Question

1. A stock price is currently $50. It is known that at the end of 1 year it will be either $40 or $60. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a one-year European CALL option with a strike price of $50? Please use

1. Non-arbitrage approach (8 points)
2. Formula approach (8 points)

Answer 1

Basic Data
Stock Price	$ 50
Exercise Price	$ 50
Expected Future spot Price on Expiry Price
Future Price 1 (FP1)	$60
Future Price 2(FP2)	$40

1. Non Arbitrage Method
Computation of Option Delta
Particulars	FP1	FP2
Future Spot Price	60	40
Position on Expiry Date (in comparison with Exercise Price)	in the money	Out of Money
Action on Expiry Date	Exercise	lapse
Value of Option on Expiry [Future Spot Price Less Exercise Price]	10	0

Option Delta=

Change in Value of Option/Change in future spot price

(10-0)/(60-40)

=0.5

Computation of amount to be invested at risk free rate

= Present Value of $40 discounted at 10% Continuous Compounding for a 1 Year Period

= 40 * e-rt

= 40 * e-0.1

=40 ÷ 1.1052

=36.192

(c) Value of Call [C]

= Option Delta X [Current Stock Price Less Amount to be invested at Risk Free Rate]

= 0.5x [50 - 36.192] = 6.904

2. Formula Aproach
Particulars	Share
Current Spot Price (SP0)	$50
Exercise Price (EP)	$50
Future Spot Price 1 (FP1)	$60
% Change (R1)	(60-50)/50=20%
Position	in the money
action	Exercise
Value on Expiry (Vc1)	FP1 - EP = $60 - $50 =$10
Future Spot Price 2 (FP2)	$40
% Change (R2)	(40-50)/50=-20%
Position	Out of Money
action	lapse
Value on Expiry (VC2)	0
Probability of FP1 [P1]	x
Probability of FP2[P2]	1-x
Probability Values	Risk Free Return = x * % Change for FP1 + [(1 - x) * % Change for
	FP2]
	→ 10%= [x * 20%] + [(1-x) * (20%)]
	→0.1 = 0.2x + [ - 0.2 + 0.2x]
	→ 0.1 = 0.2x-0.2 + 0.2x
	→ 0.1 + 0.2 = 0.4x
	→P1=x = 0.3 ÷ 0.4 = 0..75 or 75%
	→ P2 = 1 - x = 1 - 0.75 = 0.25 or 25%
Value of Call [Future Value]	→ (Vc1 * P1) + (VC2 * P2)
	→ (10 * 0.75) + (0 * 0.25) → 7.5
Present Value of Call [C]=Value of Call X e-rt
→ 7.5 x e-0.1
→ 7.5 ÷ 1.1052
→ 6.79