1. Investment Analysis and Portfolio Management Lecture 10 Gareth Myles

2. 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 + Vp – Vc • At the expiration date P = S + max{E – S, 0} – max{S – E, 0}

3. 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 + Vp – Vc= (1/(1+r)t)E

4. Valuation of Options • At the expiration date Vc = max{S – E, 0} Vp = 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 Vc and Vp uncertain • An arbitrage argument can be applied to value the options

5. 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

6. 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

7. Valuation of Options • The value of the option in the up state is Vu =max{uS – E, 0} for a call = max{E – uS, 0} for a put • The value of the option in the down state is Vd =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 V0 • This information is summarized in a binomial tree diagram

8. Valuation of Options

9. 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 –D units of the underlying stock • The cost of the portfolio at time 0 is P0 = V0 – DS

10. Valuation of Options • At the expiration date the value of the portfolio is either Pu = Vu - DuS • or Pd = Vd - DdS • The key step is to choose D so that these are equal (the hedging step) • If D = (Vu – Vd)/S(u – d) then Pu = Pd = (uVd – dVu)/(u – d)

11. 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 Pu = Pd = RP0 • This gives R[V0 – DS] = (uVd – dVu)/S(u – d)

12. Valuation of Options • Solving gives • This formula applies to both calls and puts by choosing Vu and Vd • 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

13. 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

14. Valuation of Options • This gives the value • For a put option the end point values are Vu = max{50 – 60, 0} = 0 Vd = max{50 – 45, 0} = 5 • So the value of a put is

15. Valuation of Options • Observe that 40 + 4.058 – 0.58 = 43.478 • And that (1/1.15) 50 = 43.478 • So the values satisfy put-call parity S + Vp – Vc = (1/R)E

16. Valuation of Options • The pricing formula is • Notice that • So define

17. 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

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

19. 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

20. 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