# Binary option put call parity

Scholes model, which requires dynamic replication and continual transaction in the underlying. The relationship thus only holds exactly in an ideal frictionless market with unlimited liquidity. D is the discount factor, F is the forward price of the asset, and K is the strike price. The binary option put call parity side corresponds to a portfolio of long a call and short a put, while the right side corresponds to a forward contract.

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. However, one should take care with the approximation, especially with larger rates and larger time periods. 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. 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. 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 that pays 1 dollar at maturity 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. Now assemble a second portfolio by buying one share and borrowing K bonds. 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. K bonds that each pay 1 dollar at T.

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 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. 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”. 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 work of professor Bronzin was just recently rediscovered by professor Wolfgang Hafner and professor Heinz Zimmermann.

Its first description in the modern academic literature appears to be by Hans R. Stoll in the Journal of Finance. Equivalence of calls and puts: Parity implies that a call and a put can be used interchangeably in any delta-neutral portfolio. Equivalence of calls and puts is very important when trading options. The Relationship Between Put and Call Option Prices”. Put-Call Parity of European Options, putcallparity. Put-Call Parity and Arbitrage Opportunity, investopedia.