VRM14 - Binomial Trees

Question 1

Calculate the value of an American or European option call or put using a one-step or two-step binomial tree

Answer 1

p = (exp(rT) - d) / (u - d)

F = exp(-rT) (pFu + (1-p)* Fd)

u and d are the stock price if moves up / down
Fu. Fd are the payoffs
r is risk free rate

Apply to each node in two step tree
u = exp(simga * sqrt(delta t)
d = exp(- simga * sqrt(delta t)

To extend to American, determine value of option at each node if exercised or not, maximum of the two is the value

Question 2

Describe how the value calculated using a binomial model converges as time periods are added

Answer 2

Converges to black scholes model as time steps increased

Question 3

Define and calculate the delta of a stock option

Answer 3

Delta measures sensitivity of derivatives value to the price of its stock

Question 4

Explain how the binomial model can be altered to price options on stocks with dividends, stock indices, currencies and futures

Answer 4

For dividend or using foreign risk free rate,
p = exp((r - d) * sqrt(t)) - d) / (u - d)

For futures p = (1 - d) / (u - d)