Delta hedging refers to managing the portfolio delta by entering additional positions into the portfolio. If DeltaH is the delta of the hedging instrument, the optimal number of units of the hedging instruments, NH, is given by the formula below:
Example: Suppose we know S = 100, X = 100, r = 5%, T = 1.0, σ = 30%, and δ = 5%. We have a short position in put options on 10,000 shares of stock. Based on this information, we note Deltac = 0.532, and Deltap = –0.419. Assume each stock option contract is for one share of stock.
Solution to 1:
B is correct.
The put delta is given as –0.419, thus the short put delta is 0.419. In this case, Portfolio delta = 10,000(0.419) = 4,190 and DeltaH = 1.0. Thus, the number of hedging units is –4,190 [= –(4,190/1)] or short sell 4,190 shares of stock.
Solution to 2:
A is correct. Again, the Portfolio delta = 4,190 but now DeltaH = 0.532. Thus, the number of hedging units is –7,875.9 [= –(4,190/0.532)] or sell 7,876 call options.
A few key points related to gamma are as follows: