Question

The current price of a non-dividend paying stock is $50. Use a two-step tree to value a European put option on the stock with a strike price of $48 that expires in 12 months. Each step is 6 months, the risk free rate is 5% per annum, and the volatility is 50%. What is the value of the option according to the two-step binomial model. Please enter your answer rounded to two decimal places (and no dollar sign).

Answer 1

K strike price = 48

r: risk free rate = 5% = 0.05

s: standard deviation = 50% = 0.5

dt: lenght of time step = 6months = 0.5 years

u: up factor

where e is natural exponent

u = e^(0.5 * 0.5^0.5) = 1.424

d: down factor = 1/u

d = 1/1.424 = 0.702

p: probability of up movement

p = (1.025 - 0.702)/(1.424 - 0.702) = 0.447

q: probability of down movement

q = 1 - p = 1 - 0.447 = 0.553

T = 0	T = 0.5	T =1	Put Payoff	Probability
		71.2*u = 101.38	0	p*p (2 consecutive ups) = 0.447^2 = 0.199
	50 * u = 50*1.424 = 71.2
Stock price = 50		71.2*d = 49.98	0	2*p*q (1 up 1 down in any sequence) = 0.494
	50 * d = 50*0.702 = 35.1
		35.1*d = 24.64	48 - 24.64 = 23.36	q*q (2 consecutive downs) = 0.305

Payoff put option at (T =1) = Sum[Probability*Payoff] = 0.199*0 + 0.494*0 + 0.305 * 23.36 = 7.144

Present value of payoff (by discounting for 1 year) = e^(-0.05*1) * 7.144 = 6.796