101 Concepts for the Level I Exam

Essential Concept 81: Delta Hedging and Gamma Risk

Delta hedging refers to managing the portfolio delta by entering additional positions into the portfolio. If Delta_H is the delta of the hedging instrument, the optimal number of units of the hedging instruments, N_H, is given by the formula below:

$\noindent N${}_{H}$ = $\frac{\mathrm{-}\mathrm{\ Or}\mathrm{iginal\ Portfolio\ delta}}{{\mathrm{Delta}}_{\mathrm{H}}}$$

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 Delta_c = 0.532, and Delta_p = –0.419. Assume each stock option contract is for one share of stock.

Assuming the hedging instrument is a stock, how many stocks should be bought/sold for a delta hedge?
Assuming the hedging instrument is a call option, how many call options should be bought/sold for a delta hedge?

Solution to 1:

B is correct. $\ N${}_{H}$ = $\frac{\mathrm{-}\mathrm{\ Original\ Portfolio\ delta}}{{\mathrm{Delta}}_{\mathrm{H}}}$$

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 Delta_H = 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 Delta_H = 0.532. Thus, the number of hedging units is –7,875.9 [= –(4,190/0.532)] or sell 7,876 call options.