Arbitrage via Put-Call Parity

This quant prep question is about internal consistency in option markets and how derivative prices must align with the underlying and the risk-free asset. It uses a very simple numerical setup to isolate the core logic of no-arbitrage pricing, without distractions like dividends, stochastic rates, or path dependence. Candidates see directly how vanilla calls and puts embed information about the underlying stock and the bond.

It trains your ability to detect mispricing quickly and translate a theoretical relationship into a concrete trading strategy. You practice mapping payoffs across scenarios, interpreting option quotes, and turning a violation of parity into a specific portfolio of longs and shorts. This reinforces intuition about replication, static hedging, and the connection between payoffs and prices.

It matters for quant interviews because put-call parity underpins almost everything in options quant work: pricing, model calibration, and risk checks. Interviewers use this type of question to see if you can go from a clean mathematical identity to a real-world arbitrage trade, and to test whether your understanding of option pricing is genuinely structural rather than purely formula-based.

What it tests

The core structure here is the enforcement of no-arbitrage through put-call parity, which links the prices of European calls, puts, the underlying, and risk-free bonds. The principle states that holding a call and selling a put with the same strike and expiry is economically equivalent to holding the underlying and borrowing the strike amount (or equivalently, buying a bond that pays the strike at expiry). This equivalence arises because both portfolios guarantee the same payoff in every scenario at expiry, so their prices must match in a market free of arbitrage. Whenever the relationship $C - P = S - K$ (for zero rates) fails, it signals a mispricing that can be exploited by constructing offsetting positions in these assets. The reason this pattern holds is that the payoffs of the two sides are identical across all possible future stock prices, so any price difference is pure arbitrage.