The Implied Distribution for Stocks of Companies with Warrants and/or Executive Stock Options

This paper sets out to provide a risk-management tool (namely the distribution of the stock price of a warrant-issuing firm) and at the same time resolves an outstanding issue between the theory and the empirical evidence of the warrant pricing literature. In their seminal article on warrant pricing, Galai and Schneller (1978) make the following statement: “…if the distribution of the firm’s liquidation value is lognormal, the value of its share price is not lognormally distributed”. On the other hand recent empirical studies suggest that assuming lognormality for the stock price distribution of a warrant-issuing firm gives a very good approximation for the value of a warrant (this is the so-called “option-like” warrant valuation approximation). We show that despite of the fact that the (risk-neutral) distribution of a warrant-issuing firm and a non-warrant issuing firm is different, valuation by taking expectations of the discounted payoff of the warrant over the two different risk-neutral distributions produces warrant prices very close to each other for a large number of cases. Exceptions occur for deep-out-of-the-money and close to maturity out-of-the-money warrants in general. In such cases the “option-like” approximation will significantly overprice warrants.