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.
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.
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.
Put–call parity can be stated in a number of equivalent ways, most tersely as: where
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.
At expiry, the intrinsic value of options vanish so both sides have payoff
[1] 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 Note that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price
We will suppose that the put and call options are on traded stocks, but the underlying can be any other tradeable asset.
The ability to buy and sell the underlying is crucial to the "no arbitrage" argument below.
Then one could purchase (go long) the cheaper portfolio and sell (go short) the more expensive one.
We will derive the put-call parity relation by creating two portfolios with the same payoffs (static replication) and invoking the above principle (rational pricing).
Note the payoff of the latter portfolio is also S(T) - K at time T, since our share bought for S(t) will be worth S(T) and the borrowed bonds will be worth K. By our preliminary observation that identical payoffs imply that both portfolios must have the same price at a general time
, the following relationship exists between the value of the various instruments: Thus given no arbitrage opportunities, the above relationship, which is known as put-call parity, holds, and for any three prices of the call, put, bond and stock one can compute the implied price of the fourth.
In the case of dividends, the modified formula can be derived in similar manner to above, but with the modification that one portfolio consists of going long a call, going short a put, and going long D(T) bonds that each pay 1 dollar at maturity T (the bonds will be worth D(t) at time t); the other portfolio is the same as before - long one share of stock, short K bonds that each pay 1 dollar at T. The difference is that at time T, the stock is not only worth S(T) but has paid out D(T) in dividends.
Michael Knoll, in The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage, describes the important role that put-call parity played in developing the equity of redemption, the defining characteristic of a modern mortgage, in Medieval England.
In the 19th century, financier Russell Sage used put-call parity to create synthetic loans, which had higher interest rates than the usury laws of the time would have normally allowed.
[citation needed] Nelson, an option arbitrage trader in New York, published a book: "The A.B.C.
of Options and Arbitrage" in 1904 that describes the put-call parity in detail.
Henry Deutsch describes the put-call parity in 1910 in his book "Arbitrage in Bullion, Coins, Bills, Stocks, Shares and Options, 2nd Edition".
Mathematics professor Vinzenz Bronzin also derives the put-call parity in 1908 and uses it as part of his arbitrage argument to develop a series of mathematical option models under a series of different distributions.
The original work of Bronzin is a book written in German and is now translated and published in English in an edited work by Hafner and Zimmermann ("Vinzenz Bronzin's option pricing models", Springer Verlag).
Its first description in the modern academic literature appears to be by Hans R. Stoll in the Journal of Finance.