# Binary option put call parity

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.

However, one should take care with the approximation, especially with larger rates and larger time periods. When valuing European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes:.

We can rewrite the equation as:. 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 opportunities the prices are arbitrage-free , 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 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 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. 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. 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.

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: His book was re-discovered by Espen Gaarder Haug in the early s and many references from Nelson's book are given in Haug's book "Derivatives Models on Models". 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 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 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. 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.

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.

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: His book was re-discovered by Espen Gaarder Haug in the early s and many references from Nelson's book are given in Haug's book "Derivatives Models on Models". Engham Wilson but in less detail than Nelson Mathematics professor Vinzenz Bronzin also derives the put-call parity in 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.

Its first description in the modern academic literature appears to be by Hans R. Stoll in the Journal of Finance. From Wikipedia, the free encyclopedia. Options, Futures and Other Derivatives 5th ed. Energy derivative Freight derivative Inflation derivative Property derivative Weather derivative. Retrieved from " https: Finance theories Mathematical finance Options finance.