Fx option put call parity
Both sides have payoff max S T , K 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-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.
The investor on the other side of the trade is in effect selling a put option on the currency. To eliminate residual risk, match the foreign currency notionals, not the local currency notionals, else the foreign currencies received and delivered don't offset. Corporations primarily use FX options to hedge uncertain future cash flows in a foreign currency. The general rule is to hedge certain foreign currency cash flows with forwards , and uncertain foreign cash flows with options.
This uncertainty exposes the firm to FX risk. This forward contract is free, and, presuming the expected cash arrives, exactly matches the firm's exposure, perfectly hedging their FX risk. If the cash flow is uncertain, a forward FX contract exposes the firm to FX risk in the opposite direction, in the case that the expected USD cash is not received, typically making an option a better choice.
As in the Black—Scholes model for stock options and the Black model for certain interest rate options , the value of a European option on an FX rate is typically calculated by assuming that the rate follows a log-normal process.
In Garman and Kohlhagen extended the Black—Scholes model to cope with the presence of two interest rates one for each currency. The results are also in the same units and to be meaningful need to be converted into one of the currencies. A wide range of techniques are in use for calculating the options risk exposure, or Greeks as for example the Vanna-Volga method.
Although the option prices produced by every model agree with Garman—Kohlhagen , risk numbers can vary significantly depending on the assumptions used for the properties of spot price movements, volatility surface and interest rate curves.
After Garman—Kohlhagen, the most common models are SABR and local volatility [ citation needed ] , although when agreeing risk numbers with a counterparty e. From Wikipedia, the free encyclopedia. Retrieved 21 September Energy derivative Freight derivative Inflation derivative Property derivative Weather derivative. Retrieved from " https: