A warrant is an option-like security on equity, but it is issued by the same company which has issued the
equity, and when a warrant is exercised, a new stock is issued. This new stock is issued at a the
warrant strike price, which is lower than the current stock price (If it wasn't the warrant would not be
exercised.) Since the new stock is a a fractional
right to all cashflows, this stock issue waters out, or dilutes, the equity in a company.
The degree of dilution is a function of how many warrants are issued.
Let be the strike price, the number of shares outstanding and the number of warrants issues.
Assume each warrant is for 1 new share, and let be the current asset value of firm.
Suppose all warrants are exercised simultaneously. Then
the assets of the firm increasy by the number of warrants times the strike price of the warrant.
but this new asset value is spread over more shares, since each exercised warrant is now an equity. The
assets of the firm is spread over all shares, hence each new share is worth:
making each exercised warrant worth:
If we knew the current value of assets in the company, we could value the warrant in two steps:
If we let be the warrant value, the above arguments are summarized as:
- Value the option using the Black Scholes formula and as the current stock price.
- Multiply the resulting call price with .
where is the Black Scholes formula.
However, one does not necessarily observe the asset value of the firm. Typically one only observes the
equity value of the firm. If we let be the current stock price, the asset value is really:
Using the stock price, one would value the warrant as
Note that this gives the value of as a function of . One need to solve this equation
numerically to find .
The numerical solution for is done using the Newton-Rhapson method. Let
Starting with an initial guess for the warrant value , the Newton-Rhapson method is that
one iterates as follows
where signifies iteration , until the criterion function is below some given
accuracy . In this case
An obvious starting value is to set calculate the Black Scholes value using the current stock price, and
multiplying it with .
Code 7.1 implements this calculation.
McDonald (2002) and Hull (2003) are general references. A problem with warrants is that exercise of all
warrants simultaneously is not necessarily optimal.
Press et al. (1992) discusses the Newton-Rhapson method for root finding.