A stock is trading at $80. You hold a delta-hedged portfolio in which you are short a call and long $\Delta$ units of the stock. The delta of the call is 0.65 and the gamma of the call is 0.06. If the stock registers an unexpected price decrease of $4, the value of your delta-hedged portfolio will
A) not change.
B) decrease by approximately $0.12.
C) increase by approximately $0.48.
D) decrease by approximately $0.48.
E) increase by approximately $0.12.

Question

Quizplus · Accepted Answer

The value of a delta-hedged portfolio changes primarily due to gamma when the stock price moves. The change in portfolio value due to a small change in the underlying stock price can be approximated by $ \frac{1}{2} \times \text{Gamma} \times (\Delta S)^2 $, where $ \Delta S $ is the change in stock price. Here, $ \Delta S = -4 $, Gamma = 0.06, so the change in value is $ \frac{1}{2} \times 0.06 \times (-4)^2 = \frac{1}{2} \times 0.06 \times 16 = 0.48 $. Since the portfolio is short a call, the decrease in stock price, which benefits the short call position, leads to an increase in portfolio value, but since we are considering the change in value from the perspective of maintaining a delta-hedged position (which involves adjusting the stock position to remain delta-neutral), this benefit is offset by the need to adjust the hedge, resulting in a net decrease in portfolio value.

A Stock Is Trading at $80 Δ\DeltaΔ Units of the Stock

A Stock Is Trading at $80 $\Delta$ Units of the Stock