Skip to main content
Random Walks, Non-Cooperation Games, and the Complex Mathematics of Patent Pricing
Scholarly Works
  • F. Russell Denton, King and Spaulding, LLP
  • Paul J. Heald, University of Georgia School of Law
Rutgers Law Review, Vol. 55, No. 4 (Summer 2003), pp. 1175-1288. Originally uploaded at SSRN
Publication Date

Current patent valuation methods have been described charitably as “inappropriate,” “crude,” “inherently unreliable,” and a “guesstimate.” This article provides a more rational and systematic tool than any we have found in the existing literature or relevant case law. We believe our approach to patent valuation will be useful in improving investment decisions, in facilitating licensing negotiations, and in reducing error costs in litigation. An improved valuation metric also promises to make patents easier to take as collateral and to reduce the amount of “Blue Sky” in mergers and acquisitions involving high tech corporations. To the extent that valuation problems have prevented a more efficient market for patents from emerging, we hope to facilitate the development of such a market.

In Part I, we will review the standard approaches to valuing patents. Although the cost, income, and market methods of valuation each lend some insight, none are adequate to encompass the multiple factors that bear on patent value. Not surprisingly, some sophisticated firms have abandoned the traditional approaches and adapted the Black-Scholes equation from its original use in valuing stock options to valuing patents as a kind of option. Although use of Black-Scholes is a critical improvement over previous valuation methods, its treatment in the patent valuation literature (when covered at all) tends to involve either narrow special cases or grossly oversimplified approaches.

In Part II, we show how a similar valuation problem involving stock options was solved using a novel and straightforward approach -- the Black-Scholes equation -- that has wide utility for patents and patent licenses. We explain how stock options are like patents and demonstrate how the Black-Scholes equation is applied to price real options.

In Part III, we prove that a more comprehensive and advanced form of the Black-Scholes equation can be applied to value 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 basis 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.

In Part IV we note that insights provided by game theory, in particular the Nobel Prize-winning contributions of Professor John Nash, help justify the choices that underlie the Denton adaptation of the Black-Scholes equation. In the licensing context, considerations of the effect of bargaining position are unavoidable, and we show how our assumptions are consistent with game theoretical paradigms.

In Parts V and VI, we explore the implications of our refined approach to patent valuation. The statistically definable imprecision or diffuseness in patent value can be traced to economic inefficiency in the present market for patents. We examine, therefore, how patent valuation problems currently hinder efficient transfer of technology. One telling sign in the current market is the lack of patent arbitrage. Patents, like used cars, can be transferred directly from owner to user. But unlike the market for used cars, no equivalent of the used car lot has emerged in the context of patented technology. Although patents, like cars, are numerous and, in the aggregate, hugely valuable, no significant group of arbitragers currently facilitates a market by buying and reselling them. Potential buyers and sellers of technology, therefore, are left to seek each other out directly, usually with little independent guidance as to the value of what they seek to exchange. Similar information deficits plague internal corporate decisions whether to fund particular research projects and decisions by financial institutions on how to value a patent as collateral. Given the undeniable value of patented technology, the market we describe is surprisingly inefficient. In addition, we argue that patent attorneys are uniquely situated to serve as midwives to a “thick” new technology market. Defining the legal scope of the patent, and therefore its usefulness in excluding competition, is currently the central job of the patent attorney. Attorneys also advise on whether options to a license are infringing or not. We hope to hurry the emergence of a new generation of patent attorneys armed with more sophisticated valuation tools and a better awareness of the scientific and business side of the patent industry.

In Part VI, we show 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. This is an area where math-based solutions have long evaded the courts. In the lost profits context, we offer our approach as a means by which courts can recognize a patent owner's ability to recover direct damages to the market value of its patent, in addition to lost profits suffered during the period of infringement. Development of this remedy, which has an analogy in cases of trademark infringement and in patent cases recognizing “market spoilage,” has been stifled by the absence of a method to measure confidently the diminishment in market value of a patent after the infringement ends. Finally, we note the wide applicability of our approach in the compulsory licensing context.

Citation Information
F. Russell Denton and Paul J. Heald. "Random Walks, Non-Cooperation Games, and the Complex Mathematics of Patent Pricing" (2003)
Available at: