Option Strategies 1

Note on Notation Here, T denotes time to expiry as well as time of expiry, i.e. we use T to denote indifferently T and δ = T – t Less accurate but handier this way, I think 2

3 Types of Strategies Options are prime examples of contingent claims, which can be combined at leisure Static portfolios: Take a position in the option and the underlying Take a position in 2 or more options of the same type (a spread) Take a position in a mixture of calls & puts (a combination) These are prime examples of complex payoffs that should be and to a large extent are easily priced based on LOOP alone.

4 Positions in an Option & the Underlying Profit STST K STST K STST K STST K (a) (b) (c)(d) Basis of Put-Call Parity (under LOOP)…e.g., from (c), P + S = C + Cash (  Ke -rT )

5 Bull Spread Using Calls K1K1 K2K2 Profit STST

Bull Spread Using Calls Example Create a bull spread on IBM using the following 3- month call options on IBM: Option 1: Strike:K 1 = 102 Price:C 1 = 5 Option 2: Strike:K 1 = 110 Price:C 2 = 2

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

Payoff: Long call (K 1 ) + short call (K 2 ) = Bull Spread: { 0, +1, +1} + {0, 0, -1} = {0, +1, 0 } = 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 ) = = K 2 - K 1 + (C 2 - C 1 )if S T > K 1 > K 2 ‘Break-even’: S BE = K 1 + (C 1 – C 2 ) = = 105

9 Bear Spread Using Puts K1K1 K2K2 Profit STST

10 Bull Spreads with puts & Bear Spreads with Calls Of course can do bull spreads with puts and bear spreads with calls (put-call parity) Figured out how? Suppose their price is different from their counterpart seen earlier? Is there a Type A arbitrage? How about a Type B?

11 Bull Spread Using Puts K1K1 K2K2 Profit STST

12 Bear Spread Using Calls K1K1 K2K2 Profit STST

You already hold stocks but you want to limit downside (buy a put) but you are also willing to limit the upside if you can earn some cash today (by selling an option, i.e. a call) COLLAR = long stock + long put (K 1 ) + short call (K 2 ) {0,+1,0} = {+1,+1,+1} + {-1,0,0} + {0,0,-1} Equity Collar

+1 00 Long Stock Long Put Short Call Equity Collar plus equals Equity Collar: Payoff Profile

S T K 2 Long SharesS T S T S T Long Put (K 1 )K 1 – S T 0 0 Short Call (K 2 )00 – (S T – K 2 ) Gross PayoffK 1 S T K 2 Net Profit K 1 – (P – C) S T – (P – C) K 2 – (P – C) Net Profit = Gross Payoff – (P – C) Equity Collar Payoffs

Short Put plus Long Call equals Long Forward A Basic Combination: A Synthetic Forward/Futures This is basis for Put-Call parity (remember?): c + Ke -rT = p + S 0 In fact, with ATM-fwd options (i.e., K = F 0 ): c − p = S 0 − F 0 e -rT = 0

17 Box Spread A combination of a bull call spread and a bear put spread If all options are European a box spread is, under NA, worth the present value of the difference between the strike prices Check it out Is there a Type A or Type B Arbitrage otherwise? If they are American this is not necessarily so (see Business Snapshot 11.1)

Volatility Combinations Mainly Straddle Strangles These are strategies that show the true ‘character’ of options But also Strip Straps Etc.

19 A Straddle Combination Profit STST K

Long (buy) Straddle Data: K = 102P = 3C = 5C + P = 8 profit long straddle:  = Max (0, S T – K) - C + Max (0, K – S T ) – P = 0 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

Straddles and HF Fung and Hsieh (RFS, 2001) empirically show that many hedge funds follow strategies that resemble straddles: ‘Market timers’ returns are highly correlated with the return to long straddles on diversified equity indices and other basic asset classes

22 A Strangle Combination K1K1 K2K2 Profit STST

23 Straddle vs Strangle Suppose an ATM straddle costs less than a strangle centred on the spot, i.e. with strikes of the constituent call and put equally distant from the spot price (assume zero interest rate). Is there a Type A arbitrage? How about a Type B?

24 KSTST KSTST StripStrap Strip & Strap (not joking!) Profit

Time Decay Combinations Calendar (or horizontal) spreads 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 (i.e. S 0 = K) this spread will make money from time decay, but will loose money if the stock price moves substantially

‘Quasi-Elementary’ Securities Arrow(-Debrew) introduces so called Arrow- Debrew elementary securities, i.e. contingent claims with $1 payoff in one state and $0 in all other states These can be seen as “bet” options Butterflies look a lot like them The all process of creating and trading strategies boils down to an attempt to finely ‘cover’ all states of the world, it is called completing the market

Complete Markets The all process of creating and trading strategies boils down to an attempt to finely ‘cover’ all states of the world The process of introducing new securities that allow the trading of previously untraded payoffs, thereby spanning with available securities a wider set of states of the world, is called completing the market. This is an important function that contributes to allocational efficiency of capital markets.

28 Butterfly Spread Using Calls K1K1 K3K3 STST K2K2 Profit

29 Butterfly Spread Using Puts K1K1 K3K3 Profit STST K2K2

Butterflies Replication Butterfly requires: sale of 2 ‘inner-strike price’ call options (K2) purchase of 2 'outer-strike price’ call options (K1, K3) Butterfly is a ‘bet’ on a small change in price of the underlying in either direction Potential downside of the ‘bet’ is offset by ‘truncating’ the payoff by buying some options

Short Butterflies Replication Short butterfly requires: purchase of 2 ‘inner-strike price’ call options (K2) sale of 2 'outer-strike price’ call options (K1, K3) 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 Could also buy (go long) a bull and a bear (call or put) spread, same result

32 Short Butterfly Spread Using Calls K1K1 K3K3 Profit STST K2K2