© K.Cuthbertson and D.Nitzsche 1 Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE Futures Options

© K.Cuthbertson and D.Nitzsche 2 Contracts: quotes, delivery,settlement procedures, Put-call parity relationship. Pricing futures options using Black’s formula. Merits of options on spots assets and options on futures contracts. Topics

© K.Cuthbertson and D.Nitzsche 3 Contracts: Quotes, delivery, settlement procedures

© K.Cuthbertson and D.Nitzsche 4 Futures Options contracts on T-bond and T-Note futures (CBT) Eurodollar futures (CME) EURIBOR contract (LIFFE) Japanese Yen (CME) Deutschmark (CME) Canadian dollar (CME) Crude oil and other oil products (NYM) Gold (CMX), corn (CBT) and sugar (CSCE) Contracts: Futures Options

© K.Cuthbertson and D.Nitzsche 5 long call option is exercised acquire a long position in the underlying futures contract (with a delivery price of K) plus a cash amount equal to F T – K, ( where F T is the futures price at expiry of the option) long put option is exercised acquire a short position in the underlying futures contract (with a delivery price of K) plus a cash amount equal to K - F T. Payoffs

© K.Cuthbertson and D.Nitzsche 6 Call premium: Quotes(US T-Bond Futures Option) C= (The ‘39’ represents 39/64ths) Hence: C = per $100 nominal. Delivers one futures contract, with face value=$100,000 Invoice Price of T-Bond Futures Option = ( / 100) $100,000 = $2, Payoff The long call on the September-100 T-Bond futures receives: One long T-Bond futures contract at a price K=100 plus cash of $3,000 [= ( ) ($100,000)/100] Quotes and Payoff

© K.Cuthbertson and D.Nitzsche 7 Put-Call Parity

© K.Cuthbertson and D.Nitzsche 8 Portfolio A: Long Futures + Long Put {+1, +1} + {-1, 0} ={0, +1} Portfolio B: Long Call + Bonds (or cash) {0, 1} + {0, 0} ={0, +1} Put-Call Parity

© K.Cuthbertson and D.Nitzsche 9 Portfolio-A: Long futures payoff= (F T - F) Put payoff = max[0, K - F T ]. Payoff for A: either F T - F or K - F. Portfolio-B: Bond has a face value of K - F at t=T while long call pays off max[0, F T - K]. Payoff for B is either F T – F for F T > K or K – F for F T < K ~ the same as for Portfolio-A. Two portfolios with same payoff at T,worth same today. Value of the long futures is zero at outset, hence : [12.7]P e = C e + (K - F)e -rT Put-Call Parity

© K.Cuthbertson and D.Nitzsche 10 Pricing Futures Options

© K.Cuthbertson and D.Nitzsche 11 [12.8]F = S e  T e.g.stock options, pay continuous dividends  = r-  Can show  F =  S = . Hence: Black-Scholes formula (for dividend paying stocks) replace q by r and S by F to give Black’s (1976) formula [12.9a]C = [F N(d 1 ) – K N(d 2 )]e -rT [12.9b]P = [K N(-d 2 ) - F N(-d 1 )]e -rT d 1 = d 2 = d 1 - Pricing: a la Black-Scholes

© K.Cuthbertson and D.Nitzsche 12 Let F 0 = 100 U = 1.15 D = 0.9, F u = 115 F d = 90 Long call option on the futures, K = 100 C u = max[F u - K, 0] =15 and C d = max[F d - K, 0] = 0. Pricing: BOPM (One period)

© K.Cuthbertson and D.Nitzsche 13 [12.10a]V u = C u – h(F u – F 0 ) [12.10b]V d = C d – h(F d – F 0 ) Choose h so payoffs are equal : [12.11] h = Cost of the hedge-portfolio at t=0, equals C since it cost nothing at t=0 to enter the futures market. Portfolio of one long call and h-short futures is riskless then we must earn the risk-free rate r = 10%, hence : [12.12]Ce rT = V u = C u – h(F u – F 0 ) Ce 0.10(1) = 15 –0.6( ) Therefore C = Pricing: BOPM

© K.Cuthbertson and D.Nitzsche 14 Algebraically Substituting for h from equation [12.11] in [12.12] [12.13]C = e -rT [q*C u + (1-q*)C d ] q* = (1-U) / (U-D) C u = max[0, F u – K] and C d = max[0, F d – K] Compare with formula for option on a dividend paying stock q = (e (r-  )T -D)/(U-D). For an option on a futures, cost of carry for the futures =0, hence set  =r S replaced by F and we get Pricing: BOPM

© K.Cuthbertson and D.Nitzsche 15 Greater liquidity in the futures market than spot market. Transactions costs of insurance (or hedging) with options on stock index futures are lower than for options on underlying stock indices. This is also true for futures on agricultural products,metals since these options deliver a futures contract which can be closed out at low cost. In contrast, options on the spot asset, deliver the underlying asset and this may be costly or inconvenient. Popularity of futures options

© K.Cuthbertson and D.Nitzsche 16 Payoffs at Maturity

© K.Cuthbertson and D.Nitzsche 17 [12.14]  = max[0, F T – K] - C = - Cif F T K = F T - K - C if F T > K F BE = K + C = = 482. For the S&P500 call option on futures, z = $500 Fig 12.2 gives the possible outcomes for K = 475 If, at expiry, F T = 485 (> K = 475), [12.15]  =(F T - K - C) $500=( ) $500 = $1,500 If F T < K = 475, call expires worthless Cost is the put premium C = 7, Total cost $3,500 (= $500 C) LONG CALL

© K.Cuthbertson and D.Nitzsche 18 Figure 12.2 : Call on futures option (S&P500, K = 475) FTFT Profit (per unit of index) 3 C = 7 F T = K = Breakeven : F BE = K + C = 482

© K.Cuthbertson and D.Nitzsche 19 [12.16]  = max[0, K - F T ] - P = K - F T - Pif F T < K = -Pif F T K F BE = K - P If P = 5 then, invoice price is zP = $2,500. Figure 12.3 gives the profit profile at expiry. If F T = 462 and K = 475 then Payoff is (K - F T - P) $500 = 8 ($500) = $4,000. LONG PUT

© K.Cuthbertson and D.Nitzsche 20 Figure 12.3 : Put on futures option (S&P500, K = 475) FTFT 8 P = K = Profit (per unit of index) Breakeven : F BE = K - P = 470

© K.Cuthbertson and D.Nitzsche 21 = Long Futures + Written Call on Futures Option Profit, long futures and short call (futures option): [12.17a]  F = F T – F 0 and [12.17b]  c = - max[0, F T – K] + C giving a combined payoff for the covered call of: [12.18]  = (F T – F 0 ) – max[0, F T – K] + C = K – F 0 + Cif F T > K = F T – F 0 + Cif F T K Covered call, futures options

© K.Cuthbertson and D.Nitzsche 22 Fig12.4 : Covered call, futures options FTFT C = Profit (per unit of index) -477 F T = 490 F 0 = 477 F BE = F 0 - C = Long futures F T = 460 Covered call K = 475, C = 8, F 0 = 477

© K.Cuthbertson and D.Nitzsche 23 END OF SLIDES