You are short a 100-strike call worth 8.00 with delta 0.50 on a stock at 100, hedged with 50 shares per contract. The stock gaps to 120 overnight and the call reprices to 20.50. What is your profit and loss per share, and would hedging more often have helped?

You are short a 100-strike call worth 8.00 with delta 0.50 on a stock at 100, hedged with 50 shares per contract. The stock gaps to 120 overnight and the call reprices to 20.50. What is your profit and loss per share, and would hedging more often have helped?

Approach: Value the option leg and the stock leg separately across the jump, then ask what a continuous hedging argument requires of the price path.

-$2.50 per share. The short call loses 12.50 as it reprices from 8.00 to 20.50, while the 0.50 share hedge gains 0.50 of the 20-point move, or 10.00, leaving a loss of 2.50 per share. Hedging more often would not have helped at all, because between the Friday close and the Monday open there was no price at which to trade: the delta hedging argument requires continuous paths, and a gap is exactly the case where the stock never passes through the intermediate levels the hedge needs. This is the gap risk that makes short gamma positions dangerous on takeover candidates and earnings names, where the loss is a single number set by the size of the jump rather than by the frequency of rebalancing. The market charges for it in the wings, which is why the volatility surface of a biotech into a trial result carries a smile no continuous model produces.

Follow-up: What does the at-the-money implied volatility become under a model with a 5% chance of a 20% jump?

Key concepts: jump risk, delta hedging, gap risk, gamma, continuous paths.