Put-Call Parity for ATM Options

This question focuses on European call and put options that are struck at the current spot price in a Black–Scholes setting with zero rates and no dividends. It asks the candidate to reconcile an intuitive payoff-based view, where the call seems "better" because of its unlimited upside, with the formal statement of put-call parity, which implies the two must be worth the same in this special case. The setup forces you to distinguish between raw payoff diagrams and properly priced contingent claims, using the specific simplifications of the model to strip away distractions like discounting and carry costs.

To answer it well, you need a solid grasp of no-arbitrage arguments, replication, and risk-neutral pricing. The interviewer is looking for recognition that equality of prices follows from equality of state-contingent payoffs, not from heuristic comparisons of payoff shapes. They watch whether you can construct or describe equivalent portfolios, interpret put-call parity as a structural identity rather than a rule of thumb, and explain clearly how probability weights and discounting neutralize the apparent asymmetry between call and put payoffs.

What it tests

The core structure behind problems like this is the concept of risk-neutral valuation and the symmetry imposed by put-call parity. In a risk-neutral world, the value of a derivative is determined by the expected value of its payoff, discounted at the risk-free rate, under a probability measure where all assets earn the risk-free rate. Put-call parity is not just a formula, but a consequence of the absence of arbitrage: if two portfolios have identical payoffs in all future states, they must have the same price today. This principle holds regardless of the skewness or shape of the underlying asset's distribution, as long as the market is frictionless and no arbitrage is possible. The apparent asymmetry in payoffs (unbounded for calls, bounded for puts) is exactly offset by the probabilities assigned to those payoffs under the risk-neutral measure.