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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

- Black and Scholes pricing formula for a European call option on a non-dividend-paying stock:

- 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).

The Black and Scholes Model

- In the Black-Scholes model, N(d1) is the call option’s delta, the amount by which the option price changes given a $1 increase in the underlying stock price.
- Closer to 1 for in-the-money options
- Closer to 0 for out-of-the money options
- 1/N(d1) is the hedge ratio, the number of options needed to offset the fluctuations of a single share of stock.

Black and Scholes Put Values

- Put-call parity states that

S + P = Xert + C,

so

P = Xert + [SN(d1) – Xe rtN(d2)] – S.

- Manipulating this expression algebraically gives

P = Xert[1 – N(d2)] S[1 – N(d1)],

where d1 and d2 are defined as before.

Volatility

- Of the five inputs required by the Black and Scholes model, only the underlying stock’s volatility (σ) is unobservable.
- If traders can observe the market price of an option directly, they can “invert” the Black-Scholes equation to calculate the volatility implied by the option’s price.
- The value of σ obtained in this manner is called an option’s implied volatility.

Options Embedded in Other Securities

- Plain Vanilla Stocks and Bonds
- Warrants
- Convertibles

Options Embedded in Ordinary Corporate Bonds

- 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:

Value of risky debt = Value of risk-free debt – Put value

Options Embedded in the Stock of a Levered Firm

- Owning a firm’s stock is like having a call option on its assets, where the exercise price is the amount owed to bondholders.
- Since a call option’s value increases with the volatility of the underlying asset, making riskier investments would benefit shareholders at the expense of bondholders.
- Lenders usually insist on loan covenants to prevent firms from taking on excess risk.

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
- Volatility

Call options are contracts between investors who are not necessarily connected to the underlying firm.Differences Between Warrants and Calls

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.

Differences Between Warrants and Calls

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.

Warrants

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.

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.
- The conversion ratio is the number of shares for which the bond can be traded.
- Conversion price = Bond price / Conversion ratio
- Conversion value = Stock price × Conversion ratio

Options Embedded in Capital Investments – Real Options

- 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.

Options Embedded in Capital Investments – Real Options

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.

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