Question

A binomial tree with three-month time steps is used to value a currency option. The domestic and foreign risk-free rates are 4% and 6% respectively. The volatility of the exchange rate is 12%. What is the probability of an up movement?

Answer 1

We have following information

Volatility of the exchange rate σ =12% or 0.12

Time period T = 3 months or 3/12 = 0.25 year

The domestic risk-free rate rd = 4% or 0.04

And foreign risk free rate rf = 6% or 0.06

Now we can use following formula to calculate probability of an up movement

Probability of an up movement P = (a –d)/ (u –d)

Where, u = e ^ (σ *√T) = e ^ (0.12 * √0.25) = 1.06184

d = 1/u = 1/ 1.06184 = 0.94176

And a = e ^ (rd –rf)*T = e ^ (0.04 -0.06) *0.25 = 0.99501

Probability of an up movement P = (a –d)/ (u –d)

= (0.99501 - 0.94176) / (1.06184 - 0.94176)

=0.4435

The probability of an up movement is 0.4435