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

Financial Engineering: Basic Payoffs Spread Trades Bull Spread /Bear Spread Straddle Strangle Butterfly Horizontal Spread Covered Positions Stock Index Options Topics

Financial Engineering: Basic Payoffs

Combine options to obtain the payoff structures you prefer This usually involves a mixture of speculation (since you can make and lose money) and ‘psuedo-hedging’, since these synthetic positions limit possible outcomes. Financial Engineering

© K. Cuthbertson and D. Nitzsche Figure 10.1 : Payoff for calls A. Buy (long) call B. Write (short) call

© K. Cuthbertson and D. Nitzsche Figure 10.2 : Payoff for puts 0 A. Buy (long) put B. Write (short) put 0

© K. Cuthbertson and D. Nitzsche Figure 10.3 : Payoff for futures +1 Payout profile on futures and on the underlying stocks are the same. Futures profits occur at same time as profits from the options. Futures do not require any “up front“ costs at t = A. Buy (long) futuresB. Sell (short) futures

© K. Cuthbertson and D. Nitzsche Figure 10.4 : Synthetic long call Long Futures plus Long Put equals Long Call

© K. Cuthbertson and D. Nitzsche Figure 10.5 : Synthetic short put Long Futures plus Short Call equals Short Put

© K. Cuthbertson and D. Nitzsche Figure 10.6 : Synthetic long futures Short Put plus Long Call equals Long Futures

© K. Cuthbertson and D. Nitzsche Spread Trades

© K. Cuthbertson and D. Nitzsche Spread Trades Vertical (or money) spread Use either calls or puts with the same expiration date but with different strike prices (K) Straddle This is a special type of spread, uses both calls and puts Horizontal (or time or calendar) spread Options, same strike price (K) but different maturity dates Diagonal spread Uses options with different maturities and different strikes

© K. Cuthbertson and D. Nitzsche Figure 10.7 : Bull spread with calls Long Call (at K 1 ) plus Short Call (at K 2 > K 1 ) equals Call Bull Spread +1 0 Profit Share Price K1K K 1 =102 K 2 =110 S BE = K2K Gamble on stock price rise and offset cost with sale of call Favourable time decay

Payoff: Long call (K 1 ) + short call (K 2 ) = Bull Spread: { 0, +1, +1} + {0, 0, -1} = {0, +1, 0 } C 1 = 5K 1 = 102C 2 = 2K 2 = 110  = Max(0, S T -K 1 ) –C 1 – Max(0, S T -K 2 ) + C 2 = C 2 - C 1 if S T  K 1  K 2 = S T - K 1 + (C 2 - C 1 )if K 1 < S T  K 2 = (S T - K 1 - C 1 ) + (K 2 - S T + C 2 )if S T > K 1 > K 2 = K 2 - K 1 + (C 2 - C 1 ) S BE = K 1 + (C 1 – C 2 ) = = 105 Figure 10.7 : Bull spread with calls

© K. Cuthbertson and D. Nitzsche Figure 10.8 : Bear spread with calls Short Call (at K 1 ) plus Long Call (at K 2 > K 1 ) equals Bear Spread 0 Profit Share Price K1K K2K2 0 0 Gamble on stock price fall and offset cost with sale of call

© K. Cuthbertson and D. Nitzsche Fig 10.9 : Payoff, volatility strategies a) Long (Buy) Straddle Profit 0 -8 STST c) Short (Sell) Butterfly Profit 0 STST 00 b) Long (Buy) Strangle Profit 0 STST +1 d) Short (Sell) Condor Profit 0 STST

© K. Cuthbertson and D. Nitzsche Figure : Long (buy) Straddle Long Call plus Long Put equals Long Straddle Profit 0 S BE = 94 S BE = 110 K = Relatively slow time when decay when there is a long time to maturity but rather vicious time decay in the last month.

© K. Cuthbertson and D. Nitzsche Figure : Long (buy) Straddle K = 102P = 3C = 5C + P = 8 profit long straddle (figure 10.10) : [10.4]  = Max (0, S T – K) - C + Max (0, K – S T ) – P for S T > K = S T - K – (C + P) = K + (C + P) = = 110 for S T < K = K - S T – (C + P) = K - (C + P) = = 94

© K. Cuthbertson and D. Nitzsche Short (sell) Straddle: Leeson Long Call plus Long Put equals Long Straddle Profit 0 S BE = 94 S BE = 110 K = Relatively slow time when decay when there is a long time to maturity but rather vicious time decay in the last month.

© K. Cuthbertson and D. Nitzsche Figure : Long (buy) strangle Long Put plus Long Call equals Long Strangle Profit KcKc KpKp

© K. Cuthbertson and D. Nitzsche Figure : Short butterfly Short butterfly requires:  sale of 2 'outer-strike price’ call options (K 1, K 3 )  purchase of 2 ‘inner-strike price’ call options (K 2 ) Short butterfly is a ‘bet’ on a large change in price of the underlying in either direction (e.g. result of reference to the competition authorities) Cost of the ‘bet’ is offset by ‘truncating’ the payoff by selling some options

© K. Cuthbertson and D. Nitzsche Figure : (a.) Short butterfly K1K1 K2K Sell Call at K 1 plus Buy 2 Calls at K 2 plus Sell 1 Call at K 3

© K. Cuthbertson and D. Nitzsche Figure : (a.) Short butterfly (Cont.) Profit Stock Price equals Short Butterfly +1

© K. Cuthbertson and D. Nitzsche Figure : (b.) Long butterfly Profit Stock Price

© K. Cuthbertson and D. Nitzsche Horizontal Spread Options, same strike price (K) but different maturity dates e.g. buying a long dated option (360-day) and selling a short dated option (180-day)~ both are at-the-money In a relatively static market (ie. S 0 = K) this spread will make money from time decay, but will loose money if the stock price moves substantially (figure 10.13).

© K. Cuthbertson and D. Nitzsche Figure : Horizontal spread Profit at expiry 0 Horizontal spread : a long position in a 1-year option and a short position in a 180 day option, both at-the-money Profit profile 30 days before expiry of short dated option Profit S 0 = K Stock price

27 Covered Positions.

© K. Cuthbertson and D. Nitzsche Figure : Covered call STST 24 Profit 0 $4 Short Call 2128 $3 25 K = Long Stock Covered call = long stock + short call

© K. Cuthbertson and D. Nitzsche Figure : Protective put strategy STST 24 Profit 0 -$4 Long Put 22 -$ Long stock Note : P = 5, K = 25, S 0 = 24 Protective put = long stock + long put Long stock+ Long Put = protective put (ie. Call payoff)

30 Stock Index Options.

31 Payoff S T = 200 (index points) K = 210 (index points) long put receives : $ Payoff = z (K-S T ) = $100 ( ) = $1,000 Quotes:Premia C = 5 (index points) One call contract costs $550 (= $100 x 5), hence: Invoice Price of one S&P100 Call contract = C z Payoff and Price Quotes

32 15 March 1997 Value of stock portfolio, TVS = £10m Portfolio beta, = 1.5 Current FTSE100 index, S0 = 4450 June 4450 Put Quote,P = 103 Value of one index point = £10 Delta of June-4450 Put,  p = 0.5 Note that for simplicity we choose K = 4450, the same as the initial stock price, S0 Static Hedge : Np = = = 337 contracts Cost of one June put contract = (103) (£10) = £1,035 Total cost 337 put contracts = 337 (£1,035) = £348,795 Percentage cost = (348,795/£10m)100 = 3.48% Dynamic Hedge : Np = Note: Initial percentage Cost of Puts = 6.97% Static and Dynamic Hedge (FTSE100)

© K. Cuthbertson and D. Nitzsche End of Slides