Fair Terms and Fair Pricing for Multiple Warrant Issues

Some firms utilize one or more tranches of warrant issues to supplement their capital base. Unlike exchange-traded options, the exercise of warrants requires the issuance of stock by the company, resulting in a form of dilution. Some previous studies of warrant valuation relied on “the value of the firm,” which is nonobservable, making it difficult to apply the corresponding valuation formula. This paper derives closed-form formulas to value single and multiple tranches of warrants based on the underlying stock price, its volatility, and other known parameter values. The paper first establishes the equivalence of the Black-Scholes formula for both call options and warrants in the case of a single tranche. Thereafter, it considers the impact on the value of previously issued warrants that results when a new tranche of warrants is subsequently issued, showing in each case that fair treatment of the first-issued warrant holders requires an adjustment (due to dilution) in the terms of those warrants and a corresponding modification in the warrants’ value once a second tranche of warrants is issued. To promote such fair treatment, terms of a warrant indenture would specify the nature of the adjustment required when future warrants are issued or exercised, analogous to the antidilution terms related, for example, to stock dividends. Unlike multiple issues of traded options, which are valued independently of one another, multiple warrant issues will be shown to have prices dependent on other warrants outstanding. Also examined is the sensitivity of the fair-value adjustment to changes in the underlying variables, and the theoretical fair-value prices are compared with Black-Scholes prices and with market prices of warrants in the case of two publicly traded companies, each with two warrant issues outstanding. As warrant issues modify the equity structure of a firm, the methodology of valuing warrants presented here will be useful to investment actuaries in situations in which a comprehensive market value for all of a firm’s securities is called for. In addition, risk management practices may sometimes include the use of warrant transactions to hedge stock positions similar to the way that call options are used for that purpose. This may include hedging the risk in equity-linked insurance contracts when the equity position includes stock in companies that have one or more warrant issues that are traded. The methods developed here are also applicable to multiple issues of executive stock options (ESOs) or to combinations of warrant issues and ESOs.