The delta of a portfolio of options or other derivatives dependent on a single asset
Whose price is S is: - ∂π/∂S
Where π is the value of the portfolio.
The delta of the portfolio can be calculated from the deltas of the individual options in the portfolio. If a portfolio consists of a quantity Wi of option i (1≤i≤n), the delta of the portfolio is given by ∆ = ∑wi∆i (i=1 to n)
Where ∆i is the delta of the ith option. The formula can be used to calculate the position in the underlying asset or in a futures contract on the underlying asset necessary to make the delta of the portfolio zero. When this position has been taken, the portfolio is referred to as being delta neutral.
Suppose a Stock broker has the following three positions in options
1. A long position in 100,000 call options (lot size-100) with strike price 55. The delta of each option is 0.533.
2. A short position in 200,000 call options with strike price 0.56.The delta of each option is -0.468.
3. A short position in 50,000 put options with strike price 0.56.The delta of each option is 0.508.
The delta of the whole portfolio is-
100,000 x 0.533 + 200,000 x (-0.468) + 50,000 x (0.508) = -14,900
This means that the portfolio can be made delta neutral with a long position of 14,900 with underlying.
