Put-call parity is an important principle in options pricing first identified by Hans Stoll in his paper, The Relation Between Put and Call Prices, in 1969. It states that the premium of a call option implies a certain fair price for the corresponding put option having the same strike price and expiration date, and vice versa. Support for this pricing relationship is based upon the argument that arbitrage opportunities would materialize if there is a divergence between the value of calls and puts. Arbitrageurs would come in to make profitable, riskless trades until the put-call parity is restored.
To begin understanding how the put-call parity is established, let's first take a look at two portfolios, A and B. Portfolio A consists of a European call option and cash equal to the number of shares covered by the call option multiplied by the call's striking price. Portfolio B consists of a European put option and the underlying asset. Note that equity options are used in this example.
Portfolio A = Call + Cash, where Cash = Call Strike Price
Portfolio B = Put + Underlying Asset
It can be observed from the diagrams above that the expiration values of the two portfolios are the same.
Call + Cash = Put + Underlying Asset
Eg. JUL 25 Call + $2500 = JUL 25 Put + 100 XYZ Stock
Under the idealized no-arbitrage assumptions, equal future cashflows imply equal present values. Per share, C + K·e^(−rT) = P + S for European options on a non-dividend-paying stock, using a consistent continuously compounded rate r and time T. With known cash dividends, replace S with S minus their present value. Actual costs, stock borrowing, funding and execution constrain how a discrepancy could be traded.
Put-Call Parity and American Options
American exercise rights can alter prices before expiration, so the simple European equality is not a universal American-option pricing rule. No-arbitrage bounds and the value of early exercise must be considered. Choosing to hold an American option does not remove the exercise rights embedded in its market price. The matched intrinsic-payoff identity still holds at expiration.
Validating Option Pricing Models
Put–call parity is a useful consistency check when the model and comparison satisfy the same assumptions. Adjust the test for dividends, rates, exercise rights and contract terms; a model should not be called flawed merely for failing an inapplicable European no-dividend equality.
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A present-value example
For compatible European options, a standard relationship with known cash dividends is C − P = S − PV(dividends) − PV(K). Assume stock is $100, strike $100, the present value of strike cash is $98 and the present value of dividends before expiration is $1. Then C − P = $1 per share. If the call is $6, a parity-consistent put is $5 under those assumptions.
Ignoring the dividend term would instead suggest $2, creating an apparent discrepancy caused by an omitted input. Present values depend on timing and discount conventions; a forecast dividend is not the same as a known payment.
The equality compares complete cash flows under model conditions. Trading an apparent difference requires executable bid/ask prices on every leg, financing, stock borrow, fees and compatible settlement. American early exercise changes the simple equality’s applicability; do not treat a quoted violation as guaranteed arbitrage.
For a standard contract, the $1 per-share difference is $100. For adjusted contracts, construct the actual deliverable and exercise cash before applying a familiar formula. An option that delivers shares plus cash cannot be valued as if it necessarily delivers 100 ordinary shares.
