Question

15. The following options are available on XYZ stock: | Туре Call Put $30 $30 | 1.00 1.25 Delta 0.703 -0.292 Gamma 0.036 0.032 The stock is currently priced at $32 and has a volatility (0) of 30% p.a. The continuously compounded risk-free rate is 5% p.a. The Black-Scholes option pricing model values the call at $5.60 and the put at $2.14. (a) Assume an options trader sells an XYZ call option, what position must she take in the stock to make her portfolio delta-neutral? (b) What position must she take in the stock and put options to make her portfolio delta-neutral and gamma-neutral?

Answer 1

The delta Neutral portfolio explain the net value of the portfolio remains unchanged in case of small changes in the market value of the underlying security in option trade.

Answer a)Delta neutral with spot market

Delta of XYZ call option = 0.703, selling call option.

for delta neutral portfolio , (long position) Buying 0.703 share in spot market.

i.e. 0.703*100= 70.3 shares

Answer b) Delta neutral with Put option ( total Delta Value of portfolio =0)

Delta of call portfolio = Delta of Put Option

=> 1*703 = n*-0.292

=> n = - 2.41

i.e (Short position) sell 2.41 Put option to create delta neutral portfolio.

For Gamma Neutral ( total Gamma Value of portfolio =0)

0.036 = n*0.032

=> n= 1.125

i.e (Long position) buy 1.125 Put option to create gamma neutral portfolio.