Question

A stock option market maker has a portfolio consisting of 200 shares of ABC Corp, and the following option positions on ABC Corp. For simplicity, each option is assumed to be written on a single share of the stock.

Quantity Exercise Price Option Type Price Delta Gamma 200 150 -120 100 -100 120 125 115 130 120.0000 1.0000 0.0000 4.3585 0.5362 0.0385 4.1383 0.4179 0.0268 4.19470.3289 0.0202 2.58910.2982 0.0238

The market maker is worried about stock price movements in ABC’s stock and would like to make his portfolio delta-neutral and gamma neutral over the night. The current stock price of ABC is $120, the volatility of the stock is 30%, and the risk-free rate is 5%. The market maker can form the hedge using a 3-month futures on ABC stock whose price is at full carry and either one of two options on ABC stock whose attributes are

(a) How many futures and option contracts should be bought or sold if he uses futures and call option?

(b) How many futures and option contracts should be bought or sold if he uses futures and put option?

(c) If transaction costs are $ 5 per contracts, which instruments should be selected to achieve delta-neutral and gamma-neutral?

Answer 1

Answer:

(A). Firstly, in question it is told that he hold 200 stock but in table thisis 100 only.

No of futures which will be required to hedge the portfolio will be 200 for 200 stock and 100 for 100 stock as the Delta of future is 1.

So, he will sell 200 futures for 200 stock

To hedge 200 stock price by call option with delta 0.3505, Investor will sell calls to lock the price at ATM $120

SO number of call option short will be 200*0.5362= 112.4 nearly 113 call options short. Hence near contract size is of 150 call option, he will short 1 contract consisting 150 call option

(B). To hedge the holding of 200 shares at $120 by put option, He will buy the put with strike price $120.

Hence delta of Put at stike $120 is not given, either we will calculate the Delta of Put by the formula(N(d2)-1) or by using the nearest strike price.

I see a put option of strike price $115 is given, so i will use the Delta of this 0.3289

200*0.3289=65.78 put option contract, I see the size is of 100 s we can buy 1 put contract approx.