Question

A stock currently trading at $115 pays a $4 dividend in three months and nine months. An option on the stock with an exercise price of $105 expires in ten months. Annualized yield for T-bill for this option is 11% and annualized standard deviation (volatility) of the continuously compounded return on the stock is 17% per annum.     

14. A stock currently trading at $115 pays a $4 dividend in three months and nine months. An option on the stock with an exercise price of $105 expires in ten months. Annualized yield for T-bill for this option is 11% and annualized standard deviation (volatility) of the continuously compounded return on the stock is 17% per annum. (7 points) (a) Compute adjusted stock price, S." (b) Compute di and Nd.) (c) Compute N(da) assuming dz is --2897. Use this N(da) in part (d) and (e). (d) Compute the price of the European call. — (e) Compute the price of the European put.

Answer 1

a

Adjusted stock price = 115 – 4 * e^(-11%*3/12) - 4 * e^(-11%*9/12)

= 106.54

b , c, d

	Option price=	SN(d1) - Xe-r t N(d2)
	d1 =	[ ln(S/X) + ( r+ v2 /2) t ]/ v t0.5
	d2 =	d1 - v t0.5
	Where
	Current stock price=			115
	Less: present value of dividend
	Dividends	4	4
	Paid in (months)	3	9
	Present value of dividend	-4.11	-4.34	-8.46
S=	Stock price adjusted			106.54
X=	Exercise price=		105
r=	Risk free interest rate=		11.00%
v=	Standard devriation=		17%
t=	time to expiration (in years) =		0.8333
d1 =	[ ln(106.544478845815/105) + ( 0.11 + (0.17^2)/2 ) *0.83333] / [0.17*0.83333^ 0.5 ]
d1 =	[ 0.014602 + 0.103708 ] /0.155188
d1 =	0.762369
d2 =	0.762369 - 0.17 * 0.83333^0.5
	0.607181
N(d1) =	N( 0.762369 ) =	0.77708
N(d2) =	N( 0.607181 ) =	0.72813
Option price=	106.544478845815*0.777080018511513-105*(e^-0.11*0.83333) *0.728134491693128
	13.04

Price of call option is 13.04

Price of put option is:

	Option price=	= Xe –rt × N(-d2) – S × N(-d1)
	d1 =	[ ln(S/X) + ( r+ v2 /2) t ]/ v t0.5
	d2 =	d1 - v t0.5
	Where
S=	Current stock price=	106.54
X=	Exercise price=	105
r=	Risk free interest rate=	11%
v=	Standard devriation=	17%
t=	time to expiration (in years)=	10/12 =	0.833333
d1 =	[ ln(106.544478845815/105) + ( 0.11 + (0.17^2)/2 ) *0.83333] / [0.17*0.83333^ 0.5 ]
d1 =	[ 0.0146 + 0.103708333333333 ] /0.155188
d1 =	0.7623687
d2 =	0.76237 - 0.17 * 0.83333^0.5
	0.607180673
N(-d1) =	N( - 0.76237 ) =	0.22292
N(-d2) =	N( - 0.60718 ) =	0.27187
	105 × e^(-0.11 × 0.83333) ×(1- N( 0.60718)) -106.544478845815× (1-N(0.76237))
Option price=	2.29

Price of put option is 2.29