Stop-Loss Risk in Short Calls

This question focuses on hedging a short call using a simple stop-loss rule tied to the strike, rather than a full dynamic hedge. The setup is a trader short a call option who commits to entering and exiting a stock position only when the underlying crosses the strike level. Candidates are asked to reason about how this trading rule behaves in different price paths, what happens when the stock oscillates around the trigger, and how the realized outcome of the hedge compares to the payoff of the original option. This kind of discussion is common in derivatives and exotics risk roles, and in interviews at banks and hedge funds that expect a practical understanding of option replication and hedging errors.

The question leans on ideas from dynamic hedging, replication of nonlinear payoffs, and the distinction between path-dependent strategies and path-independent payoffs. It pushes candidates to recognize slippage, gap risk, and the impact of discrete trading and transaction costs. Interviewers listen for an understanding of why continuous-time delta hedging is an idealization, and how naive rules around a single price level can systematically lose money. Strong answers identify structural weaknesses in the hedge, not just operational or implementation issues.

What it tests

Hedging an option with a stop-loss strategy exposes the fundamental challenge of discretely replicating a payoff that is inherently path-independent (the option) with a trading rule that is path-dependent (the stop-loss). The option's payoff depends only on the terminal price, but the stop-loss strategy's outcome depends on every crossing of the strike, leading to cumulative costs if the underlying oscillates. This mismatch arises because perfect replication of an option's nonlinear payoff requires continuous, infinitesimal adjustments (as in delta hedging), which are impossible in practice. Any discrete or rule-based hedging method will accumulate slippage and transaction costs whenever the underlying price moves back and forth across the critical threshold. The principle is that hedging path-independent payoffs with path-dependent rules introduces structural inefficiency, especially when the underlying is volatile near the hedge trigger.