Question

IBM stock currently sells for 49 dollars per share. Over 12 months the price will either go by 11.5 percent or down by -7.0 percent. The risk-free rate of interest is 4.5 percent continuously compounded. What is the value of a call option with strike price 51 and maturity 12 months?

	a.	0.12291
	b.	1.9353
	c.	2.1795
	d.	1.3285

Answer 1

Solution:

Sl.No.	Particulars	Notation	Value
1	Spot Price	SP0	$ 49.00
2	Strike Price	EP	$ 51.00
3	Expected future Spot price – Lower Limit - FP1 ( $ 49 * 93 % ) = $ 45.570	FP1	$ 45.570
4	Expected future Spot price – Upper Limit FP2 ( $ 49 * 111.5 % ) = $ 54.635	FP2	$ 54.635
5	Value of call at lower limit [ Action = Lapse, Since FP1 < EP. Therefore value = Nil ]	Cd	NIL
6	Value of call at upper limit [ Action = Exercise, Since FP2 > EP. Therefore value = ( $ 54.635 - $ 51.00 = $ 3.635 ) ]	Cu	$ 3.635
7	Weight for the lower scenario [FP1 / SP0 ] = ( 45.570 / 49 ) =	d	0.930
8	Weight for the upper scenario [FP2 / SP0 ] = ( 54.635 / 49 ) =	u	1.115
9	Risk free rate of Return	r	0.045
10	Duration of the call	t	1 Year
11	Future value factor (Continuous Compounding factor) = er * t = e0.045 * 1 = e0.045 = 2.71828 ( 0.045) = 1.046028	f	1.046028

Note : The value of 2.71828 ^{( 0.045)} is calculated using the excel formula =POWER(Number,Power)

=POWER(2.71828,0.045) = 1.046028

As per the Binomial Option Pricing formula the value of a call is given by the following formula:

Value of a Call = [ ( C_u * [(f-d)/(u-d) ] ) + ( C_d * [ (u-f)/(u-d) ] ) ] / f

Therefore applying the values from the table above to the formula we now have:

= [ ( 3.635*[ (1.046028 - 0.930 )/(1.115 – 0.930) ] ) + ( 0 *[ (1.115 – 1.046028 )/( 1.115 – 0.930) ] ) ] / 1.046028

= [ ( 3.635* [ (0.116028)/( 0.185 ) ] ] / 1.046028

= [ 3.635 * 0.627178 ] / 1.046028

= 2.279793 / 1.046028

= 2.179476

= 2.1795 ( when rounded off to four decimal places )

The value of a call option with strike price 51 and maturity 12 months = $ 2.1795

Thus the solution is Option c = $ 2.1795