Consistent pricing of warrants and traded options.(Statistical Data Included)

Abstract

Warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. In this paper, we investigate the effects of warrants issuance on the prices of traded options (bought and sold by third parties) already outstanding at the time of warrants issuance. We show how these options can be valued as portfolios of standard and compound options written on the stock of an otherwise similar company without warrants and derive a closed-form solution for the Black-Scholes model. An application of these results to empirical data shows that warrants issuance can have very large effects on the prices of outstanding traded options. [C] 2002 Elsevier Science Inc. All rights reserved.

JEL classification: G13; G32

Keywords: Option pricing; Warrants; Firm-value-based security pricing

1. Introduction

In the seminal papers by Black and Scholes (1973) and Merton (1973), the option pricing problem is solved assuming that the stock price follows a geometric Brownian motion. Both Black-Scholes and Merton also show how their formulae can be modified to value European warrants. (1) More detailed treatments of warrant valuation are given by Galai and Schneller (1978) and Lauterbach and Schultz (1990). The resulting implicit valuation equation is based on modelling the equity of the company (i.e., the aggregate value of stocks and warrants together) as a geometric Brownian motion.

Galai and Schneller (1978, p. 1336) note that if the (market) value of the company's equity is lognormally distributed, the distribution of the stock price after warrants issuance will not be lognormal. Bensoussan, Crouhy, and Galai (1994, p. 72) and Schulz and Trautmann (1994, p. 846) show that, even if the stock price follows a constant-volatility process before warrants issuance, volatility will be nonconstant after warrants issuance. This means that using the Black-Scholes formula for pricing options written on the issuing company's stocks is, in this case, incorrect from a theoretical point of view if the options expire after warrants issuance. In general, using any dilution-adjusted pricing model (2) for pricing warrants and using the same model (without dilution adjustment) for pricing options expiring after warrant issuance is inconsistent. Since warrant issuance changes the distribution of the stock price process, model prices of options written on these stocks also change.

Beginning with Merton (1974), derivative pricing methods have been applied in the context of capital structure models to price corporate securities. Applications include, e.g., corporate bonds (Ingersoll, 1977) and warrants (Crouhy & Galai, 1994; Galai & Schneller, 1978). In recent years, more sophisticated capital structure models have been developed featuring closed-form solutions for the prices of corporate securities (Leland, 1994) and derivatives written on these securities (Toft & Prucyk, 1997).

The aim of this paper is not to provide another firm-value-based option pricing model, but to investigate how standard assumptions made for warrant valuation ultimately affect model prices of traded options. The economic motivation for this analysis is as follows: If standard textbook warrant valuation is correct, option prices have to change as described here, otherwise arbitrage opportunities will arise.

The contributions of this paper are the following: Based on an analysis of the changes in the stock price process induced by warrants issuance, we show how standard options on stocks can be valued in this case as a portfolio of standard and compound options. Specifically, we apply Geske's compound option pricing formula to derive a closed-form solution for standard call can put option prices after warrants issuance within the Black-scholes model. Finally, we provide numerical results illustrating the magnitude of the effects described for actual warrant issues.

The arguments made here for warrants carry over to employee stock options. These options are essentially warrants, in many cases with cash settlement instead of physical delivery of newly issued shares. Moreover, these options are often subject to special provisions, e.g., the may only be exercised after several years of service, which complicates their valuation. Since the dilution effect occurs independently of the delivery mode, the same arguments put forward here apply to employee stock options.

The paper is organized as follows: Section 2 introduces the notation and the framework for our analysis. Section 3 discusses the changes in the law of the stochastic process of the stock price triggered by warrants issuance. Section 4 analyzes the problem of consistent pricing of warrants and traded options written on the same company. In Section 5, we apply our theory within the Black-Scholes model and illustrate the effects described using empirical data. Section 6 provides a summary and directions for further research.

2. Notation and framework

Throughout the paper, we will use the following notation (3.): [S.sub.t] = the stock price at time t; [V.sub.t] = the market value of the company's equity at time t, t < [t.sub.0] ([t.sub.0] denotes the time of warrants issuance) (4); [C.sub.t]([S.sub.t], x, T) = the time t price of a standard European call option on S with strike price x, expiring at time T (according to any specific pricing model satisfying the assumptions given below, e.g., the Black-Scholes model); N=the number of shares of stock outstanding; M=the number of warrants issued; [W.sub.t]([S.sub.t],x,[t.sub.0],[S.sub.[t.sub.0]],T,N,M) = the time t value of a European warrant with strike price x, issued at time [t.sub.0] (when the stock price was [S.sub.[t.sub.0]]) and expiring at time T, where (for simplicity of exposition) each warrant gives the right to purchase one share; [D.sub.t,T] = the discount factor applicable to the time period from t to T, 0 < [D.sub.t,T] [less than or equal to] 1, [D.sub.T,t] = (1/([D.sub.t,T])).

Our analysis is not limited to any specific pricing model (e.g., the Black-Scholes model). Instead, we provide results that are valid for any pricing model satisfying the following basic assumptions:

Assumption 1: Model prices of contingent claims are calculated under an equivalent martingale measure Q.

Thus, our analysis is valid …