Payoff and Replications Chapters 8, 10

Review of Option Types A call is an option to buy A put is an option to sell A European option can be exercised only at the end of its life An American option can be exercised at any time

Option Positions Long (buy) call Long (buy) put Short (write) call Short (write) put

Long Call on eBay (Figure 8.1, Page 182)—Limited liability Profit from buying one eBay European call option: option price = $5, strike price = $100, option life = 2 months Correction: Focus on payoff, not “profit” Profit ($) Terminal stock price ($)

Short Call on eBay (Figure 8.3, page 184) —Unlimited liability Profit from writing one eBay European call option: option price = $5, strike price = $ Profit ($): Change to payoff Terminal stock price ($)

Long Put on IBM (Figure 8.2, page 183) –Limited profit & liability Profit from buying an IBM European put option: option price = $7, strike price = $ Profit ($): Change to payoff Terminal stock price ($)

Short Put on IBM (Figure 8.4, page 184) –Limited liability Profit from writing an IBM European put option: option price = $7, strike price = $ Profit ($) Terminal stock price ($)

Payoffs from Options What is the Option Position in Each Case? K = Strike price, S T = Price of asset at maturity Payoff STST STST K K STST STST K K

Which of the position has limited liability? Plot the payoff Long stock Short stock Long call, put Short put Short call Short 1 call, long 1 put at the same strike Short 1 call, long 1 stock Short 1 call, short 1 put

Types of Derivative Strategies Chapter 11 Take a position in the option and the underlying Take a position in 2 or more options of the same type (A spread) Combination: Take a position in a mixture of calls & puts (A combination)

Positions in an Option & the Underlying (Figure 10.1, page 224) Profit STST K STST K STST K STST K (a) (b) (c)(d)

Bull Spread Using Calls (Figure 10.2, page 225) K1K1 K2K2 Profit STST

Bull Spread Using Puts Figure 10.3, page 226 K1K1 K2K2 Profit STST

Bear Spread Using Puts Figure 10.4, page 227 K1K1 K2K2 Profit STST

Bear Spread Using Calls Figure 10.5, page 229 K1K1 K2K2 Profi t STST

Box Spread A combination of a bull call spread and a bear put spread If all options are European a box spread is worth the present value of the difference between the strike prices If they are American this is not necessarily so. (See Business Snapshot 10.1)

Butterfly Spread Using Calls Figure 10.6, page 231 K1K1 K3K3 Profit STST K2K2

Butterfly Spread Using Puts Figure 10.7, page 232 K1K1 K3K3 Profit STST K2K2

Calendar Spread Using Calls Figure 10.8, page 232 Profit STST K

Calendar Spread Using Puts Figure 10.9, page 233 Profit STST K

A Straddle Combination Figure 10.10, page 234 Profit STST K

Strip & Strap Figure 10.11, page 235 Profit KSTST KSTST StripStrap

A Strangle Combination Figure 10.12, page 236 K1K1 K2K2 Profit STST

Standard contracts Straddles Strangles Risk reversals Binary call or put Butterfly spread

A general replication formula Prove this formula: Try to replicate the terminal payoff that pays ln(S T ) If you can replicate, you can price. Price variance swap in terms of European options, assuming continuous underlying dynamics.