Put-Call Parity

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Put-Call Parity

Key takeaways

  • Put-call parity describes a precise price relationship between European call and put options that share the same underlying asset, strike price, and expiration.
  • The standard parity equation is C + PV(X) = P + S, where PV(X) is the present value of the strike discounted at the risk-free rate.
  • Deviations from parity indicate arbitrage opportunities; in practice they are rare because markets and algorithms quickly close them.
  • Adjustments are required for dividends and other carry costs.

What put-call parity is

Put-call parity is a fundamental options-pricing identity for European options. It states that a portfolio made of a long call and a risk-free bond that pays the strike at expiration has the same value as a portfolio made of a long put and one share of the underlying. If the equality holds, the market is in equilibrium; if not, arbitrageurs can construct offsetting positions to lock in a risk-free profit.

Formula

C + PV(X) = P + S

Where:
* C = price of the European call
* P = price of the European put
* S = current spot price of the underlying
* PV(X) = present value of the strike price X, discounted at the risk-free rate to today

Equivalently, C − P = S − PV(X).

Intuition

A long call plus cash equal to PV(X) replicates the payoff of holding the underlying stock but with limited downside. Similarly, holding the stock and a long put (a protective put) also creates the same terminal payoff. Because both constructed portfolios have identical payoffs in every state at expiration, they must have the same price today; that is put-call parity.

Simple numeric example

Suppose:
* S = $50
* Strike X = $55
* PV(X) = $54.46 (present value of $55 discounted at risk-free rate)
* C = $3

Apply parity:
C + PV(X) = 3 + 54.46 = 57.46
Therefore P = 57.46 − S = 57.46 − 50 = $7.46

So the fair price of the six-month European put is $7.46.

Arbitrage when parity is violated

If market prices diverge, construct positions that sell the expensive side and buy the cheap side. Two cases:

  1. If C + PV(X) < P + S (puts and/or stock are expensive relative to calls)
  2. Arbitrage trade: sell the put, short the stock, buy the call, invest PV(X) in the risk-free bond.

  3. Net initial cash inflow = P + S − C − PV(X) > 0. At expiration payoffs cancel and the initial inflow is the arbitrage profit (ignoring costs).

  4. If C + PV(X) > P + S (calls and/or bond are expensive relative to puts and stock)

  5. Arbitrage trade: buy the put, buy the stock, sell the call, finance by borrowing PV(X).
  6. Initial cash inflow exists and net payoffs at expiration eliminate risk.

Worked mismatch example (from above): if P trades at $8 instead of $7.46, then
* Receive $8 selling the put, receive $50 shorting the stock, pay $3 to buy the call, pay $54.46 to buy the bond. Net inflow = 8 + 50 − 3 − 54.46 = $0.54. That $0.54 is a risk-free profit at inception (ignoring transactions, margin, and taxes).

Note: transaction costs, bid-ask spreads, margin requirements, and dividend uncertainty often eliminate small theoretical arbitrage profits in practice.

Dividends and interest-rate effects

If the underlying pays known dividends, expected dividends reduce the underlying’s forward price and must be included when forming the parity relation. A common adjusted form is:
C − P = S − PV(X) − PV(dividends)
Higher interest rates raise PV(X) discount effects and change relative call/put values because of the cost of carry.

Protective put vs fiduciary call

  • Protective put: long stock + long put. It provides downside protection for holding the stock.
  • Fiduciary call: long call + cash equal to PV(X). It ensures funds are available to exercise the call at expiration.
    Both strategies yield identical payoffs at expiration, illustrating put-call parity: protective put = fiduciary call.

Conclusion

Put-call parity is a cornerstone of options theory that links European call and put prices to the underlying and the present value of the strike. It provides a practical way to detect mispricings and to understand equivalences between option-based strategies. In real markets, small deviations can appear briefly, but trading frictions and dividend/interest considerations determine whether theoretical arbitrage can be profitably realized.