Put-Call Parity and Formula Diagnostics

Why Put–Call Parity Is a Useful Check

A European call and a European put with the same strike and maturity have payoffs that differ by the terminal stock price minus the strike. This elementary identity has a strong pricing consequence: in the Black–Scholes model, the difference between their current prices is determined by the current stock price and the present value of the strike. It does not depend on volatility. That makes parity both a derivation and a diagnostic: a pair of computed prices that violates it cannot both be correct under the stated model.

We work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\), with an equivalent risk-neutral measure \(\mathbb{Q}\). Under \(\mathbb{Q}\), \(B_t^{\mathbb{Q}}\) is Brownian motion and the ex-dividend stock price satisfies \(dS_t=(r-q)S_t\,dt+\sigma S_t\,dB_t^{\mathbb{Q}}\), where \(r\) is the constant risk-free rate, \(q\) is the constant continuous dividend yield, and \(\sigma>0\) is constant volatility. In integral form, the stochastic integral has progressively measurable integrand \(\sigma S_t\), with \(\mathbb{E}^{\mathbb{Q}}\int_0^T \sigma^2S_t^2\,dt<\infty\) for each finite \(T\). We use risk-neutral pricing: the time-\(t\) price of a European payoff \(X_T\) is \(e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[X_T\mid\mathcal{F}_t]\). The measure \(\mathbb{P}\) is the physical measure; it is \(\mathbb{Q}\), not \(\mathbb{P}\), that appears in this pricing relation.

Matching the Terminal Payoffs

Proof. For any terminal stock value \(x\ge0\), consider the cases \(x\ge K\) and \(x<K\). If \(x\ge K\), then \((x-K)^+=x-K\) and \((K-x)^+=0\), so their difference is \(x-K\). If \(x<K\), then \((x-K)^+=0\) and \((K-x)^+=K-x\), so their difference is again \(x-K\). At \(x=K\), both payoffs are zero and the same identity holds. Therefore \(C_T-P_T=S_T-K\) in every state. Applying the risk-neutral pricing relation and linearity of conditional expectation gives \(C-P=e^{-r\tau}\mathbb{E}^{\mathbb{Q}}[S_T-K\mid\mathcal{F}_t]\). The stock dynamics have the explicit solution \(S_T=S_t\exp((r-q-\tfrac12\sigma^2)\tau+\sigma(B_T^{\mathbb{Q}}-B_t^{\mathbb{Q}}))\). The Brownian increment is independent of \(\mathcal{F}_t\) and is normal with mean zero and variance \(\tau\); using the normal moment-generating function yields \(\mathbb{E}^{\mathbb{Q}}[S_T\mid\mathcal{F}_t]=S_t e^{(r-q)\tau}\). Also, \(\mathbb{E}^{\mathbb{Q}}[K\mid\mathcal{F}_t]=K\). Substitution gives \(C-P=S_t e^{-q\tau}-K e^{-r\tau}\), as claimed. \(\square\)

The stock term is discounted by the dividend yield, not by the risk-free rate. This is because \(S_t e^{-q\tau}\) is the value contribution of the stock under continuous dividends; the strike is a fixed cash amount paid at maturity and is discounted at \(r\). If there are no dividends, the stock term is simply \(S_t\). The identity concerns prices at the same time, for the same strike and the same maturity; changing any of these inputs changes the comparison.

Checking the Black–Scholes Formulas

For \(S_t>0\), \(K>0\), \(\tau>0\), and \(\sigma>0\), define \(d_1=[\ln(S_t/K)+(r-q+\tfrac12\sigma^2)\tau]/(\sigma\sqrt{\tau})\) and \(d_2=d_1-\sigma\sqrt{\tau}\). Let \(\Phi\) be the standard normal cumulative distribution function. The Black–Scholes formulas are \(C=S_t e^{-q\tau}\Phi(d_1)-K e^{-r\tau}\Phi(d_2)\) and \(P=K e^{-r\tau}\Phi(-d_2)-S_t e^{-q\tau}\Phi(-d_1)\).

Proof. Subtract the displayed put expression from the call expression and group the stock and strike terms: \(C-P=S_t e^{-q\tau}[\Phi(d_1)+\Phi(-d_1)]-K e^{-r\tau}[\Phi(d_2)+\Phi(-d_2)]\). For every real \(z\), symmetry of the standard normal density gives \(\Phi(-z)=1-\Phi(z)\), and hence \(\Phi(z)+\Phi(-z)=1\). Applying this identity to \(d_1\) and \(d_2\) reduces the grouped expression to \(S_t e^{-q\tau}-K e^{-r\tau}\). \(\square\)

Delta as a Second Diagnostic

For \(\tau>0\), differentiating the Black–Scholes prices with respect to the current stock price gives \(\Delta_C=e^{-q\tau}\Phi(d_1)\) and \(\Delta_P=-e^{-q\tau}\Phi(-d_1)\). Equivalently, \(\Delta_P=e^{-q\tau}(\Phi(d_1)-1)\). These are partial derivatives with all other model inputs fixed. Differentiating the parity identity with respect to \(S_t\) gives \(\Delta_C-\Delta_P=e^{-q\tau}\), which matches the two delta formulas because \(\Phi(d_1)+\Phi(-d_1)=1\). Thus a call delta and put delta should not generally differ by one when dividends are present; their difference is \(e^{-q\tau}\). When \(q=0\), that difference is one.

Common Formula Diagnostics

Parity is an internal consistency check, not a substitute for checking the model assumptions. Before comparing prices, confirm that both options are European, have the same strike and maturity, and are valued at the same time under the same interest rate and dividend assumptions. An American option can be exercised before maturity, so the European payoff argument alone does not establish the same equality for its price.

A frequent sign error is to write \(P-C=S_t-Ke^{-r\tau}\). The correct reversal is \(P-C=Ke^{-r\tau}-S_t\) when \(q=0\), or \(P-C=Ke^{-r\tau}-S_t e^{-q\tau}\) with continuous dividends. A frequent discounting error is to discount the stock by \(e^{-r\tau}\) in the continuous-dividend version, or to leave the strike undiscounted. A frequent maturity error is to use the time elapsed since inception rather than the remaining time \(\tau=T-t\). The exact example above shows that these changes alter the parity value.

Put–call parity matters because it connects two separately computed option prices to a simple portfolio identity. It can reveal implementation errors even when each price looks numerically plausible, and it supplies an independent check on the sign and dividend adjustment in the delta formulas. It does not require calculating \(d_1\), \(d_2\), or \(\Phi\), so it is particularly useful as a final check after a formula or numerical routine has been evaluated.