The General Kelvin Transform Symmetry

The General Kelvin Transform Symmetry is a measurement tool that identifies financial equivalence (provides exact measurement of relative values) between reciprocal options on opposite sides of a multi-currency, security or commodity environment. Until now it has been inconceivable that options on opposite sides of a multi security portfolio had any such relevance, let alone a direct kinship. The symmetry states that every option has a reciprocal and the reciprocal option must give up the same wealth, in this case measured in dollars, at a simultaneous expiration. For example, an in-and-up barrier option must give up the same wealth as its reciprocal, a down-and-out option. This is a startling and dramatic discovery.

The transform symmetry is substantially different from any other existing option pricing and will generate entirely new markets, as well as revolutionize the direction of the derivatives industry. By way of inference, the fact that portfolios have now been proven to be controlled by certain laws of physics, that previously have never been applied to finance, suggests that the entire methodology underlying most derivative products may be based on erroneous assumptions. Certainly, anyone who has wondered at the efficacy of using stochastic tools, like option pricing models, which are accurate only over a million observations, in single-event mediums like each individual market application, will be pleased to know that a tool to arbitrage the inconsistencies of distribution models has arrived. Further, this tool has been proven beyond refutation to expose the innate intelligence which exists in markets because no matter what the security is that is being traded it must ultimately be translated into a tradable wealth. Hence, an option on soybeans or its reciprocal option on gold must give up the same wealth when triangulated through a mediational wealth unit, usually, but not necessarily, a currency.

Until now, the gauge needed to measure the relative values of options has not existed. In fact, arbitrage opportunities are created by inconsistencies in distribution models used to price options in various markets. The role of models in creating arbitrage opportunities is unclear because few market participants know that options on opposite sides of a multi-security portfolio are literally tied together by a physical relationship.

Previously, J. Grabbe (1983) conjectured that equivalence exists between puts and calls in a two currency environment, but he did not know why. After some observation others believed it did too, so it became a standard in certain two currency environments to trade reciprocal puts and calls on cer tain currencies as equivalents (Swiss franc and Deutsche marks, for example). But no one ever observed or conjectured that there is an equivalence between other more exotic options, like barriers for example, or that this equivalence exists on all options and their reciprocals and in an environ ment of the most severe complexity; multi-commodity, currency or security.

The results are dramatic, with various applications that provide a substantial competitive advantage. Of these applications two stand out. The first is the arbitrage of two seemingly unrelated options which must provide the same wealth at expiration and the second is a previously unseen tool for identifying wealth in unit-to-unit relationships.

The first program needed to process data regarding the unit to unit relationships of options in a two currency environment has recently been completed to identify arbitrage opportunities. Market practitioners advised the building of this program would be a waste of money because the currency markets are too efficient to yield arbitrage opportunities. Well, we have news for these advisers. This seemingly efficient market has arbs big enough to drive a truck through because there has never been a tool to gauge financial equivalence. Granted, pricing of puts and calls in active two currency markets is very tight, but barriers and other exotics are rarely priced accurately. And when one considers three or more currencies, no one has a tool to price anything accurately. The currency basket option market (USD40 billion notional last year) is completely mispriced and no one knows it. Beyond currencies, try and get a quote on baskets including gold, electricity and a currency. The bid-ask spread is so wide that it provides no realistic competency. In other words, these markets don't exist. However, with the transform symmetry these markets will develop, providing entirely new products for traders, producers and consumers.

To create an arbitrage profit using the transform symmetry in a two-currency environment, simply take the price of an option offered in the marketplace, any option, apply the transform symmetry to it and immediately get the price its reciprocal must trade at. If the reciprocal is not trading at that price, simply buy one and sell the other, as long as they have simultaneous expirations (if they don't they can be manipulated to replicate simultaneous expiration.) By no later than when they expire they must give up the same wealth, as measured in dollars. They must physically be at a zero-sum. The result is a true arbitrage profit, not based on probability, but on something never before seen: a guaranteed financial equivalence.

Investment banks and other traders have an understand able fixation on arbitrage profit, but perhaps the transform symmetry's most practical application is the identification of the most direct path to achieve desired results. In the world of mathematics there is an old problem of the most direct pathway available to a salesperson as he or she visits outlying offices. Once more than three outlying offices are scheduled, the number of possible pathways to them and home multi plies exponentially, providing a mind-boggling amount of potential variations. Hedging and trading opportunities are the same. More than three securities in the portfolio provide a large amount of potential pathways to achieve desired results. The transform symmetry is the only tool known that instantly provides the most direct pathway to achieve hedging and trading results. This is achieved by being able to rotate the portfolio and view it from any perspective, something only the transform symmetry can achieve.

This will allow a producer or consumer to view its portfolio from the perspective of its own wealth. No longer will a natural gas producer have to translate its wealth into dollars it can hedge its risk from the perspective of its own wealth medium. With this approach it can choose between currency or any other wealth medium to achieve its goals. At any given time, options on corn, for example, may actually provide a better hedge on natural gas than an option on the gas itself. This, of course, is a far-fetched but not unrealistic comparison designed to expose the fact that when wealth is the objective and when all elements of a portfolio are identified in a unit-to-unit relationship around wealth with a physical tool that provides an accurate measurement of the result of this relationship, portfolios will be managed more universally and more efficiently.

This week's Learning Curve was written by Steve Rideout, managing director of Integrated Energy Services. The transform symmetry was invented by Valery Kholodnyi, v.p.-research and development at IES, in association with John Price, an independent mathematician.

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