Binomial Option Pricing with American Options: Practice Problems and Solutions

This is Section 20 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.

With American options, it is possible for the option to be exercised early. Thus, to determine the option's price at any given "node" in a binomial tree, it is necessary to compare its value if it is held to expiration to the gain that could be realized upon immediate exercise. The higher of these is the American option price.

So for an American put,

P(S, K, t) = max(K - S, e^-rh[P(uS, K, t+h)p* + P(dS, K, t+h)(1-p*)]), where

p* = (e^(r-∂)h - d)/(u - d)

For an American call,

C(S, K, t) = max(S - K, e^-rh[C(uS, K, t+h)p* + C(dS, K, t+h)(1-p*)]),

Definitions of variables:

r = annual continuously-compounded risk-free interest rate.

∂ = annual continuously-compounded dividend yield.

h = one time period in the binomial model.

t = the time equivalent to some "node" in the binomial model.

S= stock price at time t

K = option strike 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,

σ = the annualized standard deviation of the continuously compounded stock return.

P(S, K, t) = price of an American put with strike price K and underlying stock price S.

C(S, K, t) = price of an American call with strike price K and underlying stock price S.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, p. 329.

Problem BOPWAO1. The stock of Predictable Co. is currently worth $100 per share. In one year, this price can either be $120 or $90. Predictable Co. stock does not pay dividends. The annual continuously compounded risk-free interest rate is 5%. The strike price of a one-year American put option on Predictable Co. stock is $130. Using, the one-period binomial option pricing model, find the price today of one such Americanput option on Predictable Co. stock.

Solution BOPWAO1. To find the American put price, we will take the maximum of (K - S) and the otherwise equivalent European put price using the binomial option pricing model.

We note that K - S = 130 - 100 = 30

Now we calculate the otherwise equivalent European put price:

First, we consider the put option price tree

P - - - P_u

P - - - P_d

In one year, if the stock is worth $120, the put option will be worth P_u = 130 - 120 = 10.

If the stock is worth $90, the put option will be worth P_d = 130 - 90 = 40.

We are given ∂ = 0, r = 0.05, S = 100, h = 1, u = 1.2, and d = 0.9.

We can still use the same formula for the risk-neutral probability of the stock price's increase next year:

p* = (e^(r-∂)h - d)/(u - d) = (e^0.05 - 0.9)/(1.2 - 0.9) = p* = 0.5042369879

We also note that

P = e^-rh[p*P_u + (1 - p*)P_d] = e^-0.05[0.5042369879*10 + (1 - 0.5042369879)40] =

P = $23.65982519

We note that 23.65982519 < 30, so our American put option price is $30.

Problem BOPWAO2. 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 the stock price being $130 in one year is 0.77. What is the current stock price for which the one-year American put option on Reputable LLC stock with a strike price of $160 will have the same value whether calculated by means of the binomial option pricing model or by taking the difference between the stock price and the strike price?

Solution BOPWAO2.

We are essentially looking for the stock price S where

P = K - S = e^-rh[P(uS, K, t+h)p* + P(dS, K, t+h)(1-p*)])

First, we find P:

We note that P_u = 160 - 130 = P_u = 30 and

P_d = 160 - 124 = P_d = 36.

Here, p* = 0.77, h = 1, and r = 0.11.

So P = e^-rh[p*P_u + (1 - p*)P_d] = e^-0.11[0.77*30 + (1 - 0.77)36] = P = $28.11127517

We know that K = 160, so

S = K - P = 160 - 28.11127517 = S = $131.8887248.

Problem BOPWAO3. Complicated, Inc., pays dividends on its stock at an annual continuously compounded yield of 0.06. The annual effective interest rate is 0.09. Complicated, Inc., stock is currently worth $100. Every two years, it can change by a factor of 0.7 or 1.5. Using a two-period binomial option pricing model, find the price two years from today of one four-year American call option on Gregarious, Inc., stock with a strike price of $80 in the event that the stock price increases two years from today.

Solution BOPWAO3. We are essentially asked to find C_u and compare it to S - K if the stock price increases next year. If the stock price increases next year, the stock will be worth $150, so S - K = 150 - 80 = 70

Now we find C_u:

We are given that r = 0.09, ∂ = 0.06, and h = 2. Thus,

(r-∂)h = (0.09 - 0.06)*2 = 0.06.

We find

S_uu = 1.5²*100 = 225, which implies that C_uu = 145

S_ud = S_du = 1.5*0.7*100 = 105, which implies that C_ud = 25

p* = (e^(r-∂)h - d)/(u - d). Here, for every time period, p* = (e^0.06 - 0.7)/(1.5 - 0.7) = p* = 0.4522956832.

We now use the formula C = e^-rh[p*C_u + (1 - p*)C_d].

Thus, C_u = e^-0.09*2[0.4522956832*145 + (1 - 0.4522956832)25] = C_u = 66.21644859.

We note that 66.21644859 < 70, so the American call price next year if the stock price goes up would be $70.

Problem BOPWAO4. Complicated, Inc., pays dividends on its stock at an annual continuously compounded yield of 0.06. The annual effective interest rate is 0.09. Complicated, Inc., stock is currently worth $100. Every two years, it can change by a factor of 0.7 or 1.5. Using a two-period binomial option pricing model, find the price two years from today of one four-year American call option on Gregarious, Inc., stock with a strike price of $80 in the event that the stock price decreases two years from today.

Solution BOPWAO4. We are essentially asked to find C_d and compare it to S - K if the stock price decreases next year. If the stock price decreases next year, the stock will be worth $70, so S - K = 70 - 80 = -10

We are given that r = 0.09, ∂ = 0.06, and h = 2. Thus,

(r-∂)h = (0.09 - 0.06)*2 = 0.06.

We find

S_ud = S_du = 1.5*0.7*100 = 105, which implies that C_ud = 25

S_dd = 0.7²*100 = 49, which implies that C_dd = 0.

p* = (e^(r-∂)h - d)/(u - d). Here, for every time period, p* = (e^0.06 - 0.7)/(1.5 - 0.7) = p* = 0.4522956832.

We now use the formula C = e^-rh[p*C_u + (1 - p*)C_d].

C_d = e^-0.09*2[0.4522956832*25 + (1 - 0.4522956832)0] = C_d = 9.444727773

Since 9.444727773 > -10, the American call price next year if the stock price decreases would be $9.444727773.

Problem BOPWAO5. Complicated, Inc., pays dividends on its stock at an annual continuously compounded yield of 0.06. The annual effective interest rate is 0.09. Complicated, Inc., stock is currently worth $100. Every two years, it can change by a factor of 0.7 or 1.5. Using a two-period binomial option pricing model, find the price today of one four-year American call option on Gregarious, Inc., stock with a strike price of $80.

Solution BOPWAO5. We use the formula

C(S, K, t) = max(S - K, e^-rh[C(uS, K, t+h)p* + C(dS, K, t+h)(1-p*)]).

We note that today, S - K = 100 - 80 = 20

From Solution BOPWAO3, we get C(uS, K, t+h) = 70.

From Solution BOPWAO4, we get C(uS, K, t+h) = 9.444727773

From both solutions, we get p* = 0.4522956832.

Thus, e^-rh[C(uS, K, t+h)p* + C(dS, K, t+h)(1-p*)] =

e^-0.09*2[0.4522956832*70 + (1-0.4522956832)* 9.444727773] = 30.766602222

Since 30.766602222 > 20, the American call price today is $30.766602222.

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