Mechanics of Call Futures Options Futures Options When a call futures option is exercised the holder acquires Chapter 16 1. A long position in the futures 2. A cash amount equal to the excess of the futures price at previous settlement over the strike price Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 2 Mechanics of Put Futures Option The Payoffs If the futures position is closed out When a put futures option is exercised the immediately: holder acquires Payoff from call = F – K 1. A short position in the futures Payoff from put = K – F 2. A cash amount equal to the excess of the strike price over the futures price at previous where F is futures price at time of exercise settlement Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 3 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 4 Potential Advantages of Futures Put-Call Parity for European Options over Spot Options Futures Options (Equation 16.1, page 347) Futures contract may be easier to trade than Consider the following two portfolios: underlying asset 1. European call plus Ke-rT of cash Exercise of the option does not lead to delivery of the underlying asset 2. European put plus long futures plus cash equal to F e-rT Futures options and futures usually trade in 0 adjacent pits at exchange They must be worth the same at time T so Futures options may entail lower transactions that costs -rT -rT c+Ke =p+F0 e Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 5 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 6 Other Relations Binomial Tree Example -rT -rT A 1-month call option on futures has a strike price of F0 e – K < C – P < F0 – Ke 29. -rT c > (F0 – K)e Futures Price = $33 Option Price = $4 -rT p > (F0 – K)e Futures price = $30 Option Price=? Futures Price = $28 Option Price = $0 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 7 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 8 Setting Up a Riskless Portfolio Valuing the Portfolio ( Risk-Free Rate is 6% ) Consider the Portfolio: long futures short 1 call option The riskless portfolio is: long 0.8 futures 3–4 short 1 call option The value of the portfolio in 1 month is -2 –1.6 Portfolio is riskless when 3–4 = –2 or The value of the portfolio today is = 0.8 –1.6e –0.06 = –1.592 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 9 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 10 Generalization of Binomial Tree Valuing the Option Example (Figure 16.2, page 349) The portfolio that is A derivative lasts for time T and is long 0.8 futures dependent on a futures short 1 option F u is worth –1.592 0 ƒu The value of the futures is zero F0 ƒ The value of the option must therefore be 1.592 F0d ƒd Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 11 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 12 Generalization (continued) Generalization (continued) Consider the portfolio that is long futures and short 1 derivative Value of the portfolio at time T is F0u F0 – ƒu F0u –F0 –ƒu Value of portfolio today is – ƒ F0d F0– ƒd Hence The portfolio is riskless when -rT ƒ = – [F0u –F0– ƒu]e ƒ f ud Fu00 Fd Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 13 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 14 Generalization Growth Rates For Futures Prices (continued) A futures contract requires no initial investment Substituting for we obtain In a risk-neutral world the expected return –rT ƒ = [ p ƒu + (1 – p )ƒd ]e should be zero The expected growth rate of the futures where price is therefore zero 1d The futures price can therefore be treated p like a stock paying a dividend yield of r ud This is consistent with the results we have presented so far (put-call parity, bounds, binomial trees) Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 15 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 16 Black’s Model Valuing European Futures (Equations 16.7 and 16.8, page 351) Options The formulas for European options on futures We can use the formula for an option on a are known as Black’s model stock paying a continuous yield rT c e F0 N (d1 ) K N (d 2 ) Set S0 = current futures price (F0) rT Set q = domestic risk-free rate (r ) p e K N (d 2 ) F0 N (d1 ) 2 Setting q = r ensures that the expected ln( F0 / K ) T / 2 where d1 growth of F in a risk-neutral world is zero T 2 ln( F0 / K ) T / 2 d 2 d1 T T Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 17 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 18 How Black’s Model is Used in Using Black’s Model Instead of Practice Black-Scholes (Example 16.5, page 352) European futures options and spot options Consider a 6-month European call are equivalent when future contract option on spot gold matures at the same time as the otion. 6-month futures price is 620, 6-month This enables Black’s model to be used to risk-free rate is 5%, strike price is 600, value a European option on the spot price and volatility of futures price is 20% of an asset Value of option is given by Black’s model with F0=620, K=600, r=0.05, T=0.5, and =0.2 It is 44.19 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 19 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 20 American Futures Option Prices vs American Spot Option Prices Futures Style Options (page 353-54) A futures-style option is a futures contract on the If futures prices are higher than spot prices option payoff (normal market), an American call on futures Some exchanges trade these in preference to is worth more than a similar American call on regular futures options spot. An American put on futures is worth less The futures price for a call futures-style option is than a similar American put on spot F0 N(d1) KN(d2 ) When futures prices are lower than spot The futures price for a put futures-style option is prices (inverted market) the reverse is true KN(d2)F0N(d1) Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 21 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 22 Put-Call Parity Results: Summary Summary of Key Results from Chapters 15 and 16 Nondividen d Paying Stock : We can treat stock indices, currencies, rT c K e p S 0 & futures like a stock paying a Indices : rT qT continuous dividend yield of q c K e p S 0 e Foreign exchange : For stock indices, q = average rT rf T c K e p S 0 e dividend yield on the index over the Futures : option life rT rT c K e p F0 e For currencies, q = rƒ For futures, q = r Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 23 Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 24.