Question

a) Consider a one period binomial model with S(0) = 100, u = 1.2, d = 0.9, R = 0, pu = 0.6 and pd = 0.4. Determine the price at t = 0 of a European call option X = max{S(1) − 104, 0}.

b) If R > 0, motivate why the inequality (1 + R) > u would lead to arbitrage.

Answer 1

Part a)

Information gien:

Spot price - S0 - 100

Up step - u - 1.2

Down step - d - 0.9

Risk free rate - 0

Probability of up step - pu - 0.6

Probability of down step - du - 0.4

First, finding out the possible prices at time t = 1:

Upstep: 1.2*100 = 120$

Downstep: 0.9*100 = 90$

Given the possible prices, the payoff from the option:

Upstep: max(120-104,0) = 16$

Downstep: max(90-104,0) = 0$

Thus, price of the option at time t = 0 is:

= (pu*u + du*d) * exp(-R*t)

= (0.6*16 + 0.4*0) * exp(0*1)

= 9.6$

Part b)

If R is greaterr than 0 and the rate is greater than the upstep, this would indicate that the money would grow faster in a bank compared to in the stock price. In this case an arbitrage case is possible.

The arbitrage would be possible by selling the stock. The money recieved is put in the bank where it grows faster than the stock, thus at time t=1, it can be withdrawn and used to buy the stock at a price lower than the bank return. The difference is the profit from the arbitrage.