Factors That Influence Black-Scholes Warrant Dilution

A warrant is like an option, except it is issued by a company. The warrant gives the holder the right to buy stock from the company at a specified price within a designated time period. When an investor exercises a warrant, he or she buys stock from the company and those proceeds are a source of capital for the firm. While warrants aren't that common, it's important to understand what they are, and how to value them, in case a company you own shares in offers warrants, or may in the future.

Like an option, a warrant does not represent actual ownership in the stock of the company; it is simply the right (but not the obligation) to buy shares at a certain price in the future. A warrant typically has a much longer life than a call option, with an expiry extending out five or 10 years. Some warrants are even perpetual.

Although warrants are similar to options, there are several important differences. First, options are written by other investors or market makers, while warrants are typically issued by companies. Warrants are often traded over-the-counter and do not have the standardized features of option contracts. The company can create any sort of contract they want, whereas an options writer cannot. Also, options are not dilutive to current shareholders, while warrants are. This is because when a warrant is exercised, new stock is issued.

Although there are several kinds of warrants, the most common types are detachable and naked. Detachable warrants are issued in conjunction with other securities (like bonds or preferred stock) and may be traded separately from them. Naked warrants are issued as is and without any accompanying securities.

Other less common types of warrants include wedded warrants, which can only be exercised if the attached bond/preferred stock is surrendered, and put warrants which may be used to hedge employee option programs.

The most common reason for a company to issue warrants is to provide a "sweetener" for a bond or preferred stock offering. By adding the warrants, the company hopes to obtain better terms (lower rates) on the debt or preferred stock. Moreover, warrants represent a potential source of capital in the future, and can thus offer a capital-raising option to companies that cannot, or prefer not to, issue more debt or preferred stock.

In addition, there are certain accounting benefits. Issuers can use the treasury stock method to calculate earnings per share, and amortized warrant value can be used to increase interest expense and tax benefits.

Less commonly, warrants are issued as part of the recapitalization plan of a bankrupt company. While the holders of common stock are typically wiped out in a bankruptcy, issuing warrants for soon-to-be-worthless shares gives the company a future source of equity capital (if shareholders exercise those warrants) and preserves some goodwill in the former shareholder base.

Although there are several possible methods for valuing a warrant, a modified version of the Black-Scholes model is commonly used. This formula is for European-style options and, though American-style options are theoretically worth more, there is not much difference in price in practice.

In the Black-Scholes model, the valuation of a call option is expressed as:

$\begin{aligned} &C=SN\left (d{1} \right )-Xe-rTN \left( d2 \right ) \\ &\textbf{where:}\\ &C=\text{Call option}\\ &S=\text{Price of the underlying asset}\\ &N=\text{Standard normal distribution}\\ &X=\text{Option strike price}\\ &T=\text{Time to expiration}\\ &d=\text{Dividend}\\ &r=\text{Risk free interest rate}\\ &e=\text{Exponential term}\\ \end{aligned}$​C=SN(d1)−Xe−rTN(d2)where:C=Call optionS=Price of the underlying assetN=Standard normal distributionX=Option strike priceT=Time to expirationd=Dividendr=Risk free interest ratee=Exponential term​

Because of the dilution that warrants represent, the value of that call needs to be divided by (1 + q) where q is the ratio of warrants to outstanding shares, assuming each warrant is worth one share.

The formula gives the theoretical value of an option. What it is trading at in the real world may differ. To find current warrant prices, do a symbol lookup for the stock warrant you are interested in on NYSE.com or nasdaq.com. Click on the warrant symbol provided to get the current price. The following example shows warrant pricing information for Ambac Financial Group, Inc. (AMBC).

As shown by the price snapshot, warrants can be traded like stocks or options. As long as there is someone else to buy or sell from, the warrant can be traded at any time up to expiry.

Beyond the calculation above, investors should consider the following factors when evaluating the price of a warrant.

Underlying Security Price: The higher the price of the underlying security, the more valuable the warrant becomes. After all, if the price of the stock is below the strike price of the warrant, there is no reason to exercise the warrant as it is cheaper to buy the stock on the open market.

Days to Maturity: Generally speaking, options and warrants are worth less as time goes on and expiration approaches. This phenomenon is also called "time decay," and it will accelerate as expiration approaches if the strike price is above the current price.

Dividend: Warrant-holders are not entitled to receive dividends, and the corresponding reduction in the stock price when a dividend is issued to common shareholders reduces the value of the warrant.

Interest Rate/Risk-Free Rate: Higher interest rates increase the value of warrants.

Implied Volatility: The higher the volatility, the higher the odds that the warrant will eventually be in-the-money and the higher the value of the warrant will be.

Dilution: Because the exercise of a warrant will increase a company's outstanding shares, this dilution adds a twist to valuation that is not present in normal option valuation. Potential dilution may hamper the price of the common stock from rising.

Premium: Warrants can be issued at premiums; the lower the premium the more valuable the warrant.

Gearing/leverage: Gearing is the ratio of the share price to the warrant premium, and it reflects how much the price of the warrant changes for a given change in the stock. The higher the gearing, the more valuable the warrant.

Restrictions: Though very difficult to quantify mathematically, any restrictions on the exercise of warrants will impact the value of a warrant, typically negatively. A common restriction is the difference between American-style and European-style warrants. American-style warrants permit exercise at any time, while European-style warrants can only be exercised on the expiration date. The former is more valuable than the latter.

A warrant is basically a long-term option issued by a company. Investors need to make a few adjustments for unique factors like dilution, but a basic Black-Scholes options pricing formula will produce a reasonable assessment of the warrant's value. Current warrant prices can also be found online, such as on the NYSE or NASDAQ websites. Warrants can be bought or sold at any time, although not all warrants are actively traded, so check the volume of a warrant before opting to trade one.