Put-Call Parity l The prices of puts and calls are related l Consider the following portfolio l Hold one unit of the underlying asset l Hold one put option l Sell one call option l The value of the portfolio is P = S + p c l At the expiration date P = S + max{e S, 0} max{s E, 0} Put-Call Parity l If S < E at expiration the put is exercised so P = S + E S = E l If S > E at expiration the call is exercised so l Hence for all S P = S S + E = E P = E l This makes the portfolio riskfree so S + p c = (1/(1+r) t )E 1
l At the expiration date c = max{s E, 0} p = max{e S, 0} l The problem is to place a value on the options before expiration l What is not known is the value of the underlying at the expiration date l This makes the value of c and p uncertain l An arbitrage argument can be applied to value the options l The unknown value of S at expiration is replaced by a probability distribution for S l This is (ultimately) derived from observed data l A simple process is assumed here to show how the method works l Assume there is a single time period until expiration of the option l The binomial model assumes the price of the underlying asset must have one of two values at expiration 2
l Let the initial price of the underlying asset be S l The binomial assumption is that the price on the expiration date is l us with probability p l ds with probability 1- p l These satisfy u > d up state down state l Assume there is a riskfree asset with gross return R = 1+ r l It must be that u > R > d l The value of the option in the up state is u = max{us E, 0} for a call = max{e us, 0} for a put l The value of the option in the down state is d = max{ds E, 0} for a call = max{e ds, 0} for a put l Denote the initial value of the option (to be determined) by 0 l This information is summarized in a binomial tree diagram 3
Probability p Stock Price us Option alue u Stock Price S Option alue 0 Risk-free (gross) return R Probability 1 - p Stock Price ds Option alue d l There are three assets l Underlying asset l Option l Riskfree asset l The returns on these assets have to related to prevent arbitrage l Consider a portfolio of one option and Δ units of the underlying stock l The cost of the portfolio at time 0 is P 0 = 0 ΔS 4
l At the expiration date the value of the portfolio is either P u = u - ΔuS l or P d = d - ΔdS l The key step is to choose Δ so that these are equal (the hedging step) l If Δ = ( u d )/S(u d) then P u = P d = (u d d u )/(u d) l Now apply the arbitrage argument l The portfolio has the same value whether the up state or down state is realised l It is therefore risk-free so must pay the riskfree return l Hence P u = P d = RP 0 l This gives R[ 0 ΔS] = (u d d u )/S(u d) 5
l Solving gives 1 R d u R 0 = u + d R u d u d l This formula applies to both calls and puts by choosing u and d l These are the boundary values l The result provides the equilibrium price for the option which ensures no arbitrage l If the price were to deviate from this then riskfree excess returns could be earned l Consider a call with E = 50 written on a stock with S = 40 l Let u = 1.5, d = 1.125, and R = 1.15 Probability p Stock Price us = 60 u = max{60 50, 0} = 10 Stock Price S Option alue 0 Risk-free (gross) return R = 1.15 Probability 1 - p Stock Price ds = 45 d = max{45 50, 0} = 0 6
l This gives the value 0 = 1 1.15 0.025 10 + 0.375 0.35 0 0.375 = 0.58 l For a put option the end point values are u = max{50 60, 0} = 0 d = max{50 45, 0} = 5 l So the value of a put is 1 0.025 0.35 0 = 0 5 = 4.058 1.15 + 0.375 0.375 l Observe that 40 + 4.058 0.58 = 43.478 l And that (1/1.15) 50 = 43.478 l So the values satisfy put-call parity S + p c = (1/R)E 7
l The pricing formula is 1 R d R u d l Notice that R d u R > 0, > 0 u d u d l So define q = R d u d u R + u d 0 = u d R d u d u + u R d =1 l The pricing formula can then be written 1 0 = [ qu + [1 q] d ] R l The terms q and 1 q are known as risk neutral probabilities l They provide probabilities that reflect the risk of the option l Calculating the expected payoff using these probabilities allows discounting at the risk-free rate 8
l The use of risk neutral probabilities allows the method to be generalized q uu = max{uus E, 0} for a call q u 1 q = max{e uus, 0} for a put 0 1 q q ud = du = max{uds E, 0} for a call = max{e uds, 0} for a put d 1 q dd = max{dds E, 0} for a call = max{e dds, 0} for a put l u and d are defined as the changes of a single interval l R is defined as the gross return on the risk-free asset over a single interval l For a binomial tree with two intervals the value of an option is 1 2 2 [ q + 2q(1 q) + (1 q ] 0 = 2 uu ud ) R dd 9
l With three intervals 1 3 2 2 3 [ q + 3q (1 q) + 3q(1 q) + (1 q ] 0 = 3 uuu uud udd ) R l Increasing the number of intervals raises the number of possible final prices l The parameters p, u, d can be chosen to match observed mean and variance of the asset price l Increasing the number of periods without limit gives the Black-Scholes model ddd 10