Question

Suppose you are attempting to value a one-year maturity option on a stock with volatility (i.e., annualized standard deviation) ofσ= .40. What would be the appropriate values for u and d if your binomial model is set up using the following?

a. 1 period of one year

b. 4 sub-periods, each 3 months

c. 12 sub-periods, each 1 month

Short answer

a. u = 1.4918; d = 0.670

b. u = 1.2214; d = 0.8187

c. u = 1.1224; d = 0.8909

Step-by-step solution

Given information

σ= .40

u = exp(σ√Δt)

d = exp( - σ√Δt)

Calculation of 1 period of one year

u = exp(σ√Δt) = exp(0.40√1) = 1.4918

d = exp( - σ√Δt) = exp(- 0.40√1) = 0.670

Calculation of 4 sub-periods each of 3 months

u = exp(σ√Δt) = exp(0.40√1/4) = 1.2214

d = exp( - σ√Δt) = exp(- 0.40√1/4) = 0.8187

Calculation of 12 sub-periods each of 1 months

u = exp(σ√Δt) = exp(0.40√1/12) = 1.1224

d = exp( - σ√Δt) = exp(- 0.40√1/12) = 0.8909