berk chapter 21: option valuation

Copyright © 2011 Pearson Prentice Hall. All rights reserved.

Chapter 21

Option Valuation

Copyright © 2011 Pearson Prentice Hall. All rights reserved.21-2

Chapter Outline

21.1 The Binomial Option Pricing Model

21.2 The Black-Scholes Option Pricing Model

21.3 Risk-Neutral Probabilities

21.4 Risk and Return of an Option

21.5 Beta of Risky Debt


Learning Objectives

1. Illustrate the use of the Binomial Option Pricing Model to value an option.

2. Define the replicating portfolio for the Binomial Option Pricing Model.

3. Use the Law of One Price to explain how the Binomial Option Pricing Model provides the correct value under the assumptions made by the model.

4. Use the Black-Scholes Option Pricing formula to calculate the value of a call option on a non-dividend-paying stock.

5. Use the Black-Scholes Option Pricing formula to calculate the price of a European put option on a non-dividend paying stock.

6. Compute the value of a European option on a dividend-paying stock.


Learning Objectives (cont'd)

7. Define the Black-Scholes replicating portfolio for a call option on a non-dividend paying stock or a European put option on a non-dividend paying stock.

8. Discuss what is meant by risk-neutral probabilities, and show how these probabilities can be used to price any other asset for which the payoffs in each state are known.

9. Define and calculate the risk-neutral probability that the stock price will increase in a binomial tree.

10. Calculate and interpret the beta of an option.

11. Use the Black-Scholes formula to unlever the equity beta of a firm and find the beta of debt.


21.1 The Binomial Option Pricing Model

• Binomial Option Pricing Model– A technique for pricing options based on the

assumption that each period, the stock’s return can take on only two values

• Binomial Tree– A timeline with two branches at every date

representing the possible events that could happen at those times


A Two-State Single-Period Model

• Replicating Portfolio

– A portfolio consisting of a stock and a risk-free bond that has the same value and payoffs in one period as an option written on the same stock

• The Law of One Price implies that the current value of the call and the replicating portfolio must be equal.


A Two-State Single-Period Model (cont'd)

• Assume

– A European call option expires in one period and has an exercise price of $50.

– The stock price today is equal to $50 and the stock pays no dividends.

– In one period, the stock price will either rise by $10 or fall by $10.

– The one-period risk-free rate is 6%.



• The payoffs can be summarized in a binomial tree.



• Let be the number of shares of stock purchased, and let B be the initial investment in bonds.

• To create a call option using the stock and the bond, the value of the portfolio consisting of the stock and bond must match the value of the option in every possible state.



• In the up state, the value of the portfolio must be $10.

• In the down state, the value of the portfolio must be $0.

60 1.06 10B

40 1.06 0B



• Using simultaneous equations, and B can be solved for. = 0.5

– B = –18.8679



• A portfolio that is long 0.5 share of stock and short approximately $18.87 worth of bonds will have a value in one period that exactly matches the value of the call.

– 60 × 0.5 – 1.06 × 18.87 = 10

– 40 × 0.5 – 1.06 × 18.87 = 0



• By the Law of One Price, the price of the call option today must equal the current market value of the replicating portfolio.

• The value of the portfolio today is the value of 0.5 shares at the current share price of $50, less the amount borrowed.50 B 50(0.5) 18.87 6.13



• Note that by using the Law of One Price, we are able to solve for the price of the option without knowing the probabilities of the states in the binomial tree.


Figure 21.1 Replicating an Option in the Binomial Model


The Binomial Pricing Formula

• Assume:

– S is the current stock price, and S will either go up to Su or go down to Sd next period.

– The risk-free interest rate is rf .

– Cu is the value of the call option if the stock goes up and Cd is the value of the call option if the stock goes down.


The Binomial Pricing Formula (cont'd)

• Given the above assumptions, the binomial tree would look like:

• The payoffs of the replicating portfolios could be written as:

1 (1 ) and (1 ) u f u d f dS r B C S r B C


The Binomial Pricing Formula (cont'd)

• Solving the two replicating portfolio equations for the two unknowns and B yields the general formula for the replicating formula in the binomial model.– Replicating Portfolio in the Binomial Model

• The value of the option is:– Option Price in the Binomial Model

and

1 u d d d

u d f

C C C SB

S S r

C S B


Textbook Example 21.1


Textbook Example 21.1 (cont'd)


A Multiperiod Model

• Consider a two-period binomial tree for the

stock price.


A Multiperiod Model (cont'd)

• To calculate the value of an option in a multiperiod binomial tree, start at the end of the tree and work backwards.

– At time 2, the option expires, so its value is equal to its intrinsic value.

• In this case, the call will be worth $10 if the stock price goes up to $60, and will be worth zero otherwise.



• The next step is to determine the value of the option in each possible state at time 1. – If the stock price has gone up to $50 at time 1,

the binomial tree will look like:

• The value of the option will be $6.13 (just as in the single period example.)



• The next step is to determine the value of the option in each possible state at time 1.

– If the stock price has gone down to $30 at time 1, the binomial tree will look like:

• The value of the option will be $0 since it is worth $0 in either state.



• The final step is to determine the value of the option in each possible state at time 0.

– Solving for and B: 6.13 0

0.3065 and 50 30

0 30(0.3065) 8.67

1 1.06

u d

u d

d d

f

C C

S S

C SB

r



• The final step is to determine the value of the option in each possible state at time 0.

– Thus, the initial option value is: 40(0.3065) 8.67 $3.59C S B



• Dynamic Trading Strategy

– A replication strategy based on the idea that an option payoff can be replicated by dynamically trading in a portfolio of the underling stock and a risk-free bond




– In the two-period example above, the replicating portfolio will need to be adjusted at the end of each period.

• The portfolio starts off long 0.3065 shares of stock and borrowing $8.67.

• If the stock price drops to $30, the shares are worth $9.20 and the debt has grown to $9.20.

– $30 × 0.3065 = $9.20 and $8.67 × 1.06 = $9.20

• The net value of the portfolio is worthless and the portfolio is liquidated.




– If the stock price rises to $50, the net value of the portfolio rises to $6.13.

• The new of the replicating portfolio is 0.5. Therefore 0.1935 more shares must be purchased.

– 0.50 − 0.3065 = 0.1935

• The purchase will be financed by additional borrowing.

– 0.1935 × $50 = $9.67

• At the end the total debt will be $18.87.– $8.67 × 1.06 + 9.67 = $18.87– This matches the value calculated previously.


Making the Model Realistic

• Although binary up or down movements are not the way stock prices behave on an annual or even daily basis, by decreasing the length of each period, and increasing the number of periods in the stock price tree, a realistic model for the stock price can be constructed.


Figure 21.2 A Binomial Stock Price Path


21.2 The Black-Scholes Option Pricing Model

• Black-Scholes Option Pricing Model

– A technique for pricing European-style options when the stock can be traded continuously. It can be derived from the Binomial Option Pricing Model by allowing the length of each period to shrink to zero and letting the number of periods grow infinitely large.


The Black-Scholes Formula

• Black-Scholes Price of a Call Option on a Non-Dividend-Paying Stock

– Where S is the current price of the stock, K is the exercise price, and N(d) is the cumulative normal distribution

• Cumulative Normal Distribution

– The probability that an outcome from a standard normal distribution will be below a certain value

1 2 ( ) ( ) ( )C S N d PV K N d


The Black-Scholes Formula (cont'd)

– Where is the annual volatility, and T is the number of years left to expiration

1 2 1

ln[ / ( )]

2

S PV K Td and d d T

T


Figure 21.3 Normal Distribution



• Note: Only five inputs are needed for the Black-Scholes formula.

– Stock price

– Strike price

– Exercise date

– Risk-free rate

– Volatility of the stock


Table 21.1 JetBlue Option Quotes


Alternative Example 21.3

• Problem

– Assume:• CLW Inc. does not pay dividends.

• The standard deviation of CLW is 45% per year.

• The risk-free rate is 5%.

• CLW stock has a current price of $24.

– Using the Black-Scholes formula, what is the price for a ½ year American call option on CLW with a strike price of $30?



• Solution

( 0.4658) 0.3207 ( 0.7480) 0.2165N N

1

2

ln[$24 /$29.277] .45 .50.4658

2.45 .5

0.4658 .45 .5 0.7840

d

d

1/ 2

$30$29.277

1.05PV(K)

$24 0.3207 $29.277 0.2165 $1.36C


Figure 21.4 Black-Scholes Value on July 24, 2009 of the December 2009 $6 Call on JetBlue Stock



• European Put Options

– Using put-call parity, the value of a European put option is:

– Black-Scholes Price of a European Put Option on a Non-Dividend-Paying Stock

2 1 ( )[1 ( )] [1 ( )]P PV K N d S N d



• Problem

– Assume:• CLW Inc. does not pay dividends.

• The standard deviation of CLW is 45% per year.

• The risk-free rate is 5%.

• CLW stock has a current price of $24.

– Using the Black-Scholes formula, what is the price for a ½ year American put option on CLW with a strike price of $30?



• Solution

( 0.4658) 0.3207 ( 0.7480) 0.2165N N

1

2

ln[$24 /$29.277] .45 .50.4658

2.45 .5

0.4658 .45 .5 0.7840

d

d

1/ 2

$30$29.277

1.05PV(K)

$29.277[1 0.2165] $24[1 0.3207] $6.64P


Figure 21.5 Black-Scholes Value on July 24, 2009 of the January 2010 $5.00 Put on JetBlue Stock



• Dividend-Paying Stocks

– If PV(Div) is the present value of any dividends paid prior to the expiration of the option, then:

• Where Sx is the price of the stock excluding any dividends

( )xS S PV Div




– Because a European call option is the right to buy the stock without these dividends, it can be evaluated by using the Black-Scholes formula with Sx in place of S.




– A special case is when the stock will pay a dividend that is proportional to its stock price at the time the dividend is paid. If q is the stock’s (compounded) dividend yield until the expiration date, then:

/ (1 )xS S q



• Problem– FPA, Inc. has a current stock price of $42.40 per

share. The company will pay an annual dividend yield of 6% on its stock. If FPA’s returns have a standard deviation of 25% and the risk-free rate is 4%, what is the value of a one-year call option on the stock with a strike price of $40?



• Solution– The price of the call can be found using the

Black-Scholes model where the stock price is adjusted for the dividend yield:

– Thus,

1

2

$42.40 $40.00ln[ / ] .25 11.06 1.04 0.2819

2.25 1

.2819 .25 1 .0319

d

d

1 2( ) ( ) ( ) ( ) $40 0.6110 $38.46 0.5127 $4.72C S SN d PV K N d


Implied Volatility

• Of the five required inputs in the Black-Scholes formula, only is not observable directly.

– Practitioners use two strategies to estimate the value of .

• Use historical data

• “Back out” the implied volatility– Implied Volatility

» The volatility of an asset’s returns that is consistent with the quoted price of an option on the asset


The Replicating Portfolio

• Given:– Option Price in the Binomial Model

• Then:– Black-Scholes Replicating Portfolio of a Call

Option

1

2

( )

( ) ( )

N d

B PV K N d

C S B


The Replicating Portfolio (cont'd)

• Option Delta ()

– The change in the price of an option given a $1 change in the price of the stock. The number of shares in the replicating portfolio for the option.

– Note: Because is always less than 1, the change in the call price is always less than the change in the stock price.


Figure 21.6 Replicating Portfolio for the Call Option in Example 21.7



• Note: The replicating portfolio of a call option always consists of a long position in the stock and a short position in the bond.

– The replicating portfolio is a leveraged position in the stock.

• A leveraged position in a stock is riskier than the stock itself, which implies that call options on a positive beta stock are more risky than the underlying stock and therefore have higher returns and higher betas.



• The replicating portfolio of a put option is calculated as:– Black-Scholes Replicating Portfolio of a Put

Option

• Note: The replicating portfolio of a put option always consists of a long position in the bond and a short position in the stock, implying that put options on a positive beta stock will have a negative beta.

1

2

[1 ( )]

( )[1 ( )]

N d

B PV K N d


21.3 Risk-Neutral Probabilities

• If all market participants are risk neutral, then all financial assets (including options) have the same cost of capital, the risk-free rate of interest.


A Risk-Neutral Two-State Model

• Assume a world consisting of only risk-neutral investors, and consider the original

two-state example.

– The stock price today is equal to $50.

– In one period it will either go up by $10 or go down by $10.

– The one-period risk-free rate of interest is 6%.


A Risk-Neutral Two-State Model (cont'd)

• If is the probability that the stock price will increase, then (1 – ) is the probability that it will go down.

– The value of the stock today must equal the present value of the expected price next period discounted at the risk-free rate.

• Solving for yields = .65

60 40(1 )50

1.06



• The call option had an exercise price of $50, so it will be worth either $10 or nothing at expiration. The present value of the expected payouts is:

10(0.65) 0(1 0.65) 6.13

1.06



• This is precisely the value calculated using the Binomial Option Pricing Model where it was not assumed that investors were risk neutral.

– Because no assumption on the risk preferences of investors is necessary to calculate the option price using either the Binomial Model or the Black-Scholes formula, the models work for any set of preferences, including risk-neutral investors.


Implications of the Risk-Neutral World

• The Binomial and Black-Scholes models give the same option price no matter what the actual risk preferences and expected stock returns are.

– In the real world, investors are risk averse and require a positive risk premium to compensate for risk, while in a hypothetical risk-neutral world, investors do not require compensation for risk.


Implications of the Risk-Neutral World (cont'd)

• In other words, is not the actual probability of the stock price increasing.

– Rather, it represents how the actual probability would have to be adjusted to keep the stock price the same in a risk-neutral world.



• Risk-Neutral Probabilities

– The probability of future states that are consistent with current prices of securities assuming all investors are risk neutral.

• Probabilites and (1 − ) are risk-neutral probabilities.

– Also known as State-Contingent Prices, State Prices, or Martingale prices



• Assume from the previous example that the current price of $50 has a true probability of 75% of increasing to $60 and a true probability of 25% of decreasing to $40.



• This stock’s true expected return is therefore:

– Given the risk-free interest rate of 6%, this stock has a 4% risk premium. But as calculated earlier, the risk-neutral probability that the stock will increase is 65%, which is less than the true probability.

• Thus, the expected return of the stock in the risk-neutral world is 6%.

– (60 × 0.65 + 40 × 0.35 ) ∕ 50 − 1 = 6%

60 0.75 40 0.25 1 10%

50



• To ensure that all assets in the risk-neutral world have an expected return equal to the risk-free rate, relative to the true probabilities, the risk-neutral probabilities overweight the bad states and underweight the good states.


Risk-Neutral Probabilities and Option Pricing

• Consider again the general binomial stock price tree.

• Compute the risk-neutral probability that makes the stock’s expected return equal to the risk-free interest rate:

(1 ) 1 u d

f

S Sr

S


Risk-Neutral Probabilities and Option Pricing (cont'd)

• Solving for the risk-neutral probability yields:

– The value of the option can be calculated by computing its expected payoff using the risk-neutral probabilities, and discount the expected payoff at the risk-free interest rate.

(1 )

f d

u d

r S S

S S



• Derivative Security– A security whose cash flows depend solely on

the prices of other marketed assets

• The probabilities in the risk-neutral world can be used to price any derivative security.



• Monte Carlo Simulation

– A common technique for pricing derivative assets in which the expected payoff of the derivative security is estimated by calculating its average payoff after simulating many random paths for the underlying stock

• In the randomization, the risk-neutral probabilities are used and so the average payoff can be discounted at the risk-free rate to estimate the derivative security’s value.


21.4 Risk and Return of an Option

• To measure the risk of an option, we must compute the option beta.

– This can be accomplished by computing the beta of the replicating portfolio.

• Where βS is the stock’s beta and βB is the bond’s beta. Since the bond is riskless, βB = 0 and the option beta is:

– Beta of an Option

option S B

S B

S B S B

option S

S

S B


21.4 Risk and Return of an Option (cont'd)

• Leverage Ratio

– A measure of leverage obtained by looking at debt as a proportion of value, or interest payments as a proportion of cash flows

– It is calculated as:• S ∕ (S+ B)


Figure 21.7 Leverage Ratios of Options


Figure 21.8 Security Market Line and Options


21.5 Beta of Risky Debt

• When the beta of debt is zero, then:

– Where E is the beta of equity and U is the beta of unlevered equity

– However, for companies with high debt-to-equity ratios, the assumption that the beta of debt is zero is unrealistic as these companies have a positive probability of bankruptcy.

1 E U

D

E


21.5 Beta of Risky Debt (cont'd)

• When the beta of debt is not zero, then:

– Where A is the value of the firm’s assets, E is the value of equity and D is the value of debt

• And

• And

1 E U U

A D

E E

(1 ) (1 ) 1 D U U

A E

D D

1

EU D

E


Figure 21.9 Beta of Debt and Equity


Agency Costs of Debt

• Recall from Chapter 16, leverage creates an asset substitution problem because the value of the equity call option increases with the firm’s volatility. – Thus, equity holders may have an incentive to

take excessive risk.


Textbook Example 21.11 (cont’d)


Chapter Quiz

1. What is the key assumption of the binomial option pricing model?

2. What are the key assumptions of the Black-Scholes option pricing model?

3. What is the implied volatility of a stock?4. What are risk-neutral probabilities and how do

they relate to valuing options?5. How is the beta of a call option related to the

beta of the underlying stock?6. How can one estimate the beta of debt?

berk chapter 21: option valuation

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