Fair Terms and Fair Pricing For Multiple Warrant issues
by P. W. A. Dayananda
Abstract: For many years corporations have raised capital using bond issues with warrants attached or have issued warrants separately as a way to keep bond yields low or simply to raise funds. Warrants have features similar to exchange-traded call options, with one important exception being that when a warrant is exercised, new shares are issued by the company, contributing to the ‘dilution’ of equity interests.
The work by and Schultz and Trautmann (1994) and Sidenius (1996) points out that the options-pricing methods used for warrant valuation may be equivalent to the call-on-the-value-of-the-firm method if the value of the firm is considered to be market value, which is observable in many cases as the combined value of all outstanding equity issues. The starting point for the results reported here is the confirmation, in the case of a company whose equity consists entirely of its common stock and a single issue of warrants, that the valuation of a warrant as a call option on the market value of the firm and the valuation of a stock option with terms identical to those of the warrant are the same. This observation results from the recursive nature of the warrant. That is, a warrant is a call on the market value of the firm, which value includes the value of the warrant.
We assume that the stock price process is governed by a geometric Brownian motion with constant drift and volatility parameters per unit time. In the presentation, we will consider the case of a single tranche of warrants, followed by an investigation of the interaction between (among) two or more warrant issues. Particular attention will be paid to questions of fair terms and fair pricing when a newly issued warrant has implications for holders of warrants that are already outstanding, regardless of the relative exercise times of the two warrant issues. We will show that ‘fairness’ requires some adjustments for older-issued warrants when a new issue of warrants is created and will develop specific formulas for the adjusted values. Pricing formulae are derived when two tranches of warrants have been issued and the methodology can be extended to multiple issues of warrants. We also investigate the sensitivity of the warrant value/s to the parameters. The applications of the formulae derived are studied and compared with the market values of some warrants in the market.