppose 1 contract is exercised when the index level is 580 What is the payoff?,2020/9/16,10,Using Index Options for Portfolio Insurance,Suppose the value of the index is S0 and the strike price is X If a portfolio has a b of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars held If the b is not 1.0, the portfolio manager buys b put options for each 100S0 dollars held In both cases, X is chosen to give the appropriate insurance level,2020/9/16,11,Example 1,Portfolio has a beta of 1.0 It is currently worth $5 million The index currently stands at 1000 What trade is necessary to provide insurance against the portfolio value falling below $4.8 million?,2020/9/16,12,Example 2,Portfolio has a beta of 2.0 It is currently worth $1 million and index stands at 1000 The risk-free rate is 12% per annum(每3个月计一次复利） The dividend yield on both the portfolio and the index is 4%(每3个月计一次复利） How many put option contracts should be purchased for portfolio insurance?,2020/9/16,13,If index rises to 1040, it provides a 40/1000 or 4% return in 3 months Total return (incl. dividends)=5% Excess return over risk-free rate=2% Excess return for portfolio=4% Increase in Portfolio Value=4+3-1=6% Portfolio value=$1.06 million,Calculating Relation Between Index Level and Portfolio Value in 3 months,2020/9/16,14,Determining the Strike Price,An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value,2020/9/16,15,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,2020/9/16,16,Currency Options,Currency options trade on the Philadelphia Exchange (PHLX) There also exists an active over-the-counter (OTC) market Currency options are used by corporations to buy insurance when they have an FX exposure,2020/9/16,17,The Foreign Interest Rate,We denote the foreign interest rate by rf When a U.S. company buys one unit of the foreign currency it has an investment of S0 dollars The return from investing at the foreign rate is dollars This shows that the foreign currency provides a “dividend yield” at rate rf,2020/9/16,18,Valuing European Currency Options,A foreign currency is an asset that provides a continuous “dividend yield” equal to rf We can use the formula for an option on a stock paying a continuous dividend yield : Set S0 = current exchange rate Set q = r,2020/9/16,19,Formulas for European Currency Options,2020/9/16,20,Alternative Formulas,Using,2020/9/16,21,Mechanics of Call Futures Options,Most of Futures options are American. The maturity date is usually on, or a few days before, the earliest delivery date of the underlying futures contract. When a call futures option is exercised the holder acquires 1. A long position in the futures 2. A cash amount equal to the excess of the futures price over the strike price,2020/9/16,22,Mechanics of Put Futures Option,When a put futures option is exercised the holder acquires 1. A short position in the futures 2. A cash amount equal to the excess of the strike price over the futures price,2020/9/16,23,The Payoffs,If the futures position is closed out immediately: Payoff from call = F0-X Payoff from put = X-F0 where F0 is futures price at time of exercise,2020/9/16,24,Why Futures Option instead of Spot Option?,Futures is more liquid and easier to get the price information Can be settled in cash Lower transaction cost,2020/9/16,25,Put-Call Parity for Futures Option,Consider the fo