Put-Call Parity

Motivation: why this matters in quant finance

Traders use it constantly:

• Quote verification. If the bid-ask spreads on call and put quotes don't reconcile to the spot and discount factor, one of them is wrong.
• Synthetic positions. A call can be synthesised from a put plus stock plus cash. This matters when the call is illiquid but the put is liquid.
• Implied dividends. For equity options, put-call parity is how implied dividends are extracted from the market.
• Sanity bound on any model. Any option pricer — binomial, Black-Scholes, Heston, Monte Carlo — must reproduce put-call parity to the cent. If yours doesn't, it's wrong.

This note states the identity cleanly, proves it by the replication argument, and lists the important generalisations.

The informal idea

A forward contract to buy at $K$, valued today, is $S_0 - Ke^{-rT}$ (spot minus present-value of strike).

Conclusion: $C_0 - P_0 = S_0 - Ke^{-rT}$. That's put-call parity.

Formal statement

• Portfolio A: long one call, short one put, both strike $K$. Cost today: $C_0 - P_0$.
• Portfolio B: long one share of stock, borrow $Ke^{-rT}$ cash. Cost today: $S_0 - Ke^{-rT}$.

At expiry $T$:

• Portfolio A: if $S_T \ge K$, call pays $S_T - K$, put pays $0$. Net: $S_T - K$. If $S_T < K$, call pays $0$, put (short) pays $-(K - S_T) = S_T - K$. Net: $S_T - K$ in both cases.
• Portfolio B: sell the stock for $S_T$, repay the loan $K$ (since $Ke^{-rT}$ grew to $K$). Net: $S_T - K$.

Both portfolios have the same payoff $S_T - K$ in every state. By no-arbitrage, they must have the same cost today:

Key properties

• Model-free. The proof uses no distribution assumption on $S_T$. It works under Black-Scholes, Heston, jump-diffusion, regime-switching, anything.
• Dividends modify it. If the stock pays continuous dividend yield $q$: $C_0 - P_0 = S_0 e^{-qT} - Ke^{-rT}$. For discrete dividends of total PV $D$: $C_0 - P_0 = S_0 - D - Ke^{-rT}$.
• American puts break it (one-sided). For American options, $C_{Am} - P_{Am} \ge S_0 - K$ and $C_{Am} - P_{Am} \le S_0 - Ke^{-rT}$ (on non-dividend stocks). The inequality is tight for the call (no early-exercise incentive) but not for the put.
• Implied dividends. From quoted $C_{\text{mkt}}, P_{\text{mkt}}, S_0, r, T, K$: solve $C - P = S_0 - D - Ke^{-rT}$ for $D$.
• Implied forward. Equivalently, extract the forward price $F$ of the stock: $F = K + e^{rT}(C - P)$. For multiple strikes, the forward should be the same — a cross-check on market-maker quotes.

Worked example

$S_0 = 100$, $r = 0.05$, $T = 1$, $K = 100$, no dividends.

Market quotes: $C = 10.45$, $P = 5.57$.

Check: $C - P = 4.88$. $S_0 - Ke^{-rT} = 100 - 100 e^{-0.05} = 100 - 95.12 = 4.88$. ✓

If instead the market quoted $C = 11.00$, $P = 5.00$: $C - P = 6.00$, but $S_0 - Ke^{-rT} = 4.88$. Arbitrage of $6.00 - 4.88 = 1.12$ per unit:

• Sell the expensive combo (long call + short put) for $+6.00$.
• Buy the cheap combo (long stock + borrow $95.12$) for $+4.88$ (net inflow after borrowing).
• Net inflow today: $6.00 - 4.88 = 1.12$ per unit.
• At expiry: payoff from combo A is $S_T - K$; payoff from combo B is $K - S_T$ (opposite sign because we bought this one). They cancel.
• Locked-in profit: \1.12$ today, no further cash flows.

Common confusions and pitfalls

• Only European, not American. American puts can be optimally exercised early; the payoff "in every state" argument fails.
• Dividend handling. Careless discounting of dividends is the single most common source of small pricing errors.
• Settlement conventions. Physical vs cash settlement, early vs late delivery — all enter the discount factor. For exchange-traded options with one standard expiry day the details usually don't matter; for OTC bespoke contracts they do.
• Short-sale constraints. The arbitrage proof assumes you can short the stock. In practice short-sale constraints on illiquid names can cause put-call parity violations of several percent — which is not a pricing mistake but a real cost of shorting.
• Synthetic put is not a perfect put. "Long call + short stock + cash" replicates a put's payoff at expiry, but the in-between hedging dynamics differ. For American or early-exercised products this matters.

Where this goes next

• Black-Scholes formula — one consistency check on any BS implementation.
• Implied vol curves — put and call implied vols at the same strike must agree under parity; deviations are data errors or dividend-yield errors.
• American options — where parity weakens to a bound.