Question

Given a single-period binomial model with A(0) = 10, A(T) = 20, S(0) = 100 and S(T) = 210 with probability 0.5 and S(T) = 90 with probability 0.5. Assuming no arbitrage exists, find the price C(0) of a call option with strike price X = 150. Please show work

Answer 1

A(0) = 10 and A(T) = 20 where A(0) represents the value of the initial investment and A(T) respresents the value of the investment at time T. The ratio of these two values lead to a function known as the accumulation function expressed in terms of time T. The accumulating function for compound interest is (1+r)^(T), for simple interest is (1+r x T) and so on and so forth. In other words the accumulating function is the function thata determines the accmulation of interest on an initial investment say A(0) and leads to final investment value A(T).

In this context:

Accumulation Function = A(T) / A(0) = 20 / 10 = 2

Assuming continuous compounding we get:

2 = EXP [ r x T] where r is the continuously compounded interest rate and T is offourse the duration of our one-step binomila model (call option maturity period).

Current Asset Price = S(0) = $ 100 and Option Strike Price = X = $ 150

State 1: Price becomes S(T) = $ 210 with probability 0.5 and option payoff P1 = S(T) - X = 210 - 150 = $ 60

State 2: Price becomes S(T) = $ 90 with probability 0.5 and option payoff P2 = S(T) - X < 0 hence $ 0

Expected Payoff of Call Option = 0.5 x 60 + 0.5 x 0 = $ 30

Call Option Price at t = 0 will be = Expected Payoff of Call Option / EXP (r x T) (Present Value of Expected Payoff at t=0)

Call Option Price = 30 / EXP[r x T] = 30 / 2 = $ 15 where EXP[r x T] = 2 from the earlier definition of the accumulation function.