Chapter 19: Black and Scholes and Beyond. Corporate Finance, 3e Graham, Smart, and Megginson. The Black and Scholes Model. The Black-Scholes model is based on the same intuition as the binomial model, but it presumes that stock prices can move at every instant. The Black and Scholes Model.
Corporate Finance, 3e
Graham, Smart, and Megginson
S + P = Xert + C,
P = Xert + [SN(d1) – Xe rtN(d2)] – S.
P = Xert[1 – N(d2)] S[1 – N(d1)],
where d1 and d2 are defined as before.
Value of risky debt = Value of risk-free debt – Put value
Warrants are issued by firms.
2. When investors exercise warrants, the number of outstanding shares increases and the issuing firm receives the strike price.
When investors exercise call options, no change in outstanding shares occurs and the firm receives no cash.
3. Most options expire in just a few months.
3. Warrants are often issued with expiration dates several years into the future.
Although options trade as stand-along securities, firms frequently attach warrants to their bonds, preferred stock, and sometimes even common stock. Warrants that are attached to other securities in this manner are called equity kickers.
A warrant’s value is the same as the equivalent call option’s value, multiplied by a dilution factor. N1 is the number of “old shares” outstanding, and N2 represents the number of new shares issued due to the warrants being exercised.
NPVcalculations often understate the value of an investment, but pricing corporate growth options using decision trees leads to overvaluation errors.
Analysts must always value real options with the appropriate technology—that is, an option-pricing model such as the binomial or the Black and Scholes.