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

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.

Problem RNPBOP1. At the end of 1 year, the stock price of Digital Co. can change by either a factor of 1.5 or by a factor of 0.7. The annual continuously-compounded risk-free interest rate is 0.12, and the stock's annual continuously-compounded dividend yield is 0.09. Using the one-period binomial option pricing model, calculate the risk-neutral probability of the increase in Digital Co.'s stock price.

Solution RNPBOP1. We use the formula p* = (e^(r-∂)h - d)/(u - d), where h = 1, u = 1.5, d = 0.7, r = 0.12, and ∂ = 0.09. Thus, p* = (e^(0.12-0.09) - 0.7)/(1.5 - 0.7) = p* = 0.4130681674.

Problem RNPBOP2. Atypical, Inc., pays dividends on its stock at an annual continuously-compounded yield of 0.05. In two months, its stock price could either be twice what it is now or half what it is now. The risk-neutral probability of the increase in Atypical, Inc.'s stock price is 0.56. Using the one-period binomial option pricing model, what is the annual continuously-compounded risk-free interest rate?

Solution RNPBOP2. We use the formula p* = (e^(r-∂)h - d)/(u - d) and rearrange it thus:
p*(u - d) + d = e^(r-∂)h. Here, h = 1/6, u = 2, d = 0.5, ∂ = 0.05, and p* = 0.56. So

e^(r-0.05)/6 = 0.56(2-0.5) + 0.5 = 1.34

ln(1.34) = 0.292669614 = (r - 0.05)/6

r = 6*0.292669614 + 0.05 = r = 1.806017684 (No, this is not a typo. This just means that the annual continuously compounded risk-free interest rate is about 180.6%. Atypical, I know.).

Problem RNPBOP3. The stock of Reputable LLC will sell for either $130 or $124 one year from now. The annual continuously compounded interest rate is 0.11. The risk-neutral probability of an increase in the stock price (to $130) is 0.77. Using the one-period binomial option pricing model, find the current price of a call option on Reputable LLC stock with a strike price of $122.

Solution RNPBOP3. We use the formula C = e^-rh[(p*)C_u + (1 - p*)C_d]. In one year, if the stock is at $130, the call will be worth C_u = 130 - 122 = $8. If the stock is at $124, the call will be worth C_d = 124 - 122 = $2. p* = 0.77, h = 1, r = 0.11. Thus,

C = e^-0.11[0.77*8 + (1 - 0.77)2] = C = $5.930421976.

Problem RNPBOP4. The probability that the stock of Respectable Co. will be $555 one year from now is 0.6. The probability that the stock of Respectable Co. will be $521 one year from now is 0.4. Using the one-period binomial option pricing model, what is the price today of a one-year forward contract on Respectable Co. stock?

Solution RNPBOP4. We use the formula (p*)uS + (1 - p*)dS = F_{t, t+h}, where uS = 555, dS = 521, and p* = 0.6. Thus, 0.6*555 + 0.4*521 = F_{t, t+1} = 541.4

Problem RNPBOP5. The stock of Discrete Co. will either be $43 - with probability 0.2 - or $49 - with probability 0.8 - in two years. The annual continuously-compounded interest rate is 0.05, and the annual continuously-compounded dividend yield is 0.02. Using the one-period binomial option pricing model, find the expected price of Discrete Co. stock today.

Solution RNPBOP5. Here, h = 2, r = 0.05 and ∂ = 0.02 (so r - ∂ = 0.03), p* = 0.8, uS = 49, and dS = 43. We use the formula e^(r-∂)hS = (p*)uS + (1 - p*)dS

Thus, e^0.03*2S = 0.8*49 + 0.2*43 = 47.8

S = 47.8e^-0.06 = S = $45.01634471

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