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Profit Profiles for Long Calls Payoff Spot Price (S T ) 0 Long Call X S T – X – C C

Profit Profiles for Short Calls Payoff Spot Price (S T ) 0 Short Call X S T – X – C C

Call Options - Zero Sum Game Payoff Spot Price (S T ) 0 Short Call Long Call X

Profit Profiles for Long Puts 0 Payoffs Spot Price (S T ) Long Put X P X – S T – P

Profit Profiles for Short Puts 0 Payoffs Spot Price (S T ) Short Put X P – (X – S T ) P

Put Options - Zero Sum Game 0 Payoffs Spot Price (S T ) Short Put Long Put X

In the Money - exercise of the option would be profitable Call: market price>exercise price (S t > X) Put: exercise price>market price (S t < X) Out of the Money - exercise of the option would not be profitable Call: market price<exercise price (S t < X) Put: exercise price X) At the Money - exercise price and asset price are equal (S t = X) Market and Exercise Price Relationships

Bull Spread Using Calls X1X1 X2X2 Profit STST Positon: Long 1 call at X 1 Short 1 call at X 2

Straddle Combination Position: Long 1 call at X Long 1 put at X Profit STST X

Option Strategies involving stock Protective Put Long Stock Long Put Profit STST X ) All you are doing is buying calls

Option Strategies involving stock Covered Call Long Stock Short Call Profit STST X ) All you are doing is writing puts

Chapter 15: Pricing Options Put-Call Parity Consider the following portfolio: – Going LONG one European call option, – Going SHORT one European put option – LENDING the present value of the exercise price PV(X) = X/(1+r) T at the interest rate r. What are the payoffs of this portfolio? Notation: – c = price of call – p = price of put

Put-Call Parity ActionToday (cost) At maturity (cash flows: what you get) S T < XS T > XS T =X Buy a call Write a put Buy bond with face value = X Total cost c -p X/(1+r) T c – p + X/(1+r) T 0 S T - X -(X – S T ) 0 X X STST STST 0 0 X (S T ) STST

Put-Call Parity The future cash flows of this transaction are identical to the value of the stock S T. You can get the same cash flows by buying the stock. No arbitrage implies that the price of the stock equals the price of the portfolio: S 0 = c – p + PV(X) or S 0 + p = c + PV(X) (S 0 = current price of stock)

Example: Put-Call Parity Suppose you are a trader on the floor of the CBOE, you notice the following: – Price of IBM stock = 100 – Call(X=110,T=1) price = – Put(X=110,T=1) price = 7.45 – Risk-free rate is 5% – Note: T is time to maturity Is there an arbitrage opportunity? How would you take advantage of it?

Example: Put-Call Parity Step 1: Write down PCP equation S 0 + p = c + PV(X) Step 2: Plug in numbers on each side – S 0 + p = = – c + PV(X)= /(1.05)= Step 3: Go long the cheap side and short the expensive side – Long: stock and put – Short: call and bond with FV=X

Example: Put-Call Parity ActionToday (cost) At maturity (flows: what you get) S T < XS T > XS T =X short a call short PV(X) Long put Long Stock Net STST STST STST (S T -110) (110-S T )0 000

Binomial Option Pricing: Call Option on Dell The current price is S 0 = $60. After six months, the stock price will either grow to $66 or fall to $54. – Pick what ever probabilities you want. The annual risk-free interest rate is 1%. – Assume yield curve is flat What is the value of a call option with a strike price of $65 that expires after 6 months?

Binomial Option Pricing: Call Option on Dell Stock Price Tree Option Price Tree ? 1 0 Find value of a corresponding call option with X=65: H L

Binomial Option Pricing: Call Option on Dell Claim: we can use the stock along with a risk-free bond to replicate the option Replicating portfolio: – Position of  shares of the stock If  is positive, that means you “own” the stock If  is negative, that means you are “short” the stock – Position of $B in bonds (B=present value) If B is positive, that means you “own” the bond If B is negative, that means you are “short” the bond

Binomial Option Pricing: Call Option on Dell Strategy: If we know that holding  shares of stock and $B in bonds will replicate the payoffs of the option, then we know the cost of the option is  S 0 + B Example: Suppose the stock is currently $60, and we find that holding 1 share of stock and shorting $55 in bonds will give us the exact same payoffs as the option (in either state). Then we know the price of the option is ________ = 5

Binomial Option Pricing: Call Option on Dell We want to find  and B such that  and  54 are the payoffs from holding  shares of the stock B(1.01) 1/2 is the payoff from holding $B of the bond Mathematically possible – Two equations and two unknowns

Binomial Option Pricing: Call Option on Dell Shortcut to finding  : Subscripts: – H – the state in which the stock price is high – L – the state in which the stock price is low

Binomial Option Pricing: Call Option on Dell Once we know , it is easy to find B So if we – buy 1/12 shares of stock – Short $4.48 of the bond – Then we have a portfolio that replicates the option

Binomial Option Pricing: Call Option on Dell Do we know how to price the replicating portfolio? Yes: We know the price of the stock is $60 – 1/12 shares of the stock will cost $5 When we short $4.49 of the bond – we get $4.48 Total cost of replicating portfolio is – = 0.52 This is the price of the option. Done.

Binomial Option Pricing: Put Option On Dell Stock Price Tree Option Price Tree ? 0 11 Find value of a corresponding put option with X=65:

Binomial Option Pricing: Put Option on Dell We want to find  and B such that  and  54 are the payoffs from holding  shares of the stock B(1.01) 1/2 is the payoff from holding B shares of the bond Mathematically possible – Two equations and two unknowns

Binomial Option Pricing: Put Option on Dell Shortcut to finding  : Subscripts: – H – the state in which the stock price is high – L – the state in which the stock price is low

Binomial Option Pricing: Put Option on Dell Once we know , it is easy to find B So if we – short 11/12 shares of stock – buy $60.20 of the bond – Then we have a portfolio that replicates the option

Binomial Option Pricing: Put Option on Dell Do we know how to price the replicating portfolio? Yes: The price of the stock is $60 – When we short 11/12 shares of the stock we will get $55.00 To buy $60.20 of the bond – This will cost $60.20 Total cost of replicating portfolio is – = 5.20 This is the price of the option. Done.

Insights on Option Pricing The value of a derivative – Does not depend on the investor’s risk-preferences. – Does not depend on the investor’s assessments of the probability of low and high returns. – To value any derivative, just find a replicating portfolio. – The procedures outlined above apply to any derivative with any payoff function