Question

The volatility of a non-dividend-paying stock whose price is $45, is 20%. The risk-free rate is 3% per annum (continuously compounded) for all maturities. Use a two-step tree to calculate the value of a derivative that pays off [max(?! − 48, 0)]" where ST is the stock price in four months?

Answer 1

The payoff of the derivative max(St-48,0) represents the payoff of a $48 strike European call and hence the price of the $48 strike Euroepan call option has to be found

One period or step is for 2 months (t=2/12)

u= exp(s*t^0.5) = exp(0.2*(1/6)^0.5) s- standard deviation,

=1.08508

d= 1/u = 0.92159

The stock lattice with above u and d (where stock can go up by factor of u or down by a factor of d) is as shown below

		52.98
	48.83	45.00
45.00	41.47	38.22
t=0	t=1	t=2

At t=2, when the option matures, the payoff when stock price is $52.98 = $4.98

payoff when stock price is $45 = 0 and payoff when stock price is $38.22 = 0

the Risk neutral probability p = (exp(rt)-d)/(u-d)

= (exp(0.03*2/12)-0.92159)/(1.08508-0.92159)

= 0.5103

1-p =0.4897

Value of the option at each of the nodes of t=1 and t=0 are given by

Value of option at each node = (p*value of option at upside in next period+ (1-p)*value of option at downside in next period)*exp(-0.03*2/12)

The option lattice calculated using above formula is as shown below

		4.98
	2.53	0.00
1.28	0.00	0.00
t=0	t=1	t=2

Value of derivative is $1.28