Question

Current stock price is $150; volatility is 20% per annum. An at-the-money European put option on the stock expires in 3 months. Risk free rate is 5% per annum, continuously compounded. There is no dividend expected over the next 3 months. Use a 3-step CRR model to price this option.

Answer 1

Stock Price = 150 ; Volatility = 20% ; Risk Free Rate = 5% ;

Now Calculate upward and downward probabilities

u	1.059434	eσ√t
d	0.9439	e-σ√t
a	1.004175	ert
p	0.52171	(a-d)/(u-d)
(1-p)	0.47829

For put option the value of the option will be MAX(K-S,0), where S = stock price, K = exercise price

For node A the value = $150 * u = $150 * 1.059434 = 158.92, node B = $150 * 0.9439 = $141.59

For node D the value = $158.92 * 1.059434 = 168.36, node E = $158.92 * 0.9439 = $150.00, node F = $141.59 * 0.9439 = $133.64

Option value is calculated as at F = MAX(150-133.64,0) = $ 16.36, at E = MAX(150-150,0)=0

Option value is calculated as at B as = (Option Value at E * P) + (Option Value at F *(1-p))

=(0 * 0.52171) + (16.36 * 0.47829) = 7.82 now convert this to present value = 7.79

Option value is calculated as = (Option Value at A* P) + (Option Value at B*(1-p))

= (0*0.52171) + (7.79 * 0.47829) = 7.35 converted to present value = 7.32 this is the answer