In: Finance
A fund manager has just sold a call option on 100 shares of a stock. The stock price is $87 and its volatility is 20% per annum. The strike price of the option is $89 and it matures in 6 months. The risk-free rate is 6% per annum (continuously compounded).
a) What position should the fund manager take in the stock to achieve delta neutrality?
b) Suppose after the fund manager sets up the delta neutral position, the stock price suddenly jumps to 80. Should she buy or sell shares to maintain delta neutrality? Why? Did she gain money, lose money or achieve no gain/loss on her position? Why?
a). Delta = ratio of change in the price of the option to change in the price of the stock
Volatility (v) = 20% and risk-free rate (r) = 6%; T (time to expiry) = 6/12 = 0.5; current stock price (S0) = 87
Price if stock price moves up (Su) = S0*e^(v*T^0.5) = 87*e^(20%*0.5^0.5) = 100.216
If price at expiry is 100.216 then option payoff becomes max(Su - K, 0) where K (strike price) = 89
Option payoff (Pu) = max(100.216-89, 0) = 11.216
Price if stock price moves down (Sd) = S0*e^(-v*T^0.5) = 87*e^(-20%*0.5^0.5) = 75.527
Then option payoff (Pd) is max(Sd-K, 0) = max(75.527-89, 0) = 0
Risk-free probability (p) = [e^(r*T) - e^(-v*T^0.5)]/[e^(v*T^0.5) - e^(-v*T^0.5)] (applying the formula from Binomial model)
p = [e^(6%*0.5) - e^(-20%*0.5^0.5)]/[e^(20%*0.5^0.5) - e^(-20%*0.5^0.5)] = 57.20%
So, 1-p = 1-57.20% = 42.80%
Option price now (P0) = ((p*Pu) + ((1-p)*Pd))*e^(-r*T)
= ((57.20%*11.216) + (42.80%*0))*e^(-6%*0.5) = 6.226
Delta = (6.226 - 0.0)/(87 - 75.527) = 0.5427
In order to be delta neutral, the fund manager should buy 0.5427 of stock.
b). Again, using the same formulas as above with S0 = 80, we get a delta of
(1.750-0)/(80-69.45) = 0.1659
This means that the fund manager needs to sell (0.5427 - 0.1659) of stock so she will now hold 0.1659 of stock.
Loss on 100 stocks = 100*(0.5427 - 0.1659)*(87-80) = 263.75