Hedged Call After Stock Drop

This question describes a delta-hedged long call on a single stock that suddenly experiences a sharp negative price shock due to extreme news about the company. The setup forces the candidate to think about what happens to the option's sensitivities when the underlying moves significantly and discontinuously, rather than by a small smooth change. It also implicitly probes whether the candidate really understands the meaning of "delta-hedged" for a long call position and how that hedge is implemented in practice via borrowing or lending against stock. Variants of this style of question often show up in equity derivatives and exotics interviews.

Conceptually, the problem leans on option Greeks, especially delta and gamma, and their behavior after a large move in the underlying. It tests whether the candidate can reason directionally about how the call's delta responds when the stock price falls, and translate that into concrete trading actions: adjusting the number of shares held and the corresponding financing position. Interviewers are watching for clear understanding of replicating portfolios, the link between delta and stock holdings, and the intuition for rebalancing after a jump rather than mechanically invoking formulas.

What it tests

Delta-hedging is built on the principle that the price sensitivity of an option to the underlying asset (its `delta`) determines the number of shares needed in the replicating portfolio to offset small price changes. For a long call, the replicating portfolio is long `delta` shares of stock and short a risk-free bond. As the underlying price changes, the option's `delta` changes nonlinearly, especially near the strike price, due to the convexity of the option's payoff. The hedge must be dynamically rebalanced to match the new `delta`, which always reflects the instantaneous rate of change of the option's value with respect to the underlying. This dynamic adjustment ensures that the portfolio remains locally riskless to first order, but requires constant monitoring as `delta` is path-dependent and sensitive to both price and volatility shifts.