Question

Consider a binomial pricing economy. There are two dates only, date 0 and 1. A European call written on the market portfolio expires on date 1; its exercise price is $100. The price of this call at date 0 is $8. The market portfolio has a price of $100 at date 0 and its future price at date 1 will be either $120 or $95, with probability 0.6 and 0.4 respectively.

a) Compute the expected returns of the call option and the market portfolio.

b) What is the risk-free rate implied by this call option and the market portfolio?

Answer 1

Given information:

S0 = $100 at date 0

X = $100

c= European call option premium = $8

The price at date 1 can be either $ 120 with probability 0.6 or $80 with probability 0.4

probability of upside = p = 0.6

probability of downside = 1-p = 0.4

We will calculate the option pay off at date 1 for both upside and downside. Then discount the option payoff to the date 0 to determine the risk free rate.

Sub part a:

At Date 1, expected price of the market portfolio = probability of upside * price at upside + probability of downside * price at downside = 0.6*120 + 0.4*95 = 110

Since the expected price of the market portfolio at date 1 is 110 which is higher than price at date 1, the call option will be exercised and the investor will have to deliver the market portfolio at $100

Total gain at date 2 = premium collected from European call option + capital gain from holding the market porfolio

= 8 + (100-100) = $8

Amount invested at date 0 = $100

Total gain at date 1 = $8

Hence return = Total gain at date 1 / Amount invested at date 0 = 8/100 = 8%

Sub part b:

Option pay off for a European call option is given by max(0, (St-X/(1+R_f)^t) ) where t is the time left to expiry, Rf is the risk free rate, St is the spot rate at time t and X is the exercise price.

At date 1, for upside, option payoff = max (0,(120-100)) since t= 0 the denominator is 1

= 20

discount this payoff to date 0 using interest rate Rf. Hence payoff = 20/(1+Rf)

At date 1, for downside, option payoff = max (0,(95-100)) since t= 0 the denominator is 1

= 0

The price of the option = probability of upside * option payoff at upside + probability of downside * option payoff at downside

= 0.6 * 20/(1+Rf) +0.4 * 0

The option price is given as $8

Hence, 8 = 0.6 * 20/(1+Rf)

(1+Rf) = 0.6 * 20/8= 1.5

Rf = 0.5