Warrant Pricing Using Observable Variables

Abstract

The classical warrant pricing formula requires knowledge of the firm value and of the firm-value process variance. When warrants are outstanding, the firm value itself is a function of the warrant price. Firm value and firm-value variance are then unobservable variables. I develop an algorithm for pricing warrants using stock prices, an observable variable, and stock return variance. The method also enables estimation of firm-value variance. A proof of existence of the solution is provided. JEL Classifications: G13, G63I would like to thank Allan Eberhart, Jonathan E. Ingersoll Jr., William T. Moore (the editor), Meir Schneller (the reviewer), and Matthew Spiegel for helpful comments.1 Phelps, Moore, and Roenfeldt (1991) show that warrants can represent potentially large additions to outstanding equity. In their sample, the median ratio of the number of new shares issued in the event of full exercise to the number of shares outstanding before issuance is 18%.

The risk-shifting effect and the value of a warrant

“…According to these equity-based models, the value of a warrant is equal to the value of a call option written on a share of the firm's equity and corrected for the dilution effect. This approach allows us to ignore the shape of the stock volatility, but it requires unobservable variables, and therefore the solution has to be obtained through a numerical algorithm (Ukhov 2004). In contrast to the equity-based approach, some studies suggest pricing a warrant as a plain-vanilla call option on the firm's stock.…”

“…Dilution and time-to-maturity are other important variables explaining the extent of the pricing bias. For any level of moneyness, either a dilution increase or a yThis value is also computable through the numerical algorithm proposed by Ukhov (2004). Nevertheless, in our simulation this method is not necessary since we assume that the asset value and its volatility are both known.…”

“…These drawbacks preclude obtaining a closedform formula for the warrant price. A feasible solution, suggested by Ukhov (2004), involves the numerical solution of an iterative algorithm based on observable variables (stock price and stock return volatility).…”

Pricing Multiple Stock Warrants with Strategic Exercise

“…At a node k that is on the last slice of the tree, we know that the intrinsic value of each warrant is: (14) We can solve each of these equations independently again. We repeat the above process until we get to the initial node 1 = k .…”

“…We repeat the above process until we get to the initial node 1 = k . After iteration, we have all of the values [14].…”