The section on PAT in a two-asset environment introduced the idea of financially equivalent payoffs on the opposite sides of different markets, and the use of these financially equivalent payoffs as alternatives in the place of 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. 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 allowing risk management professionals the ability to view the value of their multi-asset portfolios from a single asset perspective.

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

• PAT in a three-asset environment provides new true arbitrage opportunities by identifying financially equivalent payoffs that do not seem to be equal if PAT is not used to identify them. These could be, for example, a single put option, a single call option, and a spread option. 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, in this situation it is more economical for them to hedge their positions using alternative options that are financially equivalent to the originally desired options but that cost less. We note that in a three-asset environment there is a greater choice, as there are more alternative 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 opposite sides of different markets. If a model used independently to generate option prices for these financially equivalent payoffs does not provide financially equivalent results, this reveals an inconsistency in the model that must be fixed. For example, a spread option on assets 1 and 2 should have a value that is financially equivalent to either one of its financially equivalent options: a put, denominated in Asset 1, on Asset 2, and a call, denominated in Asset 2, on Asset 1.

• PAT also provides a unifying way to manage an entire comprehensive portfolio by enabling a manager to convert various different payoffs in different assets back to one single payoff in the domestic asset. In this way, the different possible risk/reward scenarios can be viewed and hedged from the viewpoint of a single (domestic) asset to provide maximum profitability. For example, electric power producers could view their input (Natural Gas), output (electricity) and the currency of the markets (US Dollar) from the perspective of any one of these wealth units. In this way, power producers can have a clear picture of their potential risks and profits. It also becomes possible for them to hedge their positions from the perspective of their electric power producing capacity. Trying to do this based on various statistical models can be very misleading. But since there are no limits to the number of assets that PAT can handle, this feature could be very useful for electric power producers dealing with numerous different assets and risks such as input fuels, electrical energy, weather derivatives, BTU markets, etc.

Again, there are probably other uses for PAT that are not included in this section. PAT is a revolutionary new tool for the fields of finance and risk management, and it will continue to evolve and provide additional benefits as a tool based upon an underlying symmetry in the marketplace. We believe that traders and risk managers will be constantly finding new applications for PAT.