In financial mathematics, putcall 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 putcall parity relationship, the assumption is that the options are not exercised before expiration day, which necessarily applies to European options. Putcall parity can be derived in a manner that is largely model independent.
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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:
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:
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 riskless profit and represents an arbitrage opportunity.
Thus the following relationship exists between the value of the various instruments at a general time t:
where
If the bond interest rate, r, is assumed to be constant then
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:
Where D(t) represents the present value of the dividends to be paid out before expiration of the option.
Nelson, an option arbitrage trader in New York, published a book: "The ABC of Option Arbitrage" in 1904 that describes the putcall parity in detail. His book was rediscovered by Espen Gaarder 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 putcall 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 also derives the putcall 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).
Michael Knoll, in The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage, describes the important role that putcall parity played in developing the equity of redemption, the defining characteristic of a modern mortgage, in Medieval England
Russell Sage used putcall 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.
Putcall parity implies:
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.
For American options, where you have the right to exercise before expiration, this affects the B(t, T) term in the above equation. Putcall parity only holds for European options, or American options if they are not exercised early.
c + PV(x) = p + s

