The section on PAT in a two-currency environment introduced the idea of financially equivalent payoffs on the opposite sides of different markets, and also introduced the idea of using these financially equivalent payoffs as alternatives in the place of the originally desired payoffs. We saw that such financially equivalent payoffs may not always look equivalent, but are equivalent nevertheless.

This, we stated, could be a huge advantage for the market participants who adopt this technology. Currency traders could employ the true arbitrage opportunity, and risk management professionals could find alternate payoffs for hedging their operations. In addition we stated that PAT could be used for verifying the consistency of existing option pricing models and for helping money managers to view the value of their multi-currency portfolios from a single currency perspective.

Applying PAT in a three-currency environment will further extend these applications, providing risk-management and trading professionals with a greater choice in the process:

• PAT in a three-currency environment provides new true arbitrage opportunities by identifying financially equivalent payoffs that do not appear to be equal unless PAT is not used to identify them. These equivalent payoffs could be, for example, the payoffs on a single put option, a single call option, and a spread option (page 30). If such instruments are being traded at inconsistent values, traders can enter the arbitrage position by going short one and going long either one of the other two financially equivalent options, taking a profit in the process. As the payoffs are financially equal, they will cancel each other out at expiration. For risk-management professionals it is more economical to hedge their positions using alternate payoffs that are financially equivalent to the originally desired options but that cost less. We note that in a three-currency environment there is a greater choice, as there are more alternate but financially equivalent payoffs to select from.

• PAT provides a screen for testing the consistency of existing option pricing models by correctly identifying financially equivalent payoffs on the different sides of the exchange rate market. If a model used to independently generate option prices for financially equivalent options does not provide financially equivalent results, this reveals an inconsistency in the model which must be fixed. For example, a spread option on Currencies 1 and 2 should be priced equally with either one of its financially equivalent options: a put on Currency 2 denominated in Currency 1, and a call on Currency 1 denominated in Currency 2 (page 30).

• PAT also provides a unifying way to manage an entire comprehensive portfolio by enabling a manager to convert various different payoffs in different currencies back into one single payoff in the domestic currency. In this way, the different possible risk/reward scenarios can be viewed and hedged from the viewpoint of a single (domestic) currency to provide maximum profitability. For example, a trader could hold an option, denominated in Currency 3, on Currency 2. He might also be short an option on Currency 1 denominated in Currency 2. Let's say that his "domestic" currency – the currency in which he reports his profitability – is Currency 1. With the help of PAT, converting the entire portfolio into a single payoff denominated in the Currency 1 is a simple process. Trying to do this with various statistical models can be very misleading. Since there are no limits to the number of currencies that PAT can handle, this feature could be very useful for tracking the profitability of off-shore investments, or hedging the profitability of production operations when production is taking place in several different countries.

Again, there are probably other uses for PAT not included on this list. PAT is a revolutionary new tool for the fields of finance and risk management, and it will continue its evolution providing benefits as a tool based upon undelying symmetry. We believe that new applications for PAT will be coming up continuously.