The purpose of this workshop is to talk about some fundamental symmetries in foreign exchange markets. The symmetry relationships are established for the values of a wide range of foreign exchange options including European options with general payoffs, Bermudan and American options with general time-dependent payoffs, and barrier options with time-dependent, discretely and continuously activated barriers on general European, Bermudan and American options.

In the particular cases of European and American call and put options, these symmetry relationships take the form of a conjecture due to Grabbe (1983) that a call or put option with strike X on one side of a foreign exchange market is equivalent to a portfolio of X put or call options with strike 1/X on the opposite side of the market.

The symmetry relationships hold in a general foreign exchange market environment. In particular, not only is no assumption made on the nature of a probability distribution for exchange rates, we do not even assume the existence of such a distribution.

The symmetry for two-currency foreign exchange options is based on the symmetry at the level of payoffs that associate with any given payoff a financially equivalent payoff on the opposite side of the market. These equivalent payoffs represent the objective amount of wealth contingent upon the state of the market as expressed by the values of the exchange rates. It is this wealth that has an objective financial status and hence, using the terminology of modern theoretical physics, is a financial observable.

The concept of financial observables allow for a coordinate-free description of financial phenomena in foreign exchange markets. In this formalism the act of changing sides in a foreign exchange market is described as a change of basis.

In two-currency markets the change of basis uses the one-dimensional Kelvin transform. For general markets the framework is described in terms of vector lattice bundles on a graph and connections on these bundles.

Practical applications include the detection of foreign exchange option arbitrage and the development and testing of algorithms and software to value and analyze portfolios of foreign exchange options, both for in-house and commercial use.

References

• Valery A. Kholodnyi and John F. Price, Foreign Exchange Option Symmetry, World Scientific, (1998).
• Valery Kholodnyi and John Price, Foundations of Foreign Exchange Option Symmetry, Les Press, (1996).

Please feel free to bring this to the attention of interested colleagues.