# Put call parity interest rate options trading

In financial mathematicsput—call parity defines a relationship between the price of a European call option and European put optionboth with the identical strike price 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 **put call parity interest rate options trading** 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. Put—call parity is a static replicationand 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.

These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the Black—Scholes put call parity interest rate options tradingwhich requires dynamic replication and continual transaction in the underlying. Replication put call parity interest rate options trading one can enter into derivative transactions, which requires leverage and capital costs to back thisand buying and selling entails transaction costsnotably the bid-ask spread.

The relationship thus only holds exactly in an ideal frictionless market 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. 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 **put call parity interest rate options trading** price of asset, and strike price paid at expirywhich the discount factor D converts to present values. In this case the left-hand side is a fiduciary callwhich 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 putwhich is long a put and the asset, so the asset can be sold for the strike price if the spot is below put call parity interest rate options trading at expiry.

Both sides have payoff max S TPut call parity interest rate options trading at expiry i. 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. Put call parity interest rate options trading, 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-freetwo 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 Tone portfolio were cheaper than the other. Then one could purchase go long the cheaper portfolio and sell go short the more expensive.

At time Tour 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 put call parity interest rate options trading 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 Swhich 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 Tsince 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 parityholds, and for any three prices of the call, put call parity interest rate options trading, 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 Tthe 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 put call parity interest rate options trading in the early 20th century. The Early History of Regulatory Arbitragedescribes the important role that put-call parity played in developing the equity of redemptionthe 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.

All articles with unsourced statements Articles with unsourced statements from June Articles with unsourced statements from July Views Read Edit View history. This page was last edited on 4 Aprilat By using this site, you agree to the Terms of Use and Privacy Policy.

**Put call parity interest rate options trading** financial mathematicsput—call parity defines a relationship between the price of a European call option and European put optionboth with the identical strike price 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, put call parity interest rate options trading thus in either case one unit of the asset will be purchased for the put call parity interest rate options trading price, put call parity interest rate options trading 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. Put—call parity is a static replicationand thus put call parity interest rate options trading 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. These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the Black—Scholes modelwhich 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 thisand buying and selling entails transaction costsnotably the bid-ask spread. The relationship thus only holds exactly in an ideal frictionless market 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.

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 expirywhich the discount factor D converts to present values. In this case the put call parity interest rate options trading side is a fiduciary callwhich 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 putwhich 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 TK at expiry i. 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.

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-freetwo 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 Tone portfolio were cheaper than the other.

Then one could purchase go long the cheaper portfolio and sell go short the more expensive. At time Tour overall portfolio would, for any value of the share price, have zero value all the assets and liabilities have canceled out. The put call parity interest rate options trading 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 Swhich 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 put call parity interest rate options trading S T - K at time Tsince 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 parityholds, 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 put call parity interest rate options trading 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 Tthe stock is not only worth S T but **put call parity interest rate options trading** 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 Arbitragedescribes the important role that put-call parity played in developing the equity of redemptionthe 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 put call parity interest rate options trading 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. All articles with unsourced statements Articles with unsourced statements from June Articles with unsourced statements from July Views Read Edit View history.

This put call parity interest rate options trading was last edited on 13 Julyat By using this site, you agree to the Terms of Use and Privacy Policy.

A **put call parity interest rate options trading** consisting of stock and a protective put on the stock establishes a minimum amount of value for the portfolio that also has an unlimited upside potential. If the stock declines below the strike of the put, the put increases in value by a dollar for every dollar decline of the stock below the strike price.

If the stock climbs above the strike price, the put expires worthless, leaving only the stock. An equivalent portfolio can be established by buying a call and a risk-free T-bill that matures on the expiration day of the call. Because a portfolio that has a minimum value but unlimited upside potential can be established using either a put or a call, then they must be equivalent, because if they weren't, then arbitrageurs could take advantage of the price discrepancy, writing one option and buying the other until equivalence was achieved.

Similarly, the portfolio consisting of stock and a protective put would have as its minimum value the strike price of the put. Thus, if the 2 portfolios provide equal values, then they should cost the same to establish—otherwise, arbitrageurs would profit from the difference until the difference fell below trading costs, buying 1 option and selling the other.

Arbitrageurs would not have much effect on the stock price or the interest rate of the T-bill, but they would have an effect on the prices of the options, and it is this effect that would equilibrate the prices.

Thus, the stock plus put must equal the T-bill plus call. This leads put call parity interest rate options trading the put call parity interest rate options trading equation, called the put-call parity theorem:. Options are not exercised before expiration day, and stocks do not pay dividends before expiration. This relationship assumes that no dividends are paid by the stock before expiration of the put or call. However, the put-call parity equation can be extended to include dividends, if the options are European style or are held to maturity:.

Note that if the stock pays no dividends before expiration, then this equation is equivalent to the equation for the put-call parity. It can also be seen put call parity interest rate options trading in this equation that dividends increase the put premiums and decrease call premiums.

These equations assume that the options are not exercised before expiration; otherwise, the payoff of the portfolios will probably differ. Lowest Margin Interest that I could find: We can then profit from what is known as conversion arbitrage. We sell the left hand of the equation and buy the right for an immediate profit. Note that the profit is the same regardless of what the stock price is on expiration — thus, this is risk-free arbitrage. Let's change the put value in Example 1 to 1. This arbitrage is called a reverse conversionbecause it is basically the reverse of a conversion.

Now we want to buy the left side of the put-call parity equation and sell the right side. Note that the call covers the shorted stock if the stock rises above the strike, and the put is covered by the shorted stock, if the stock price is less than the strike, which explains S t - The put-call parity is used to determine the theoretical price of a put from the Black-Scholes formulaa widely used method to determine the theoretical price of calls.

Options are not exercised before expiration day.