Multi-Period Binomial Option Pricing with Recombining Trees

This is Section 18 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.

A recombining binomial option price tree is one in which an up move in the stock price for one period followed by a down move in the stock price in the next period is identical in its result (S_ud) to a down move in the first period followed by an up move in the next (S_du).

In the recombining tree, there are three possible stock prices after two time periods:
S_uu = u²S

S_ud = S_du = udS

S_dd = d²S

We recall from Section 17 that the terms u and d can be found as follows:

u = e^{(r-∂)h + σ√(h)}

d = e^{(r-∂)h - σ√(h)}

Definitions of variables:

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.

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,

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

The way to approach multi-period binomial option pricing models is to figure out the stock and option prices in the latest period and work backward from there using any and all the formulas introduced in Sections 15 through 18.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, pp. 323-328.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem MPBOPWRT. Gregarious, Inc., stock is currently worth $56. Every year, it can change by a factor of 0.9 or 1.3. The stock pays no dividends, and the annual continuously-compounded risk-free interest rate is 0.04. Using a two-period binomial option pricing model, find the price today of one two-year European call option on Gregarious, Inc., stock with a strike price of $70.

Solution MPBOPWRT1.

In one year, the stock will either be worth S_u = 1.3*56 = 72.8, or it will be worth S_d = 0.9*56 = 50.4. In two years, the stock will either be worth

S_uu = 1.3²*56 = 94.64 or S_ud = S_du = 1.3*0.9*56 = 65.52 or S_dd = 0.9*0.9*56 = 45.36.

At S_uu = 94.64, the call is worth C_uu = 94.64 - 70 = 24.64

At S_du = 65.52, the call is worth C_du = 0

At S_dd = 45.36, the call is worth C_dd = 0

Using C_uu = 24.64 and C_ud = 0, we calculate the call option value C_u at the end of one year in the event of an up move in the stock price.

We recall our formula for the risk-neutral probability in the increase in the stock price over one time period:

p* = (e^(r-∂)h - d)/(u - d). Here, for every time period, p* = (e^0.04 - 0.9)/(1.3 - 0.9) = 0.3520269355.

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

Thus, in year 1, C_u = e^-0.04[0.3520269355*24.64 + (1 - 0.3520269355)0] =

C_u = 8.333833493

Using C_du = 0 and C_dd = 0, we calculate the call option value C_d at the end of one year in the event of an up move in the stock price. This is evidently C_d = 0.

Thus, we can calculate C = e^-rh[p*C_u + (1 - p*)C_d] =

e^-0.04[0.3520269355*8.333833493 + (1 - 0.3520269355)0] = C = $2.818700515

Problem MPBOPWRT2. 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 European call option on Gregarious, Inc., stock with a strike price of $80.

Solution MPBOPWRT2. 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

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].

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

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

Thus, C = e^-0.09*2[0.4522956832*66.21644859 + (1 - 0.4522956832)9.444727773] =

C = $29.3366377.

Problem MPBOPWRT3.

The annualized standard deviation of the continuously compounded stock return on Prominent Co. is 0.23. The annual continuously compounded rate of interest is 0.12, and the annual continuously compounded dividend yield on Prominent Co. is 0.07. The current price of Prominent Co. stock is $35 per share. Using a two-period binomial model, find the price of Prominent Co. stock if it moves up twice over the course of 7 years.

Solution MPBOPWRT3.

First, we find u = e^{(r-∂)h + σ√(h)} , where, if the two-period model is applied over 7 years, it follows that one period = h = 3.5 years. Also, σ = 0.23, r = 0.12, and ∂ = 0.07. Thus,

(r-∂)h = (0.12 - 0.07)*3.5 = 0.175. Hence, u = e^{0.175 + 0.23√(3.5)} = 1.831784447.

We want to find S_uu = u²S = 1.831784447²*35 = S_uu = 117.4401991.

Problem MPBOPWRT4.

The annualized standard deviation of the continuously compounded stock return on Prominent Co. is 0.23. The annual continuously compounded rate of interest is 0.12, and the annual continuously compounded dividend yield on Prominent Co. is 0.07. The current price of Prominent Co. stock is $35 per share. Using a two-period binomial model, find the price of Prominent Co. stock if it moves up once and then down once over the course of 7 years.

Solution MPBOPWRT4.

By design, this problem has the same initial conditions as Problem MPBOPWRT3. We already know that u = 1.831784447.

It remains to find d = e^{(r-∂)h - σ√(h)} = e^{0.175 - 0.23√(3.5)} = d = 0.7746913403

We then want to find S_ud = udS = 1.831784447*0.7746913403*35 = S_ud = 49.6673642

Problem MPBOPWRT5. The annualized standard deviation of the continuously compounded stock return on Prominent Co. is 0.23. The annual continuously compounded rate of interest is 0.12, and the annual continuously compounded dividend yield on Prominent Co. is 0.07. The current price of Prominent Co. stock is $35 per share. Find the price of one 7-year European call option on Prominent Co. stock with a strike price of $40.

Solution MPBOPWRT5.

By design, this problem has the same initial conditions as Problems MPBOPWRT3-4.

There, we determined that u = 1.831784447 and d = 0.7746913403.

Also, we know that

S_uu = 117.4401991, so C_uu = 77.4401991

S_ud = 49.6673642, so C_ud = 9.6673642

It is not hard to find S_dd = d*dS = 0.7746913403²*35 = 21.00513355, in which case C_dd = 0.

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

p* = (e^0.175 - 0.7746913403)/(1.831784447 - 0.7746913403) = p* = 0.3940569412.

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

C_u = e^-0.12*3.5[0.3940569412*77.4401991 + (1 - 0.3940569412)9.6673642] =

C_u = 23.89923719

C_d = e^-0.12*3.5[0.3940569412*9.6673642 + (1 - 0.3940569412)0] = C_d = 2.503014582

C = e^-0.12*3.5[0.3940569412*23.89923719 + (1 - 0.3940569412)2.503014582] =

C = $7.184376357

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