Put-call parity

Definition

Put-call parity is the arbitrage identity linking a call, a put, the underlying, and financing: for European-style options, C − P = S − K·e^(−rT) − PV(dividends). It is model-free — no volatility assumption, no distribution assumption; only the absence of free money. Every synthetic construction, box spread, and conversion/reversal in this KB is parity rearranged, and parity DEVIATIONS are information: financing rates, borrow stress, and dividend expectations read off option prices.

How it works / structure

• The replication argument: long call + short put (same K, T) has exactly the forward payoff of the stock bought with borrowed money — two portfolios with identical payoffs must price identically, or the difference is riskless profit (Stoll 1969 formalized it).
• Rearrangements (engine-executable identities): synthetic stock = C − P (strategy-synthetic-stock); conversion = long stock + P − C (locks the parity spread); box spread = two-parity sandwich = pure financing (strategy-box-spread); protective put = call + cash (the insurance and the call are the same position).
• American-style caveat: early exercise breaks the strict equality into bounds — equity options (American) satisfy parity as an inequality band whose width is the early- exercise premium (opt-dividend-effects drives the call side).
• What deviations mean: implied financing above/below market rates (box rates), implied borrow cost on hard-to-borrow names (ms-short-locate-borrow — negative synthetic rates flag squeeze conditions), and implied dividend revisions ahead of declarations.

When it applies

Structure equivalence checking (the engine canonicalizes positions through parity — a “collar” and a “vertical plus cash” that are the same position get recognized as such); financing/borrow signal extraction; arbitrage sanity checks on quotes (violations at mid are usually stale/wide data, not free money — the friction band absorbs them).

Risk profile & failure modes

• Friction band illusions: apparent parity violations inside spread + fee + borrow costs are noise; naive scanners rediscover them daily (ms-bid-ask-spread).
• American-style misuse: applying strict parity to dividend-paying American equity options generates phantom arbitrages; the inequality form is the correct tool.
• Pin/settlement wrinkles: parity trades held to expiry cross assignment mechanics (ms-expiration-exercise-assignment).

Evidence & limits

Stoll (1969) formalized the relationship; subsequent empirical literature finds violations confined within transaction-cost bands in liquid markets — the identity’s real-world status is “enforced by arbitrage capital, minus friction.” As mathematics conditional only on no-arbitrage, it is the KB’s most secure pricing statement — more robust than any model in opt-pricing-models.

Falsifiable-thesis examples

Illustrations only, not signals:

• “X’s synthetic-implied financing rate (from 6-month ATM parity) will stay within 150bp of Treasury bills this quarter (no borrow stress)” — falsified by the implied-rate series.
• “The implied dividend in X’s parity relationship will rise ahead of next quarter’s declaration (a raise is priced)” — falsified by the implied-dividend series and the declaration.

Cross-references

• Constructions built on it: strategy-synthetic-stock, strategy-box-spread, strategy-collar
• The premium it prices without a model: contrast opt-pricing-models
• Signals in its deviations: ms-short-locate-borrow, opt-dividend-effects, greek-rho