Question

You wish to write one European Call contract on Zoom (ZM) with a strike price of $160. Zoom's current price is $158.01, has a u of 1.25, and d of 0.79. The risk free rate is 1% per period. Use a two period binomial model. How many shares of Zoom do you need to purchase to hedge your price risk when you write the call? Note: Answer in number of shares, report up to two decimal places.

Answer 1

S0 =	Stock price today	=	158.1
r=	risk free interest rate	=	1%
u=	up factor	=	1.25
d=	Down factor	=	0.79
X =	Exercise price	=	160
	We first compute the possible values of the stock at each node in the binomial tree:
t=1
S+ =	= 158.1*1.25	=	197.625
S- =	= 158.1*0.79	=	124.899
t = 2 = T
S++ =	= 158.1*1.25*1.25	=	247.03125
S+ - =	= 158.1*1.25*0.79	=	156.12375
S- - =	= 158.1*0.79*0.79	=	98.67021
	Intrinsic value of the call option at expiration
c++ =	= Max(0, S++ - X)
	= Max(0, 247.03125 - 160)	=	87.03125
c+ - =	= Max(0, S+ - - X)
	= Max(0, 156.12375 - 160)	=	0
c- - =	= Max(0, S- - - X)
	= Max(0, 98.67021 - 160)	=	0
∏=	Risk neutral probability	=	(1+r-d)/(u-d)
∏=	Risk neutral probability	=	(1+0.01-0.79)/(1.25-0.79)
		=	0.4783
	1- ∏=	=	0.5217
	Compute the value of call option at each node for t=1
c+ =	Call price t=1	=	[∏c++ + (1-∏)c+ - ]/ (1+r)
c+=	[0.4783*87.03125 + 0.5217*0] /[1+0.01 ]	=	41.21
c- =	Call price t=1	=	[∏c+ - + (1-∏)c- - ]/ (1+r)
	[0.4783*0 + 0.5217*0] /[1+0.01 ]	=	-
	Finally, value of call option
c =	Call price t=0	=	[∏c+ + (1-∏)c - ]/ (1+r)
c =	Call price today
	[0.4783*41.21 + 0.5217*0] /[1+0.01 ]	=	19.51

Call price is $19.51

Hedge ratio:

n =	[ c+ - c- ] / [ S+ - S- ]
	[ 41.21 - 0 ] / [ 197.63 - 124.9 ]
	= 0.5667