Question

Ken is interested in buying a European call option written on Southeastern Airlines, Inc., a non-dividend-paying common stock, with a strike price of $80 and one year until expiration. Currently, the company’s stock sells for $81 per share. Ken knows that, in one year, the company’s stock will be trading at either $94 per share or $68 per share. Ken is able to borrow and lend at the risk-free EAR of 4 percent.

  

a.	What should the call option sell for today? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
b.	What is the delta of the option? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
c.	How much would Ken have to borrow to create a synthetic call? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
d.	How much does the synthetic call option cost? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Answer 1

d

Current stock price = 81
Upward 1-year stock price = 94
Downward 1-year stock price = 68
Strike price = 80
Call option payoff in case of upward movement = Max(0, (94 - 81))
Call option price in case of upward movement = 13
Call option payoff in case of Downward movement = Max(0, (68 - 81))
Call option price in case of Downward movement = 0
Rteurn in case of upward movement of stock = 94/81 - 1 = 16.05%
Rteurn in case of downward movement of stock = 68/81 - 1 = -16.05%
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
0.04 = (ProbabilityRise) * (ReturnRise) + (1 - ProbabilityRise) * (ReturnFall)
0.04 = (ProbabilityRise) * (0.1605) + (1 - ProbabilityRise) * (-0.1605)
0.04 = 0.1605 * (ProbabilityRise) - 0.1605 + 0.1605 * ProbabilityRise
0.321 * ProbabilityRise = 0.2005
ProbabilityRise = 62.46%
ProbabilityFall = 1 – 62.46% = 37.54%
Expected payoff at expiration = (0.6246) * 13 + (0.3754) * 0
Expected payoff at expiration = 8.1198
PV(Expected payoff at expiration) = 8.1198/ 1.04
PV(Expected payoff at expiration) = $7.8075
Therefore call option price is $7.8075

Part B:
Delta of the option = (Swing of option) / (Swing of stock)
Delta of the option = (13 - 0) / (94 - 68)
Delta of the option = 0.5

Part C:
Firse we need to buy 0.50 shares of the stock to create synthetic call.
The stock is currently trading for $81
Cost = 0.5 * 81 = $40.5
To know the amount we need to borrow, we will compare payoff of actual call option with the payoff of delta shares at expiration:
Call Option:
If the stock price rises to $94: Payoff = $13
If the stock price fall to $68: Payoff = $0
Delta Option:
If the stock price rises to $94
Payoff = $94 * 0.5 = 47
If the stock price fall to $68:
Payoff = $68 * 0.5 = 34
So we borrow PV of 34. The obligation to pay $34 in one year will reduce the payoffs so that they exactly match those of an actual call option.
We will purchase 0.5 shares of stock and borrow:
34/ 1.04 = $32.69
to create a synthetic call option with a strike price of $80 and 1 year until expiration.

Part D:
Cost of synthetic option = $40.5 - $32.69
Cost of synthetic option = $7.81