Question

A stock price is currently $30. Each month for the next two months it is expected to increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off [max(30 − ST ; 0)]2, where ST is the stock price in two months? If the derivative is American-style, should it be exercised early?

Answer 1

Given Factor	Notation	Value
Spot Price	SP0	$30
Exercise price	EP	$30
Lover future spot Price	FP1	$27
Higher future spot Price	FP2	$32.40
Risk free Rate		5%
Time	2 Month
Computed Factor	Notation	Value
Continuous Comp. factor(f) =e0.05 x 2/12	f	1.008
Extent of EP1 on SP0 = FP1/SP0 =27/30	d	0.90
Extent of EP2 on SP0 = FP2/SP0 =32.40/30	u	1.08
Probeblity of Lower limit = u-f/u-d		0.4
Probeblity of higher limit   (1-above)		0.6
	1-Month	2-Month
		34.99
	32.4
SP $30		29.16
		29.16
	27
		24.3
Expecteted value(EV) = Intrinsic value x Probability
After one month:
Expected value of Call option =	2.4 x.6 + 0 x.4    =	0.96
Expected value of Put option =	0 x.6 + 3 x.4    =	1.8
Present Value of Option:
Call option          =	Future value x e-rt
=	0.96xe-.05 x 1/12
=	.096 x.9960
=	0.96
Put option          =	Future value x e-rt
=	1.80xe-0.05x 1/12
=	3 x.9960
=	2.99
After Two month:
IV of Call	IV of Put	Prob.	EV of Call	EV of Put
34.99-30		.6 x.6	4.99 x .36	0 x .36
4.99	0	0.36	1.7964	0
	30-29.16	.6 x.4	0 x .36	.84 x .24
0	0.84	0.24	0	0.2016
	30-29.16	.4 x.6	0 x .24	.84 x .24
0	0.84	0.24	0	0.2016
	30-24.30	.4 x.4	0 x .16	5.7 x .16
0	5.7	0.16	0	0.912
Expected Value of Maturity		1.7964	1.3152
Present Value of Option:
Call option          =	Future value x e-rt
=	1.7964xe-.05 x 2/12
=	1.7964 x.9920
=	1.78
Put option          =	Future value x e-rt
=	1.3152xe-.05 x 2/12
=	1.3152 x.9920
=	1.30
If Buy Call option :
It should not exercised early becouse value of call option after one month is 0.96 & after two month is 1.78
If Buy Put option :
It should be exercised early becouse value of call option after one month is 2.99 & after two month is 1.30