Random Walks, Non-Cooperative Games, and the Complex Mathematics of Patent Pricing

Abstract

Current patent valuation methods have been described charitably as "inappropriate," "short-sighted," "inherently unreliable," and a "guestimate." This article provides a more rational and systematic tool than any found in the existing literature. We explain how patents are like stock options and demonstrate how the Black-Scholes equation for pricing real options can be applied to price patents. First, we explain the major difficulties inherent in applying the standard equation to patents and then proceed to demonstrate how it can be adapted to overcome those problems. In particular, the Denton Variation of Black-Scholes begins with fine distinctions in identifying the source of value, followed by a systematic analysis of factors, especially market forces, that influence variance and its sources over time. We show that the point-price paradigm relied upon by patent valuations to date has been flawed. Here, we leave the world of contemporary patent valuation behind. We claim that solving a single Black-Scholes equation is grossly inadequate for a risky, long-lived, infrequently traded item such as a patent. For a patent, the present value exists as a distribution curve with variously weighted probabilities, thus the apparent precision in picking a starting value by traditional patent valuation methods is illusory. The Denton Variation eliminates two historic shortcomings of the parent equation by providing a precise way to factor in transactions costs, and by quantifying the impact of the option cost on the profitability of the transaction. Moreover, the expression can accommodate a variety of patent profitability situations. For the purposes of illustration, we run through the equation to value a hypothetical patent. We also take the time to explain how insights provided by game theory help justify the choices that underlie the Denton adaptation of the Black-Scholes equation. In the patent licensing context, considerations of the effect of bargaining position are unavoidable, and we show how our assumptions are consistent game theoretical paradigms. In conclusion, we explore the implications of our refined approach to patent valuation. We examine, therefore, how patent valuation problems currently hinder efficient transfer of technology and how our enhanced version of the new Black-Scholes variant equation fits comfortably into the calculation of the reasonable royalty remedy applicable in cases of patent infringement when the patent owner cannot prove lost profits.

Keywords: patents, Black-Scholes, game theory, valuation, pricing, variance, random walk, licensing, damages, royalties, options, Nash, mathematics

JEL Classification: C00, C72, G12, G13, K19, K29