Question

In: Finance

A fund manager has just sold a call option on 100 shares of a stock. The...

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?

Solutions

Expert Solution

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


Related Solutions

Peter has just sold a European call option on 10,000 shares of a stock. The exercise...
Peter has just sold a European call option on 10,000 shares of a stock. The exercise price is $50; the stock price is $50; the continuously compounded interest rate is 5% per annum; the volatility is 20% per annum; and the time to maturity is 3 months. (a) Use the Black-Scholes-Merton model to compute the price of the European call option. (b) Find the value of a European put option with the same exercise price and expiration as the call...
Peter has just sold a European call option on 10,000 shares of a stock. The exercise...
Peter has just sold a European call option on 10,000 shares of a stock. The exercise price is $50; the stock price is $50; the continuously compounded interest rate is 5% per annum; the volatility is 20% per annum; and the time to maturity is 3 months. (a) Use the Black-Scholes-Merton model to compute the price of the European call option. (b) Find the value of a European put option with the same exercise price and expiration as the call...
Peter has just sold a European call option on 10,000 shares of a stock. The exercise...
Peter has just sold a European call option on 10,000 shares of a stock. The exercise price is $50; the stock price is $50; the continuously compounded interest rate is 5% per annum; the volatility is 20% per annum; and the time to maturity is 3 months. (a) Use the Black-Scholes-Merton model to compute the price of the European call option. (b) Find the value of a European put option with the same exercise price and expiration as the call...
A trader has a call option contract to sell 100 shares of a stock for a...
A trader has a call option contract to sell 100 shares of a stock for a strike price of $50. What is the effect on the terms of the contract of the following events? (a) A $5 dividend being paid (b) A 5-for-4 stock split (c) A 10% stock dividend being paid.
A trader has a call option contract to sell 100 shares of a stock for a...
A trader has a call option contract to sell 100 shares of a stock for a strike price of $50. What is the effect on the terms of the contract of the following events? (a) A $5 dividend being paid (b) A 5-for-4 stock split (c) A 10% stock dividend being paid.
please fast A bank has just sold 500,000 call options on shares of a stock. The...
please fast A bank has just sold 500,000 call options on shares of a stock. The strike price is 40; the stock price is 40; the continuously risk-free rate is 5%; the volatility is 30%; and the time to maturity is 0.25. What position should the company take in the stock for delta neutrality?
A short call option is sold for 1,460 dollars and covers 100 shares of Johnson Incorporated....
A short call option is sold for 1,460 dollars and covers 100 shares of Johnson Incorporated. If the Strike price of the option is 70, what is the break-even share price?
A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend...
A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stockS0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks, m = 13%The Black-Scholes-Merton value of the option is $240,000 , how do you get to $240,000 ?
Suppose that a financial institution has sold for $300,000 a European call option on 100,000 shares...
Suppose that a financial institution has sold for $300,000 a European call option on 100,000 shares of a non-dividend stock. Assume that the stock price is $49, the strike price is $50, the risk-free interest rate is 5% per annum, the stock price volatility is 20% per annum, the time to maturity is 20 weeks (0.3826 years), and the expected return from the stock is 13% per annum. (b) (Strategy 1) The financial institution decides to do nothing but deposit...
Assume you own 100 shares of Microsoft stock. Explain the difference between a hedged call option...
Assume you own 100 shares of Microsoft stock. Explain the difference between a hedged call option and a speculative put option on IBM. The stock sold on June 9, 2020 for $190. Illustrate with two hypothetical examples with assumed call and put premia both from the investor’s perspective. Are these illustrative options in-the-money or out-of-the-money?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT