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(St − 48, 0)]" where is the stock price in four months?

Answer 1

Solution:-

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

One Period or Step is for 2 Months (t=2/812),

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 a 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 the stock price is $52.98=$ 4.98,

The payoff when the stock price is $45 = 0 and payoff when the 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

The value of derivatives is $1.28.

****Please give a Positive Rating...

Thanking YOu.....