Question

Consider a world in which some stock, S, can either go up by 25% or down by 20% in one year and no other outcomes are possible. The continuously compounded risk-free interest, r, is 5.5% and the current price of the stock, S0, is $100. (a) What are the possible stock values in one year’s time, ST? (b) What are the possible payoffs of a European call option written on stock S with a strike price, X, of $100 and time-to-expiration of 1-year, T = 1? (c) Suppose you want to form a portfolio, P, consisting of short on one call option and long on some number, ?, of the stock, such that the portfolio value in one year’s time, PT , does not depend on the value of the stock, ST. What would be the appropriate value of ?? (d) What would be the (certain) portfolio value in one year’s time, PT? (e) What is the arbitrage-free value of the portfolio today, P0? (f) What is the premium of the call option today, c0, if there is no arbitrage opportunity? (g) Define p = (e rT – d)/( u ? d), and call this the risk-neutral probability that the stock price increases. What is the value of p? (h) What is the expected value of the stock in one year’s time, E(ST ), under the risk-neutral probabilities? (i) At what continuous rate would the stock price have to grow to end up at the expected value? (j) What would be the expected value of the call option in one year’s time, E(cT ), under the riskneutral probabilities? (k) At what continuous rate would the call price have to grow to end up at the expected value?

Answer 1

A) Possible stock value in a year

Upper node = (1+25%)*100 = $125

Lower Node = (1-20%)*100 = $ 80

b) Payoff after a year with strike price X=100

Payoff = Max(Stock-strike,0)

Payoff at upper node = Max(125-100,0) = 25

Payoff at Lower node = Max(80-100,0) = 0

C). Delta = Change in call payoff/Change in stock price

Delta = (25-0)/(125-80) = 25/45 = 0.555

If one call is shorted, then stock that are need to be bought = Delta = 0.555

Call Premium = (0.5*25+0.5*0)*exp(-0.055*1) = $ 11.83

D. AFter one year

Share value can be 125 or 80

So upper case, portfolio value = 125*0.555-Max(125-100,0)+11.83 = 56.21

Lower case, portfolio value = 80*0.555-Max(80-100,0)+11.83 = 56.23