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

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.

r = annual continuously-compounded risk-free interest rate

α = the annual continuously compounded expected return on the stock.

C = price of the call option.

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

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

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 11, pp. 347-348.

Problem OVUTPBM1. You know that the replicating portfolio for a call option on Dependable Co. stock consists of (2/3) shares and borrowing $45. You also know that the annual continuously compounded expected return on the stock is 0.32 and the annual continuously-compounded risk-free interest rate is 0.03. The stock currently sells for $450. Find the expected return on the call option over the course of one year using a one-period binomial model.

Solution OVUTPBM1. We use the formula e^γh = [SΔ/(SΔ + B)]e^αh + [B/(SΔ + B)]e^rh, and we want to find γ. Here, h = 1, S = 450, r = 0.03, α = 0.32, B = -45 (money is borrowed), and Δ = 2/3.

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

(450(2/3)/[450(2/3)-45])e^0.32 - 45/[450(2/3)-45]e^0.03 = 1.438305393 = e^γ. Thus, γ = ln(1.438305393) = γ = 0.3634656103 (nice return for one year!)

Problem OVUTPBM2. The stock of Dependable Co. currently sells for $450. You also know that the annual continuously compounded expected return on the stock is 0.32 and the annual continuously-compounded risk-free interest rate is 0.03. You know that in three years, the stock price will change either by a factor of 0.34 or by a factor of 3.

The annual continuously-compounded expected return on a particular call option is 0.3634656103. This option has a strike price of $465 and a time to expiration of three years. Find the price today of this option using a one-period binomial model.

Solution OVUTPBM2. First we find p = (e^αh - d)/(u - d), where α = 0.32, h = 3, d = 0.34, and u = 3. Thus, p = (e^0.32*3 - 0.34)/(3 - 3.4) = p = 0.85402124306.

If the stock price triples in 3 years, uS = 1350 and so C_u = 1350 - 465 = 885.

If the stock price declines in 3 years, the call option will be worth C_d = 0.

Also, we are given that γ = 0.3634656103.

Thus, C = e^-γh[pC_u + (1-p)C_d] = e^{-0.3634656103*3}[0.85402124306*885 + 0] =

C = $254.0845601

Problem OVUTPBM3. A call option on Artificial LLC currently sells for $32. In two years, it will sell for $35 with a real probability of 0.93 and for $13 with a real probability of 0.07. Using a one-period binomial model, find the annual continuously-compounded expected return on this option.

Solution OVUTPBM3. We use the formula C = e^-γh[pC_u + (1-p)C_d], where

e^-γh = C/[pC_u + (1-p)C_d] and C = 32, C_u = 35, C_d = 13, h = 2, and p = 0.93. Thus,

e^-2γ = 32/[0.93*35 + 0.07*13] = 0.9563658099. Thus, γ = -ln(0.9563658099)/2 =

γ = 0.0223073964

Problem OVUTPBM4. A call option on Content-of-Character Co. currently sells for $45. In three years, it will sell for $67 or for $32. The annual continuously-compounded expected return on this option is 0.12. Using a one-period binomial model, find the probability that the option will sell for $67 in three years.

Solution OVUTPBM4. We use the formula C = e^-γh[pC_u + (1-p)C_d], rearranging it to solve for p: Ce^γh = pC_u -pC_d + C_d

Ce^γh - C_d = p(C_u -C_d)

p = (Ce^γh - C_d)/(C_u -C_d)

Here, γ = 0.12, C = 45, C_u = 67, C_d = 32, h = 3, so p = (45e^0.12*3 - 32)/(67-32) =

p = 0.9285663901

Problem OVUTPBM5. By performing some financial analysis of Elusive Co. stock, Maximus concludes that for a particular call option, e^γh*(SΔ + B) = 0.05 for h = 3. Maximus also knows that the annual continuously-compounded risk-free interest rate is 0.06 and the annual continuously compounded expected return on the stock is 0.12. The delta of the replicating portfolio for this option is (3/4), and the stock is currently worth $990. How much in lending does a replicating portfolio for this call option contain?

Solution OVUTPBM5. We use the formula e^γh = [SΔ/(SΔ + B)]e^αh + [B/(SΔ + B)]e^rh , noting that e^γh*(SΔ + B) = SΔe^αh + Be^rh = 0.05. We are also given Δ = 0.75, r = 0.06, h = 3, S = 990, and α = 0.12. We want to find B = [0.05 - SΔe^αh]e^-rh =

[0.05 - 990(3/4)e^0.12*3]e^-0.06*3 = B = -888.8921286. (This means that the replicating portfolio requires one to borrow $888.8921286.)

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.