The challenges of managing a multi-currency portfolio are enormously more complicated than the challenges associated with a two-currency portfolio. The reason is the increased number of variables at play. This creates the need for tools that are capable of maintaining their integrity in an environment with any level of complexity. PAT is a tool that can handle any number of currencies, working always by the same principle – identifying financially equivalent payoffs among any number of markets. This is because PAT works by the principle of underlying symmetry, not by statistical analysis.

In the previous section we showed how PAT functions in a two-currency environment. In this section we will explain how PAT functions in a three-currency environment. From there, we can project how PAT would function in an environment with any number of currencies. We will also present some of the benefits which PAT has to offer to financial professionals and risk managers.

In order to gain a deeper understanding of the mechanics of PAT in a three-currency environment, we will again present a series of graphs of European and American options. This time the graphs consist of three three-dimensional charts for Currency 1, Currency 2 and Currency 3. Payoffs in the vertical scale are presented in the units of respective currencies, with the exchange rates presented on the horizontal axes.

On each of the charts consisting of three 3-D graphs, three financially equivalent payoffs are presented - one in each of the currencies 1, 2, and 3. These payoffs can look very different, even more so than in the case of a two-currency environment. This is because of one more variable (currency), which makes the situation more complex. However, since PAT employs the principle of an underlying symmetry relationship between the currencies, each of these payoffs is financially equivalent. Any one of these three payoffs can be substituted for either one of the remaining two payoffs, leaving the risk/reward scenarios in each case absolutely unchanged.