Investment Analysis and Portfolio Management Lecture 10 Gareth Myles

Put-Call Parity The prices of puts and calls are related Consider the following portfolio Hold one unit of the underlying asset Hold one put option Sell one call option The value of the portfolio is P = S + V p – V c At the expiration date P = S + max{E – S, 0} – max{S – E, 0 }

Put-Call Parity If S < E at expiration the put is exercised so P = S + E – S = E If S > E at expiration the call is exercised so P = S – S + E = E Hence for all S P = E This makes the portfolio riskfree so S + V p – V c = (1/(1+r) t )E

Valuation of Options At the expiration date V c = max{S – E, 0} V p = max{E – S, 0} The problem is to place a value on the options before expiration What is not known is the value of the underlying at the expiration date This makes the value of V c and V p uncertain An arbitrage argument can be applied to value the options

Valuation of Options The unknown value of S at expiration is replaced by a probability distribution for S This is (ultimately) derived from observed data A simple process is assumed here to show how the method works Assume there is a single time period until expiration of the option The binomial model assumes the price of the underlying asset must have one of two values at expiration

Valuation of Options Let the initial price of the underlying asset be S The binomial assumption is that the price on the expiration date is uS with probability p “up state” dS with probability 1- p “down state” These satisfy u > d Assume there is a riskfree asset with gross return R = 1+ r It must be that u > R > d

Valuation of Options The value of the option in the up state is V u = max{uS – E, 0} for a call = max{E – uS, 0} for a put The value of the option in the down state is V d = max{dS – E, 0} for a call = max{E – dS, 0} for a put Denote the initial value of the option (to be determined) by V 0 This information is summarized in a binomial tree diagram

Valuation of Options

There are three assets Underlying asset Option Riskfree asset The returns on these assets have to related to prevent arbitrage Consider a portfolio of one option and –  units of the underlying stock The cost of the portfolio at time 0 is P 0 = V 0 –  S

Valuation of Options At the expiration date the value of the portfolio is either P u = V u -  uS or P d = V d -  dS The key step is to choose  so that these are equal (the hedging step) If  = (V u – V d )/S(u – d) then P u = P d = (uV d – dV u )/(u – d)

Valuation of Options Now apply the arbitrage argument The portfolio has the same value whether the up state or down state is realised It is therefore risk-free so must pay the risk- free return Hence P u = P d = RP 0 This gives R[V 0 –  S] = (uV d – dV u )/S(u – d)

Valuation of Options Solving gives This formula applies to both calls and puts by choosing V u and V d These are the boundary values The result provides the equilibrium price for the option which ensures no arbitrage If the price were to deviate from this then risk- free excess returns could be earned

Valuation of Options Consider a call with E = 50 written on a stock with S = 40 Let u = 1.5, d = 1.125, and R = 1.15

Valuation of Options This gives the value For a put option the end point values are V u = max{50 – 60, 0} = 0 V d = max{50 – 45, 0} = 5 So the value of a put is

Valuation of Options Observe that – 0.58 = And that (1/1.15) 50 = So the values satisfy put-call parity S + V p – V c = (1/R)E

Valuation of Options The pricing formula is Notice that So define

Valuation of Options The pricing formula can then be written The terms q and 1 – q are known as risk neutral probabilities They provide probabilities that reflect the risk of the option Calculating the expected payoff using these probabilities allows discounting at the risk-free rate

Valuation of Options The use of risk neutral probabilities allows the method to be generalized

Valuation of Options u and d are defined as the changes of a single interval R is defined as the gross return on the risk-free asset over a single interval For a binomial tree with two intervals the value of an option is

Valuation of Options With three intervals Increasing the number of intervals raises the number of possible final prices The parameters p, u, d can be chosen to match observed mean and variance of the asset price Increasing the number of periods without limit gives the Black-Scholes model