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In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical
strike price In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set ...
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 In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties (especially cash flows). This is meant in two distinct senses: static replication, where the portfolio has t ...
, and thus requires minimal assumptions, namely the existence 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 The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black� ...
, 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 Frictionless can refer to: * Frictionless market * Frictionless continuant * Frictionless sharing * Frictionless plane The frictionless plane is a concept from the writings of Galileo Galilei. In his 1638 '' The Two New Sciences'', Galileo prese ...
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 is the (current) value of a call, P is the (current) value of a put, D is the discount factor, F is the forward price of the asset, and K is the strike price. Note that the spot price is given by D\cdot F = S (spot price is present value, forward price is future value, discount factor relates these). The left side corresponds to a portfolio of long a call and short a put, while the right side corresponds to a forward contract. The assets C and P on the left side are given in current values, while the assets F and K are given in future values (forward price of asset, and strike price paid at expiry), which the discount factor D converts to present values. Using spot price S instead of forward price F yields: : Rearranging the terms yields a different interpretation: : In this case the left-hand side is a fiduciary call, which is long a call and enough cash (or bonds) to pay the strike price if the call is exercised, while the right-hand side is a
protective put A protective option or married option is a financial transaction in which the holder of securities buys a type of financial options contract known as a " call" or a " put" against stock that they own or are shorting. The buyer of a protective o ...
, which is long a put and the asset, so the asset can be sold for the strike price if the spot is below strike at expiry. Both sides have payoff ''max''(''S''(''T''), ''K'') at expiry (i.e., at least the strike price, or the value of the asset if more), which gives another way of proving or interpreting put–call parity. In more detail, this original equation can be stated as: : where :C(t) is the value of the call at time t, :P(t) is the value of the put of the same expiration date, :S(t) is the spot price of the underlying asset, :K is the strike price, and :B(t,T) is the present value of a
zero-coupon bond A zero coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. Unlike regular bonds, it does not make periodic interest payments or have so-called coupons, hence the term zero- ...
that matures to $1 at time T. This is the present value factor for K. Note that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price ''K''. Thus one way to read the equation is that a portfolio that is long a call and short a put is the same as being long a forward. In particular, if the underlying is not tradeable but there exists forwards on it, we can replace the right-hand-side expression by the price of a forward. If the bond
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
, r, is assumed to be constant then : Note: r refers to the force of interest, which is approximately equal to the effective annual rate for small interest rates. However, one should take care with the approximation, especially with larger rates and larger time periods. To find r exactly, use r = \ln (1+i) , where i is the effective annual interest rate. When valuing European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes: : where D(t) represents the total value of the dividends from one stock share to be paid out over the remaining life of the options, discounted to
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
. We can rewrite the equation as: : and note that the right-hand side is the price of a forward contract on the stock with delivery price ''K'', as before.


Derivation

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. First, note that under the assumption that there are no
arbitrage In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between t ...
opportunities (the prices are
arbitrage-free In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between the ...
), two portfolios that always have the same payoff at time T must have the same value at any prior time. To prove this suppose that, at some time ''t'' before ''T'', one portfolio were cheaper than the other. Then one could purchase (go long) the cheaper portfolio and sell (go short) the more expensive. At time ''T'', our overall portfolio would, for any value of the share price, have zero value (all the assets and liabilities have canceled out). The profit we made at time ''t'' is thus a riskless profit, but this violates our assumption of no arbitrage. We will derive the put-call parity relation by creating two portfolios with the same payoffs (
static replication In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties (especially cash flows). This is meant in two distinct senses: static replication, where the portfolio has t ...
) and invoking the above principle ( rational pricing). Consider a call option and a put option with the same strike ''K'' for expiry at the same date ''T'' on some stock ''S'', which pays no dividend. We assume the existence of a
bond Bond or bonds may refer to: Common meanings * Bond (finance), a type of debt security * Bail bond, a commercial third-party guarantor of surety bonds in the United States * Chemical bond, the attraction of atoms, ions or molecules to form chemical ...
that pays 1 dollar at maturity time ''T''. The bond price may be random (like the stock) but must equal 1 at maturity. Let the price of ''S'' be S(t) at time t. Now assemble a portfolio by buying a call option ''C'' and selling a put option ''P'' of the same maturity ''T'' and strike ''K''. The payoff for this portfolio is ''S(T) - K''. Now assemble a second portfolio by buying one share and borrowing ''K'' bonds. 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 t, 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 ''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.


History

Forms of put-call parity appeared in practice as early as medieval ages, and was formally described by a number of authors in the early 20th century. 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 Russell Risley Sage (August 4, 1816 – July 22, 1906) was an American financier, railroad executive and Whig politician from New York. As a frequent partner of Jay Gould in various transactions, he amassed a fortune. Olivia Slocum Sage, his se ...
used put-call parity to create synthetic loans, which had higher interest rates than the usury laws of the time would have normally allowed. 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. His book was re-discovered by Espen Gaarder Haug in the early 2000s and many references from Nelson's book are given in Haug's book "Derivatives Models on Models". Henry Deutsch describes the put-call parity in 1910 in his book "Arbitrage in Bullion, Coins, Bills, Stocks, Shares and Options, 2nd Edition". London: Engham Wilson but in less detail than Nelson (1904). Mathematics professor
Vinzenz Bronzin Vinzenz Bronzin (1872 in Rovigno – 1970 in Trieste) was an Italian mathematics professor, known today for an early ("rediscovered") option pricing formula, similar to, and predating, the Black–Scholes 1973 formula; he also provided a formula ...
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 work of professor Bronzin was just recently rediscovered by professor Wolfgang Hafner and professor Heinz Zimmermann. 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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 i ...
). Its first description in the modern academic literature appears to be by Hans R. Stoll in the '' Journal of Finance''. Cited for instance in


Implications

Put–call parity implies: * ''Equivalence of calls and puts'': Parity implies that a call and a put can be used interchangeably in any
delta-neutral In finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security. Such a portfolio typically contains options and their c ...
portfolio. If d is the call's delta, then buying a call, and selling d shares of stock, is the same as selling a put and selling 1 - d shares of stock. Equivalence of calls and puts is very important when trading options. * ''Parity of implied volatility'': In the absence of dividends or other costs of carry (such as when a stock is difficult to borrow or sell short), the implied volatility of calls and puts must be identical.


See also

* Spot-future parity *
Vinzenz Bronzin Vinzenz Bronzin (1872 in Rovigno – 1970 in Trieste) was an Italian mathematics professor, known today for an early ("rediscovered") option pricing formula, similar to, and predating, the Black–Scholes 1973 formula; he also provided a formula ...


References


External links

*Put-Call parity
Put-call parity
tutorial by
Salman Khan (educator) Salman Amin Khan (born October 11, 1976), commonly known as Sal Khan, is an American educator and the founder of Khan Academy, a free online non-profit educational platform and an organization with which he has produced over 6,500 vi ...

Put-Call Parity and Arbitrage Opportunity
investopedia.com
The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage
Michael Knoll's history of Put-Call Parity *Other arbitrage relationships

Prof. Thayer Watkins
Rational Rules and Boundary Conditions for Option Pricing
( PDFDi), Prof. Don M. Chance
No-Arbitrage Bounds on Options
Prof. Robert Novy-Marx *Tools

Prof. Campbell R. Harvey {{DEFAULTSORT:Put-call parity Finance theories Mathematical finance Options (finance) Arbitrage