Comparing Risk-Neutral and Real Probabilities in the Binomial Model: Revised Practice Problems and Solutions

In a risk-neutral situation, the risk-neutral probability of the stock price increasing during a period of the binomial model must satisfy this equation:

(p*)uSe^δh + (1-p*)dSe^δh = Se^rh

With real probabilities, for a stock that does not pay dividends,

puS + (1-p)dS = Se^αh

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.

∂ = annual continuously-compounded dividend yield.

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

Thus, the formula is as follows for p:

p = (e^αh - d)/(u - d)

There is a constraint that enables the value of p to always be between 0 and 1:

d < e^αh < u

Problem CRNRPBM1. The price of Deleterious Co. stock, which does not pay dividends, will change by a factor of either 1.5 or 0.4 in 2 years. Find the upper constraint on the annual continuously compounded expected return on the stock.

Solution CRNRPBM1. We use the inequality d < e^αh < u, where we want to find the upper bound on α, given that e^αh < u. This upper bound X occurs when e^Xh = u or

X = ln(u)/h. Here, u = 1.5 and h = 2, so X = ln(1.5)/2 = X = 0.2027325541

Problem CRNRPBM2. The price of Deleterious Co. stock, which does not pay dividends, will change by a factor of either 1.5 or 0.4 in 2 years. The annual continuously compounded expected return on the stock is 0.05. Find the real probability that the stock will increase in price in two years.

Solution CRNRPBM2. We use the formula p = (e^αh - d)/(u - d), where u = 1.5, d = 0.4, h = 2, and α = 0.05. Thus, p = (e^0.05*2 - 0.4)/(1.5 - 0.4) = p = 0.641064471.

Problem CRNRPBM3. Imperious LLC stock will change by a factor of either 1.9 or 0.2 in h years. The stock pays dividends on a continuously compounded basis with annual yield of 0.06. The annual continuously compounded rate of interest is 0.09. The risk-neutral probability of the stock price's increase in h years is 0.72. Find h.

Solution CRNRPBM3. To find h, we will need to use the formula

p* = (e^(r-∂)h - d)/(u - d)

We can simplify p* using the values given for u = 1.9, d = 0.2, r = 0.09, and δ = 0.06, so (r - δ) = 0.03. Thus,

p* = 0.72 = (e^0.03h - 0.2)/1.7. Thus, e^0.03h = 1.7*0.72 + 0.2 = 1.424 and h =

ln(1.424)/0.03 = h = 11.7823271 years.

Problem CRNRPBM4. You live in a risk-neutral world, and just purchased a share of Lucrative Co. stock at a price of $1000. You do not know the annual continuously compounded risk-free interest rate (who does, in reality?), but you do know that the stock pays dividends with an annual continuously compounded yield of 0.12. You also plan to hold the stock for 13 years, at the end of which it will be either 3.4 or 0.1 of its present price. The risk-neutral probability of a stock price increase is 0.82. You could have invested the $1000 for 13 years in an account earning the annual continuously compounded risk-free interest rate. What would your account balance be at the end of 13 years if you did so?

Solution CRNRPBM4. We use the formula (p*)uSe^δh + (1-p*)dSe^δh = Se^rh, where we desire to find Se^rh. We are given that p* = 0.82, u = 3.4, d = 0.1, h = 13, ∂ = 0.12, and S = 100. Thus, Se^rh = 0.82*3.4*1000e^0.12*13 + 0.18*0.1*1000e^0.12*13 = Se^rh = $13353.25241.

Problem CRNRPBM5. At t =13 years, Lucrative Co. stopped paying dividends on its stock, and everybody suddenly became risk-averse. Lucrative Co. also changed its name to Not-So-Lucrative Co. In another 10 years, its stock price will change by a factor of 1.3 or by a factor of 0.3. You take $13353.25241and you invest it in Not-So-Lucrative Co. stock, knowing that the real probability of the stock price increasing over that time period is 0.76. You could have invested the $13353.25241 for 10 years in an account earning a return equal to the annual continuously compounded expected return on the stock, but you do not know what that return is. What would your account balance be at the end of 10 years if you did so?

Solution CRNRPBM5. We use the formula puS + (1-p)dS = Se^αh, where we desire to find Se^αh. We are given that p = 0.76, u = 1.3, d = 0.3, S = $13353.25241 (for all practical purposes, we can consider this the price of one share). Thus, Se^αh = 13353.25241(0.76*1.3 + 0.24*0.3) = Se^αh = $14154.44756.

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