# Put–call parity

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Put–call parity

In financial mathematics, put-call parity defines a relationship between the price of a call option and a put option—both with the identical strike price and expiry. To derive the put-call parity relationship, the assumption is that the options are not exercised before expiration day, which necessarily applies to European options. Put-call parity can be derived in a manner that is largely model independent.

Derivation

An example using stock options follows, though this may be generalised to other options.

Consider a call option and a put option with the same strike K for expiry at the same date T on some stock, which pays no dividend. Let S denote the (unknown) underlying value at expiration.

First consider a portfolio that consists of one put option and one share. This portfolio at time T has value: $\left\{^\left\{K if S<=K \left(the put has value \left(K-S\right) and the share has value S\right)\right\}_\left\{S if S>=K \left(the put has value 0 and the share has value S\right)\right\}$

Now consider a portfolio that consists of one call option and K bonds that each pay 1 (with certainty) at time T. This portfolio at T has value: $\left\{^\left\{K if S<=K \left(the call has value 0 and the bonds have value K\right)\right\}_\left\{S if S>=K \left(the call has value S-K and the bonds have value K\right)\right\}$

Notice that, whatever the final share price S is at time T, each portfolio is worth the same as the other. This implies that these two portfolios must have the same value at any time t before T. To prove this suppose that, at some time t, one portfolio were cheaper than the other. Then one could purchase (go long) the cheaper portfolio and sell (go short) the more expensive. Our overall portfolio would, for any value of the share price, have zero value at T. We would be left with the profit we made at time t. This is known as a risk-less profit and represents an arbitrage opportunity.

Thus the following relationship exists between the value of the various instruments at a general time $t$:

:$C\left(t\right) + K cdot B\left(t,T\right) = P\left(t\right)+S\left(t\right) ,$

where:$C\left(t\right)$ is the value of the call at time $t$,:$P\left(t\right)$ is the value of the put,:$S\left(t\right)$ is the value of the share,:$K$ is the strike price, and:$B\left(t,T\right)$ value of a bond that matures at time $T$. If a stock pays dividends, they should be included in $B\left(t,T\right)$, because option prices are typically not adjusted for ordinary dividends.

If the bond interest rate, $r$, is assumed to be constant then:$B\left(t,T\right) = e^\left\{-r\left(T-t\right)\right\} ,$.

Using the above, and given no arbitrage opportunities, for any three prices of the call, put, bond and stock one can compute the implied price of the fourth.

When valuing European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes:

:$D\left(t\right) + C\left(t\right) + K cdot B\left(t,T\right) = P\left(t\right)+S\left(t\right) ,$

Where D(t) represents the present value of the dividends to be paid out before expiration of the option.

History

Nelson, an option arbitrage trader in New York, published a book: "The ABC of Option Arbitrage" in 1904 that describes the put-call parity in detail. His book was re-discovered by Espen Haug in the early 2000 and many references from Nelson's book is 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: Eﬃngham Wilson but in less detail than Nelson (1904).

Mathematics professor Vinzenz Bronzin also derives the put-call parity in 1908 and use 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 a edited work by Hafner and Zimmermann (Vinzenz Bronzin's option pricing models, Springer Verlag).

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

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.

Its first description in the "modern" literature appears to be Hans Stoll's paper, "The Relation Between Put and Call Prices", from 1969.

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 portfolio. If $d$ is the call's delta, then buying a call, and selling $d$ shares of stock, is the same as buying a put and buying $1 - d$ 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.

Other arbitrage relationships

Note that there are several other (theoretical) properties of option prices which may be derived via arbitrage considerations. These properties define price limits, the relationship between price, dividends and the risk free rate, the appropriateness of early exercise, and the relationship between the prices of various types of options. See links below.

Put-call Parity and American Options

For American options, where you have the right to exercise before expiration, this affects the B(t, T) term in the above equation. Put-call parity only holds for American options, if they are not exercised early.

c + PV(x) = p + s

*Put-Call parity c + PV(x) = p + s
** [http://www.quantnotes.com/fundamentals/options/putcallparity.htm Put-Call Parity Relationship] , quantnotes.com
** [http://www.investopedia.com/articles/optioninvestor/05/011905.asp Put-Call Parity and Arbitrage Opportunity] , investopedia.com
** [http://lsr.nellco.org/upenn/wps/papers/49/ The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage] , Michael Knoll's history of Put-Call Parity
*Other abitrage relationships
** [http://www.sjsu.edu/faculty/watkins/arb.htm Arbitrage Relationships for Options] , Prof. Thayer Watkins
** [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN99-05.pdf Rational Rules and Boundary Conditions for Option Pricing] (PDFDi), Prof. Don M. Chance
*Tools
** [http://www.duke.edu/~charvey/Classes/ba350/optval/arbitrage/arbitrage.htm Option Arbitrage Relations] , Prof. Campbell R. Harvey

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