Option Valuation Using True Probabilities in the Binomial Model: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 27

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 27 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. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here.

In a non-risk-neutral world, where we use true probabilities instead of risk-neutral probabilities, if the expected return on a stock option is γ, then we can find γ by taking a weighted average of the return of the assets in a replicating portfolio for the option. Recall from Section 15 that a replicating portfolio on an option consists of Δ in shares of the underlying asset (here, a stock) and B in lending. The following formula enables us to compute γ.

e^γh = [SΔ/(SΔ + B)]e^αh + [B/(SΔ + B)]e^rh

In this case, the expected European call option payoff can be calculated in the binomial model as follows.

C = e^-γh[pC_u + (1-p)C_d], where p = (e^αh - d)/(u - d)

This calculation gives the same ultimate result as the calculation which involves risk-neutral probabilities. So for all practical purposes, using risk-neutral probabilities in the binomial model is just as realistic as attempting to account for true probabilities (unless you are investing anything in the real world, in which case you will lose money if you rely solely on these formulas!).

Meaning of Variables:

S = underlying asset (stock) price.

p* = (e^(r-∂)h - d)/(u - d) = risk-neutral probability of stock price increase.

p = true probability of stock price increase.

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

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

h = one time period in binomial model.