Question

The following is for a mathematical finance course, so showing things algebraically will be greatly appreciated.

Suppose the stock price is $50, and you bought 10 contracts (each
contract is 100 shares) of 3-month call option with strike $50 in the hope that
the stock will go up in three months. However, the call option comes with a
volatility risk, namely the option price will drop if the volatility drops. On the
other hand, you do not believe that the stock is going to go above $60 so you
decide to take a short position in a call with strike $60.

(a) How many contracts of this call should you sell in order that the vega of the portfolio is zero?

(b)Once the vega of the portfolio is zero, you will not worry about the volatility moves just
for the moment. But you still want the portfolio to be delta-neutral. What can
you do to maintain the portfolio to be delta-neutral? You can assume r = 0 and
= 30% at the time of pricing.

Answer 1

(a) As the vega of the call options is not given, we shall make an assumption that the vega of the $50 call option (V₅₀) is $5 and the vega of the $60 call option (V₆₀) is $2.

The current vega of the portfolio is calculated as : $5 * 10 contracts = $50

Short call options have a negative vega. Long call options have a positive vega.

To sell the $60 call option and make the portfolio vega neutral, we need to solve for the number of $60 call option contracts to be sold such that the vega of the $60 short call options equals $50, which is the current vega of the portfolio.

Number of $60 call option contracts to be sold * ($2) = $50

Number of $60 call option contracts to be sold =  25

(b) As the delta of the call options is not give, we shall make an assumption that delta of the $50 call option (D₅₀) is 0.50 and the delta of the $60 call option (D₆₀) is 0.25.

The delta of a long call option is positive and that of a short call option is negative.

The delta of the portfolio after selling the $60 call options is calculated as:

(10*0.50) - (25*0.25) , which is -1.25

To make the portfolio delta neutral, while maintaining the vega neutrality of the portfolio is a highly complex task as any changes to make portfolio delta neutral will affect the vega of the portfolio also. While for large, institutional option books this can be done with the help of computer programs, in this case we shall look at two possibilities:

1. Long 1.25 contracts of the underlying asset. This will make the portfolio delta-neutral as the delta of the underlying asset is always equal to 1
2. Long 2.5 contracts of the $50 call option. This will make the delta of the portfolio equal to zero. However, the vega of the portfolio is increased to $12.50