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22.1 Options 22.1 Options 22.2 Call Options 22.3 Put Options 22.4 Selling Options 22.5 Stock Option Quotations 22.6 Combinations of Options 22.7 Valuing Options 22.8 An Option‑Pricing Formula 22.9 Stocks and Bonds as Options 22.10 Capital-Structure Policy and Options 22.11 Mergers and Options 22.12 Investment in Real Projects and Options 22.13 Summary and Conclusions

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22.1 Options Many corporate securities are similar to the stock options that are traded on organized exchanges. Almost every issue of corporate stocks and bonds has option features. In addition, capital structure and capital budgeting decisions can be viewed in terms of options.

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22.1 Options Contracts: Preliminaries An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or perhaps before) a given date, at prices agreed upon today. Calls versus Puts Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. When exercising a call option, you “call in” the asset. Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone.

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22.1 Options Contracts: Preliminaries Exercising the Option The act of buying or selling the underlying asset through the option contract. Strike Price or Exercise Price Refers to the fixed price in the option contract at which the holder can buy or sell the underlying asset. Expiry The maturity date of the option is referred to as the expiration date, or the expiry. European versus American options European options can be exercised only at expiry. American options can be exercised at any time up to expiry.

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Options Contracts: Preliminaries In-the-Money The exercise price is less than the spot price of the underlying asset. At-the-Money The exercise price is equal to the spot price of the underlying asset. Out-of-the-Money The exercise price is more than the spot price of the underlying asset.

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Options Contracts: Preliminaries Intrinsic Value The difference between the exercise price of the option and the spot price of the underlying asset. Speculative Value The difference between the option premium and the intrinsic value of the option.

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22.2 Call Options Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today. When exercising a call option, you “call in” the asset.

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Basic Call Option Pricing Relationships at Expiry At expiry, an American call option is worth the same as a European option with the same characteristics. If the call is in-the-money, it is worth ST - E. If the call is out-of-the-money, it is worthless. CaT = CeT = Max[ST - E, 0] Where ST is the value of the stock at expiry (time T) E is the exercise price. CaT is the value of an American call at expiry CeT is the value of a European call at expiry

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Call Option Payoffs

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Call Option Payoffs

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Call Option Profits

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22.3 Put Options Put options give the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone.

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Basic Put Option Pricing Relationships at Expiry At expiry, an American put option is worth the same as a European option with the same characteristics. If the put is in-the-money, it is worth E - ST. If the put is out-of-the-money, it is worthless. PaT = PeT = Max[E - ST, 0]

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Put Option Payoffs

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Put Option Payoffs

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Put Option Profits

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22.4 Selling Options The seller (or writer) of an option has an obligation.

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22.5 Stock Option Quotations

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22.5 Stock Option Quotations

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22.5 Stock Option Quotations

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22.5 Stock Option Quotations

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22.5 Stock Option Quotations

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22.5 Stock Option Quotations

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22.6 Combinations of Options Puts and calls can serve as the building blocks for more complex option contracts. If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.

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Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiry

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Protective Put Strategy Profits

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Covered Call Strategy

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Long Straddle: Buy a Call and a Put

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Short Straddle: Sell a Call and a Put

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Long Call Spread

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Put-Call Parity

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22.7 Valuing Options The last section concerned itself with the value of an option at expiry.

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Option Value Determinants

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Market Value, Time Value, and Intrinsic Value for an American Call CaT > Max[ST - E, 0]

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22.8 An Option‑Pricing Formula We will start with a binomial option pricing formula to build our intuition.

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Binomial Option Pricing Model Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S0= $25 today and in one year S1 is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option?

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Binomial Option Pricing Model A call option on this stock with exercise price of $25 will have the following payoffs. We can replicate the payoffs of the call option. With a levered position in the stock.

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Binomial Option Pricing Model Borrow the present value of $21.25 today and buy one share. The net payoff for this levered equity portfolio in one period is either $7.50 or $0. The levered equity portfolio has twice the option’s payoff so the portfolio is worth twice the call option value.

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Binomial Option Pricing Model The levered equity portfolio value today is today’s value of one share less the present value of a $21.25 debt:

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Binomial Option Pricing Model We can value the option today as half of the value of the levered equity portfolio:

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The Binomial Option Pricing Model If the interest rate is 5%, the call is worth:

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The Binomial Option Pricing Model If the interest rate is 5%, the call is worth:

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Binomial Option Pricing Model the replicating portfolio intuition.

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The Risk-Neutral Approach to Valuation We could value V(0) as the value of the replicating portfolio. An equivalent method is risk-neutral valuation

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The Risk-Neutral Approach to Valuation S(0) is the value of the underlying asset today.

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The Risk-Neutral Approach to Valuation The key to finding q is to note that it is already impounded into an observable security price: the value of S(0):

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Example of the Risk-Neutral Valuation of a Call: Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. The risk-free rate is 5%. What is the value of an at-the-money call option? The binomial tree would look like this:

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Example of the Risk-Neutral Valuation of a Call: The next step would be to compute the risk neutral probabilities

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Example of the Risk-Neutral Valuation of a Call: After that, find the value of the call in the up state and down state.

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Example of the Risk-Neutral Valuation of a Call: Finally, find the value of the call at time 0:

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Risk-Neutral Valuation and the Replicating Portfolio This risk-neutral result is consistent with valuing the call using a replicating portfolio.

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The Black-Scholes Model The Black-Scholes Model is

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The Black-Scholes Model Find the value of a six-month call option on Microsoft with an exercise price of $150. The current value of a share of Microsoft is $160. The interest rate available in the U.S. is r = 5%. The option maturity is six months (half of a year). The volatility of the underlying asset is 30% per annum. Before we start, note that the intrinsic value of the option is $10—our answer must be at least that amount.

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The Black-Scholes Model Let’s try our hand at using the model. If you have a calculator handy, follow along.

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The Black-Scholes Model

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Another Black-Scholes Example

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22.9 Stocks and Bonds as Options Levered Equity is a Call Option. The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond. If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call, they will pay the bondholders, and “call in” the assets of the firm. If at the maturity of the debt the shareholders have an out-of-the-money call, they will not pay the bondholders (i.e., the shareholders will declare bankruptcy), and let the call expire.

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22.9 Stocks and Bonds as Options Levered Equity is a Put Option. The underlying asset comprise the assets of the firm. The strike price is the payoff of the bond. If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in-the-money put. They will put the firm to the bondholders. If at the maturity of the debt the shareholders have an out-of-the-money put, they will not exercise the option (i.e., NOT declare bankruptcy) and let the put expire.

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22.9 Stocks and Bonds as Options It all comes down to put-call parity.

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22.10 Capital-Structure Policy and Options Recall some of the agency costs of debt: they can all be seen in terms of options. For example, recall the incentive shareholders in a levered firm have to take large risks.

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Balance Sheet for a Company in Distress Assets BV MV Liabilities BV MV Cash $200 $200 LT bonds $300 ? Fixed Asset $400 $0 Equity $300 ? Total $600 $200 Total $600 $200 What happens if the firm is liquidated today?

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Selfish Strategy 1: Take Large Risks (Think of a Call Option) The Gamble Probability Payoff Win Big 10% $1,000 Lose Big 90% $0 Cost of investment is $200 (all the firm’s cash) Required return is 50% Expected CF from the Gamble = $1000 × 0.10 + $0 = $100

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Selfish Stockholders Accept Negative NPV Project with Large Risks Expected cash flow from the Gamble To Bondholders = $300 × 0.10 + $0 = $30 To Stockholders = ($1000 - $300) × 0.10 + $0 = $70 PV of Bonds Without the Gamble = $200 PV of Stocks Without the Gamble = $0 PV of Bonds With the Gamble = $30 / 1.5 = $20 PV of Stocks With the Gamble = $70 / 1.5 = $47

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22.11 Mergers and Options This is an area rich with optionality, both in the structuring of the deals and in their execution.

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22.12 Investment in Real Projects & Options Classic NPV calculations typically ignore the flexibility that real-world firms typically have. The next chapter will take up this point.

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22.13 Summary and Conclusions The most familiar options are puts and calls. Put options give the holder the right to sell stock at a set price for a given amount of time. Call options give the holder the right to buy stock at a set price for a given amount of time. Put-Call parity

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22.13 Summary and Conclusions The value of a stock option depends on six factors: 1. Current price of underlying stock. 2. Dividend yield of the underlying stock. 3. Strike price specified in the option contract. 4. Risk-free interest rate over the life of the contract. 5. Time remaining until the option contract expires. 6. Price volatility of the underlying stock. Much of corporate financial theory can be presented in terms of options. Common stock in a levered firm can be viewed as a call option on the assets of the firm. Real projects often have hidden options that enhance value.

Теги Options

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