【正文】 nstruct a July 90 – 95 – 100 butterfly spread. So, we: – Buy one call with exercise price $90, for $. – Sell two calls with exercise price $95, for $ each. – Buy one call with exercise price $100, for $. – Total cost: $. Butterfly spreads payoffs Share price at expiry ($) Payoff to long call at 90 ($) Payoff to writing two calls at 95 ($) Payoff to long call at 100 ($) Total portfolio payoff ($) % Return to portfolio 120 30 2*25 = 50 20 0 % 110 20 2*15 = 30 10 0 % 100 10 2*5 = 10 0 0 % 99 9 2*4 = 8 0 1 % 98 8 2*3 = 6 0 2 % 97 7 2*2 = 4 0 3 % 96 6 2*1 = 2 0 4 % 95 5 0 0 5 % 94 4 0 0 4 % 93 3 0 0 3 % 92 2 0 0 2 % 91 1 0 0 1 % 90 0 0 0 0 % 80 0 0 0 0 % 70 0 0 0 0 % Returns based upon investment costs of $ – 2*$ + $ = $. Butterfly spread payoff 90 Share price at expiry Spread value at expiry (PAYOFF) 100 5 95 Straddle: definition ? A straddle consists of buying both a call and a put option both at the same exercise price. ? In our example, we buy a call with an exercise price of $95 at a cost of $ and a put option at a cost of $, so a total cost of $. Straddle payoff Share price at expiry ($) Payoff from the long call ($) Payoff from the long put ($) Total portfolio payoff ($) % return 130 35 0 35 % 120 25 0 25 % 110 15 0 15 % 100 5 0 5 % 95 0 0 0 % 90 0 5 5 % 80 0 15 15 % 70 0 25 25 % 60 0 35 35 % Straddle payoff: graph 25 Share price at expiry Straddle payoff at expiry Option pricing ? It will be very tough to obtain an exact value for the options but it will be possible to establish some important bounds and relationships. ? First note that both call and put options must have positive value, . noone should have to pay you to take ownership of a call or put option. This is clearly true as the least an option is worth at expiry is 0. Call 0 Put 0 ? American options must be worth at least as much as European options because American options can be exercised at any time – . it is a European option with extra flexibility. American call European call American put European put PutCall parity ? The most useful simple relationship between call and put option prices is the put call parity. ? Consider, for the sake of an example, European put and call options on a share with current price $50 (paying no dividends), both options have exercise prices of $55 and the same maturity. ? Construct two portfolios: A: Buy one European put option and also buy the share. B: Buy one European call option and invest an amount of cash which, at expiry, will have value equal to the exercise price ($55). ? The following table demonstrates that they both have the same value at the maturity of the option. Putcall parity: cash flows Share Price ($) First Portfolio Second Portfolio Put + Share = Total Call + Safe Asset = Total 70 0 70 70 15 55 70 65 0 65 65 10 55 65 60 0 60 60 5 55 55 55 0 55 55 0 55 55 50 5 50 55 0 55 55 45 10 45 55 0 55 55 40 15 40 55 0 55 55 35 20 35 55 0 55 55 Portfolio payoffs to show Putcall parity, exercise price is 55 Putcall parity ? Notice that the first portfolio is often termed the insurance portfolio as the portfolio can never have a value less than the exercise price, $55 in this case, which means that the holder of this portfolio can always sell the share for $55 regardless of its price. ? Clearly regardless of what the share price is then these two portfolios are equal in value at the maturity of the options (regardless of how long this actually is). ? If this is true then the present value of the two portfolios must also be the same, otherwise there will be an arbitrage opportunity…. Put call parity: arbitrage ? Assume that the maturity of the options is one year and that the current share price is $50 and the 1year risk free rate is 1%. The current price of the put option is $7, and the current value of the risk free asset is 55/() = $. ? In this case the call option should have value $ (. $7 + $50 $ = $). What happens if call options are actually $4? ? Well the value of p