Put-Call Parity
The mathematical relationship between calls, puts, stock and bonds
What Is Put-Call Parity?
Put-call parity is a fundamental mathematical relationship between the prices of call and put options with the same strike and expiration. The relationship is: Call Price − Put Price = Asset Price − Present Value of the Strike. Rearranged: Call Price = Put Price + Asset Price − Present Value of the Strike. This relationship must hold if there are to be no arbitrage opportunities (riskless profit). If it is violated, traders can execute a "conversion" or "reverse conversion" arbitrage to capture riskless gains. Put-call parity is one of the most important principles in options pricing because it constrains how the relative prices of calls and puts can be set. It was formalised by Hans R. Stoll in 1969, although the relationship had been described in the literature since the early twentieth century (Nelson, 1904), and it is foundational to all modern options theory.
Put-Call Parity Explained Without Formulas
Imagine you own the shares: you are long and bullish. Now you buy a put and sell a call, both at the same strike. The put protects you below that level and the call caps you above it, so whatever price does, you end up selling the shares at exactly the strike. Your outcome is fixed in advance: you have turned a risky position into a certain payoff. That combination (long stock + long put + short call) must therefore return the risk-free rate. Hence: Stock + Put − Call = risk-free return. Rearranging: Call − Put = Stock − risk-free return. The risk-free return is typically approximated by bonds or Treasury bills. This conceptual derivation shows why parity must hold. If it did not, you could make money with no risk, which contradicts the efficient-market hypothesis. Professional arbitrageurs constantly hunt for put-call parity violations to capture quick profits.
Practical Implications of Parity
Put-call parity has several practical implications. First, if you know the call price, you can estimate the put price using the relationship (or vice versa). If a $100-strike call sells for $5 when the stock costs $102, you can estimate that the put should be worth roughly Put = Call − Underlying + Strike = $5 − $102 + $100 = $3 (more precisely, discounting the strike to present value at the risk-free rate). Second, if you see a severe parity violation in real-time data, it is probably a data error or a bid-ask spread rather than a real opportunity. Third, parity helps market makers price options; if they know the call price, they can calculate the fair put price. Fourth, parity explains why at the strike equal to the underlying’s forward price the call and the put are worth practically the same: there the term Underlying − PV(Strike) vanishes. Parity acts as a guardrail preventing call and put prices from drifting incoherently apart.
Conversion Arbitrage: Exploiting Parity Violations
If put-call parity is violated, a "conversion" arbitrage opportunity exists. For example, if in the market Call − Put > Stock − risk-free rate, then the call is relatively expensive or the put relatively cheap. In that case you can sell the call (collecting the expensive premium), buy the put (paying the cheap one) and buy the stock, financing yourself at the risk-free rate. This combined portfolio generates a riskless profit equal to the size of the parity violation. Institutional traders with low-cost borrowing, fast execution and low commissions can capture these gains. A "reverse conversion" is the opposite strategy, used when the inverse relationship holds. In modern markets, put-call parity violations are usually small and short-lived, because arbitrageurs constantly correct any imbalance, keeping prices fair.
Limitations and Extensions of Parity
Perfect put-call parity holds only under certain assumptions: no transaction costs, no market frictions, no dividends (although a dividend-adjusted version exists), and no possibility of early exercise (it is exact for European options). In practice, there are small deviations from parity due to transaction costs, bid-ask spreads and other factors. Those deviations are generally small (a few cents) because arbitrageurs constantly correct any large imbalance. When dividends are present, parity adjusts to: Call − Put = Stock − PV(Dividends) − PV(Strike). For American options, which allow early exercise, the exact equality relaxes into a bounded inequality. Despite these complications, put-call parity remains one of the most important concepts in options trading and theoretical pricing.