Pricing levered warrants with dilution using observable variables

Abstract

We propose a valuation framework for pricing European call warrants on the issuer’s own stock that allows for debt in the issuer firm. In contrast to other works that also price warrants with dilution issued by levered firms, ours uses only observable variables. Thus, we extend the models of Crouhy and Galai [J. Bank. Finance, 1994, 18, 861–880] and Ukhov [J. Financ. Res., 2004, 27(3), 329–339]. We provide numerical examples to study some implementation issues and to compare the model with existing ones.

Keywords:

• Corporate warrants
• Dilution
• Leverage
• Observable variables

Classification:

• C13
• C63
• G32

Acknowledgments

We are grateful for valuable comments from David Abad, Julio Carmona, Ariadna Dumitrescu, Louis Ederington, Stewart Hodges, Belén Nieto, David Offenberg, Gonzalo Rubio, Eduardo Schwartz, Andrey Ukhov, Ashraf Zaman, seminar participants at a number of conferences and universities and an anonymous referee. We acknowledge the financial support of the Andalusian Regional Government (grants P08-SEJ-03917 and P09-SEJ-4467) and the Spanish Ministry of Economy and Competitiveness (grants ECO2012-35946-C02-01 and ECO2012-34268). Javier F. Navas also acknowledges the hospitality of Carlos III University and Purdue University, where part of this work was done. An earlier version of this work appeared as Working Paper No. 450 in the Working Paper Series of the Fundación de las Cajas de Ahorros (FUNCAS). The usual disclaimer applies.

Notes

Other empirical studies finding correction for dilution necessary are those of Schulz and Trautmann (1994) and Darsinos and Satchell (2002).

Such as Galai and Schneller (1978), Ingersoll (1987), Galai (1989), Schulz and Trautmann (1989), (1994), and Crouhy and Galai (1991).

is usually referred to as the warrant ratio.

Obviously, despite stock returns being an observable variable, the volatility of stock returns is non-observable, thus we use the term ‘observable variables’ as in Ukhov (2004).

This is the well-known proposition I of MM.

See, for example, Ingersoll (1987), Schulz and Trautmann (1994) and Ukhov (2004).

Note that we can easily express in terms of .

Except for the BSM model, where dilution is not considered.