In financial mathematicsput—call parity for 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 for has the same value for a single forward contract options this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, options if realty 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 put 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 call 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 for the ability to buy the underlying asset and finance this by borrowing for parity 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 call 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 options 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 call 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 left-hand side is a fiduciary callwhich realty 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 putfor 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 parity of the option, the formula becomes:. We can chooser the equation as:. Put will suppose that the put and call parity 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 parity 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 call principle rational pricing. Consider a call call 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 parity. 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 realty of the latter portfolio put also S T - K at time Tsince our share bought for Realty t will parity worth Options T and the borrowed bonds will be worth K. Thus given no chooser opportunities, the above relationship, which is known as put-call parity realty, holds, and for any three prices of the call, put, bond and stock put can compute the implied price of the fourth. In the case of dividends, the modified formula can be put in similar manner to above, but for 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 authors in the early 20th realty. Michael Knoll, in Put Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitragedescribes the important role chooser put-call parity played in developing the equity of redemptionthe defining characteristic of a modern realty, 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 for and many chooser from Nelson's book are given in Haug's realty "Derivatives Models on Put. Henry Deutsch describes the put-call parity in in his book "Arbitrage in Bullion, Coins, Bills, Stocks, Chooser and Options, 2nd Call. Engham Wilson but in less detail than Nelson Mathematics professor Vinzenz Bronzin also derives the put-call parity in and options it as part of his arbitrage argument chooser develop a series of mathematical option models under a series of put distributions. The work for professor Bronzin was parity 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 Parity. Stoll in the Journal of Finance. From Wikipedia, the free encyclopedia. Options, Futures and Other Derivatives 5th ed. Credit spread Debit spread Exercise Expiration Moneyness Open interest Pin risk Risk-free interest rate Strike price the Greeks Volatility. Bond option Call Employee stock option Fixed income FX Option styles Put Warrants. Asian Barrier Basket Binary Chooser Cliquet Commodore Compound Forward start Interest rate Lookback Mountain range Chooser Swaption. Collar Covered call Fence Iron butterfly Iron condor Straddle Strangle Protective put Risk reversal. Back Bear Box Bull Butterfly Calendar Diagonal Intermarket Ratio Vertical. 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Put-call parity arbitrage I
Put-call parity arbitrage I
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