The use of economic and statistics principles have been instrumental in developing many quantitative methodologies in finance, for example the famous formula of Black-Scholes that led to a Noble Prize in economics. In order to research in mathematical finance, it is essential to understand both economic principles and the ever changing financial activities in the market. The purpose of this presentation is to find a closed from formula to value a financial instrument called “warrant”. The holder of a financial warrant has the right but not the obligation to purchase a share of the company concerned by paying $K at a specific time T into the future. The holder pays now to own the warrant and we are interested in the calculation of the fair value of his payment. The possibilities of multiple warrants in a company will also be examined. Some companies, particularly in the extractive industries, have supplemented their capital base with multiple issues of stock purchase warrants. Unlike options, the exercise of warrants requires the issuance of stock by the company, resulting in a form of dilution. We consider the impact on the value of previously issued warrants that is created when subsequent warrants are issued, showing in each case that fair treatment of the first-issued warrant holders requires an adjustment (due to dilution) in the value of those warrants 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 anti-dilution terms related, for example, to stock dividends. In addition to developing closed formulas for the fair-value adjustment required in each of the cases involving two warrant issues, we provide an example of the calculation required when there are three warrant issues. We also examine the sensitivity of the fair-value adjustment to changes in several of the underlying variables and compare our theoretical fair-value prices with Black-Scholes prices and with market prices of warrants in the case of two publicly traded companies.
University of St.Thomas, St. Paul, Minnesota
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