Gauge Theory of Arbitrage, or Financial Markets Resembling Screening in Electrodynamics

Arbitrage-image

When a mispricing appears in a market, market speculators and arbitrageurs rectify the mistake by obtaining a profit from it. In the case of profitable fluctuations they move into profitable assets, leaving comparably less profitable ones. This affects prices in such a way that all assets of similar risk become equally attractive, i.e. the speculators restore the equilibrium. If this process occurs infinitely rapidly, then the market corrects the mispricing instantly and current prices fully reflect all relevant information. In this case one says that the market is efficient. However, clearly it is an idealization and does not hold for small enough times.

The general picture, sketched above, of the restoration of equilibrium in financial markets resembles screening in electrodynamics. Indeed, in the case of electrodynamics, negative charges move into the region of the positive electric field, positive charges get out of the region and thus screen the field. Comparing this with the financial market we can say that a local virtual arbitrage opportunity with a positive excess return plays a role of the positive electric field, speculators in the long position behave as negative charges, whilst the speculators in the short position behave as positive ones. Movements of positive and negative charges screen out a profitable fluctuation and restore the equilibrium so that there is no arbitrage opportunity any more, i.e. the speculators have eliminated the arbitrage opportunity.

The analogy is apparently superficial, but it is not. The analogy emerges naturally in the framework of the Gauge Theory of Arbitrage (GTA). The theory treats a calculation of net present values and asset buying and selling as a parallel transport of money in some curved space, and interpret the interest rate, exchange rates and prices of asset as proper connection components. This structure is exactly equivalent to the geometrical structure underlying the electrodynamics where the components of the vector-potential are connection components responsible for the parallel transport of the charges. The components of the corresponding curvature tensors are the electromagnetic field in the case of electrodynamics and the excess rate of return in case of GTA. The presence of uncertainty is equivalent to the introduction of noise in the electrodynamics, i.e. quantization of the theory. It allows one to map the theory of the capital market onto the theory of quantized gauge field interacting with matter (money flow) fields. The gauge transformations of the matter field correspond to a change of the par value of the asset units which effect is eliminated by a gauge tuning of the prices and rates. Free quantum gauge field dynamics (in the absence of money flows) is described by a geometrical random walk for the assets prices with the log-normal probability distribution. In general case the consideration maps the capital market onto Quantum Electrodynamics where the price walks are affected by money flows.

Electrodynamical model of quasi-efficient financial market

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