Professor XXXXX Course Name / # © 2007 Thomson South-Western Chapter 19 Black and Scholes and Beyond

2 The Black and Scholes Model  The Black-Scholes model presumes that stock prices can move at every moment.

3 The Black-Scholes Model  The call option price equals the stock price minus the present value of the exercise price, adjusted for the probability that when the option expires, the stock price will exceed the strike price (i.e., the probability that the option expires in the money).

4 Standard Normal Distribution

5 Black and Scholes Put Values

6 Volatility  If traders can observe the market price of an option directly, they can “invert” the Black and Scholes equation to calculate the volatility implied by the option’s market price.  The value of σ obtained in this manner is called an option’s implied volatility.

7 Options Embedded in Other Securities  Plain Vanilla Stocks and Bonds  Warrants  Convertibles

8 Options Embedded in Ordinary Corporate Bonds

9  Another way to place a value on the put option: If holding a risky corporate bond is identical to holding a risk-free bond and selling a put option, then we can calculate the put value by simply comparing the market value of the firm’s debt to the market value of identical bonds that are risk free.  The difference in prices must equal the put value: Options Embedded in Ordinary Corporate Bonds

10 Options Embedded in the Stock of a Levered Firm

11 Warrants  Warrants are securities issued by firms that grant the right to buy shares of stock at a fixed price for a given period of time.  Warrants bear a close resemblance to call options.  The same five factors that influence call option values will also affect warrant prices  stock price, risk-free rate, strike price, expiration date, and volatility

12 Differences between Warrants and Calls  When investors exercise warrants, the number of outstanding shares increases and the issuing firm receives the strike price as a cash inflow.  Call options are contracts between investors who are not necessarily connected to the firm whose stock serves as the underlying asset.  Warrants are issued by firms.  Warrants are often issued with expiration dates several years in the future,  When investors exercise call options, no change in outstanding shares occurs and the firm receives no cash.  Most options expire in just a few months.

13 Differences between Warrants and Calls  Firms frequently attach warrants to public or privately placed bonds, preferred stock, and sometimes even common stock. When warrants are attached to other securities, they are called equity kickers, implying that they give additional upside potential to the security to which they are attached. When firms bundle warrants together with other securities, they may or may not grant investors the right to unbundle them and sell them separately.  Call and put options trade as stand-alone securities.

14 Convertibles  A convertible bond is essentially an ordinary corporate bond with an attached call option or warrant.  Grants investors the right to receive payment in the shares of an underlying stock rather than in cash.  Conversion ratio  Conversion value

15 The Value of a Convertible Bond

16 Options Embedded in Capital Investments – Real Options  NPV calculations 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.