In financial mathematics, the put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to (and hence has the same value as) a single forward contract at this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a forward contract. The validity of this relationship requires that certain assumptions be satisfied; these are specified and the relationship is derived below. In practice transaction costs and financing costs (leverage) mean this relationship will not exactly hold, but in liquid markets the relationship is close to exact.
Assumptions Put–call parity is a static replication, and thus requires minimal assumptions, of a forward contract. In the absence of traded forward contracts, the forward contract can be replaced (indeed, itself replicated) by the ability to buy the underlying asset and finance this by borrowing for fixed term (e.g., borrowing bonds), or conversely to borrow and sell (short) the underlying asset and loan the received money for term, in both cases yielding a self-financing portfolio. These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the Black–Scholes model, which requires dynamic replication and continual transaction in the underlying. Replication assumes one can enter into derivative transactions, which requires leverage (and capital costs to back this), and buying and selling entails transaction costs, notably the bid–ask spread. The relationship thus only holds exactly in an ideal frictionless market with unlimited liquidity. However, real world markets may be sufficiently liquid that the relationship is close to exact, most significantly FX markets in major currencies or major stock indices, in the absence of market turbulence.
Statement Put–call parity can be stated in a number of equivalent ways, most tersely as:
where C {\displaystyle C} is the (current) value of a call, P {\displaystyle P} is the (current) value of a put, D {\displaystyle D} is the discount factor, F {\displaystyle F} is the forward price of the underlying asset, and K {\displaystyle K} is the strike price. The left side corresponds to a portfolio of a long call and a short put; the right side corresponds to a forward contract. The assets C {\displaystyle C} and P {\displaystyle P} on the left side are given in present values, while the assets F {\displaystyle F} and K {\displaystyle K} are given in future values (forward price of asset, and strike price paid at expiry), which the discount factor D {\displaystyle D} converts to present values. Now the spot price S = D ⋅ F {\displaystyle S=D\cdot F} can be obtained by discounting the forward price F {\displaystyle F} by the factor D {\displaystyle D} . Using spot price S {\displaystyle S} instead of forward price F {\displaystyle F} gives us:
Rearranging the terms gives a first interpretation:
Here the left-hand side is a fiduciary call, which is a long call and enough cash (or bonds) to exercise it by paying the strike price. The right-hand side is a Married put, which is a long put paired with the asset, so that the asset can be sold at the strike price on exercise. At expiry, the intrinsic value of options vanish so both sides have payoff max ( K , S ) {\displaystyle \max(K,S)} equal to at least the strike price K {\displaystyle K} or the value S {\displaystyle S} of the asset if higher. That a long call with cash is equivalent to a long put with asset is one meaning of put-call parity. Rearranging the terms another way gives us a second interpretation:
Now the left-hand side is a cash-secured put, that is, a short put and enough cash to give the put owner should they exercise it. The right-hand side is a covered call, which is a short call paired with the asset, where the asset stands ready to be called away by the call owner should they exercise it. At expiry, the previous scenario is flipped. Both sides now have payoff min ( K , S ) {\displaystyle \min(K,S)} equal to either the strike price K {\displaystyle K} or the value S {\displaystyle S} of the asset, whichever is lower. So we see that put-call parity can also be understood as the equivalence of a cash-secured (short) put and a covered (short) call. This may be surprising as selling a cash-secured put is typically seen as riskier than selling a covered call. To make explicit the time-value of cash and the time-dependence of financial variables, the original put-call parity equation can be stated as:
where
C ( t ) {\displaystyle C(t)} is the value of the call at time t {\displaystyle t} ,
P ( t ) {\displaystyle P(t)} is the value of the put of the same expiration date,
S ( t ) {\displaystyle S(t)} is the spot price of the underlying asset,
K {\displaystyle K} is the strike price, and
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