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
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 |
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