Question

The volatility of a non-dividend-paying stock whose price is $55, is 25%. The risk-free rate is 4% per annum (continuously compounded) for all maturities. Use a two-step tree to calculate the value of a derivative that pays off [max(St − 62, 0)]" where St is the stock price in four months?

Answer 1

The information given:

Spot Price = 55$ (S0)

Pay off = max(St-62,0)

Time = 4/12 = 0.333 years (T)

Volatility = 25% (sig)

Risk free rate = 4%(r)

Tree size = 2 (n)

Now to calculate the values of the steps:

dt = T/n = 0.16667

Up step (u) = exp(sig*sqrt(dt)) = exp(0.25*sqrt(0.1667)) = 1.10745

Down step (d) = 1/u = 0.902974

Probability (q) = (exp(r*dt)-d)/(u-d) = (exp(0.04*0.16667) - 0.902974)/(1.10745-0.902974) = 0.507219

Now to form the binomial tree:

Start with the S0 value, and then the next step is S0*u and S0*d and so on. Using this we get the following tree:

Now to find the price, we calculate the payoff for each of the last values. As the payoff gives a value only if the number is above 62, there is a payoff only in the first case. We then find the present value by using the probability of that state occuring with the following formula:

= (q*up step + (1-q)* down step) * exp(-r*dt)

Using this we get the following table:

Thus, the value of the option is 1.3848$