Introduction to Corporate Finance

8: Options

8-5b WarraNTS aNd CONVerTIBleS

Warrants are securities that are issued by companies and that grant investors the right to buy shares at a

fixed price for a given period of time. Warrants bear a close resemblance to call options, and the same

factors that influence call option values (share price, risk-free rate, strike price, expiration date and

volatility) affect warrant prices, too. However, there are some important differences between warrants

and calls:

■ Warrants are issued by companies, whereas call options are contracts between investors who are not

necessarily connected to the company whose shares serve as the underlying asset.

■ When investors exercise warrants, the number of outstanding shares increases and the issuing

company receives the strike price as a capital inflow. When investors exercise call options, no change

in outstanding shares occurs, and the company receives no cash.

■ Warrants are often issued with expiration dates several years in the future, whereas most options

expire in just a few months.

■ Although call and put options trade as stand-alone securities, companies frequently attach warrants

to public or privately placed bonds, preferred shares and sometimes even ordinary shares. Warrants

that are attached to other securities are called equity kickers, implying that they give additional upside

potential to the security to which they are attached. When companies bundle warrants together with

other securities, they may or may not grant investors the right to unbundle them and sell the warrants

separately.

Even though warrants and options differ in some important respects, the Black–Scholes model can

be used to value warrants – provided an adjustment is made to account for the dilution that occurs

when companies issue new shares to warrant holders. A simple example will illustrate how to adjust for

dilution.

Assume that a small company has 1,000 shares outstanding worth $10 each. The company has no

debt, so the value of its assets equals the value of its equity: $10,000. Two years ago, when the company’s

share price was just $8, the company issued 100 ordinary shares to a private investor. Each share had an

attached warrant granting a two-year right to purchase one share for $9. The warrants are about to expire,

and the investor intends to exercise them.

What would the investor’s payoff be if she held ordinary call options (sold to her by another private

investor) rather than warrants? Because the price of the shares is $10 and the strike price is $9, the

investor would earn a profit of $1 per share, or $100 on the calls. If calls were exercised, then the

company would still have 100 shares outstanding worth $10 each. From the company’s point of view, the

call exercise would generate neither a cash inflow nor a cash outflow.

In contrast, if the investor exercises her warrants, then two changes take place. First, the company

receives cash equal to the strike price ($9) times the number of warrants exercised (100), or a total inflow

of $900. This raises the total value of the company’s assets to $10,900. Simultaneously, the company’s

outstanding shares increase from 1,000 to 1,100, so the new price per share can be calculated as follows:

New price per share = $10,900@1,100 = $9.91

The investor’s payoff on the warrants is just $0.91, compared to $1.00 on a comparable call option.

Fortunately, it’s easy to use the Black–Scholes model to value a call option with characteristics similar

to those of a warrant and then multiply the call value times an adjustment factor for dilution. If N 1

represents the number of ‘old shares’ outstanding and N 2 represents the number of new shares issued as

warrants

Securities that grant rights

similar to a call option,

except that when a warrant

is exercised, the company

must issue a new share, and it

receives the strike price as a

cash inflow

equity kickers

Warrants attached to another

security offering (usually

a bond offering) that give

investors more upside

potential

How does a price of a warrant

compare to the price of a call

with identical characteristics?

thinking cap

question