**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.**

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.

eγh = [SΔ/(SΔ + B)]eαh + [B/(SΔ + B)]erh

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

C = e-γh[pCu + (1-p)Cd], where p = (eαh – 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.

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

Cd = 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 eγh = [SΔ/(SΔ + B)]eαh + [B/(SΔ + B)]erh, 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 eγh = [SΔ/(SΔ + B)]eαh + [B/(SΔ + B)]erh =

(450(2/3)/[450(2/3)-45])e0.32 – 45/[450(2/3)-45]e0.03 = 1.438305393 = eγ. 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 = (eαh – d)/(u – d), where α = 0.32, h = 3, d = 0.34, and u = 3. Thus, p = (e0.32*3 – 0.34)/(3 – 3.4) = p = 0.85402124306.

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

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

Also, we are given that γ = 0.3634656103.

Thus, C = e-γh[pCu + (1-p)Cd] = 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[pCu + (1-p)Cd], where

e-γh = C/[pCu + (1-p)Cd] and C = 32, Cu = 35, Cd = 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[pCu + (1-p)Cd], rearranging it to solve for p: Ceγh = pCu -pCd + Cd

Ceγh – Cd = p(Cu -Cd)

p = (Ceγh – Cd)/(Cu -Cd)

Here, γ = 0.12, C = 45, Cu = 67, Cd = 32, h = 3, so p = (45e0.12*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, eγh*(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 eγh = [SΔ/(SΔ + B)]eαh + [B/(SΔ + B)]erh , noting that eγh*(SΔ + B) = SΔeαh + Berh = 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 – SΔeαh]e-rh =

[0.05 – 990(3/4)e0.12*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.**