Risk-Neutral Probability in Binomial Option Pricing: Practice Problems and Solutions

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 16 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here.

Here, we develop the one-period binomial option pricing model introduced in Section 15.

The risk-neutral probability of an increase in the stock price from S to uS in the next time period is p*, which can be expressed as follows:

p* = (e^(r-∂)h - d)/(u - d)

Then the price of a call option on the stock today using the one-period binomial option pricing model is

C = e^-rh[p*C_u + (1 - p*)C_d]

Furthermore, the expected undiscounted price of the stock today is

e^(r-∂)hS = (p*)uS + (1 - p*)dS = F_{t, t+h}

So the one-period binomial model can be used to determine the price of the forward contract on shares of the stock in question. p* can be thought of as the probability that the expected stock price is the forward price.

Meaning of variables:

C = current call option price.

C_u = the call option price if the stock price increases.

C_d = the call option price if the stock price decreases.

S = current stock price.

u = 1 + rate of capital gain on stock if stock price increases.

d = 1 + rate of capital loss on stock if stock price decreases.

r = annual continuously-compounded risk-free interest rate.

∂ = annual continuously-compounded dividend yield.

F_{t, t+h} = price of forward contract made at time t and expiring at time t + h.

h = one time period in the binomial model.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, pp. 320-321.