1

16

Option Valuation

16-2

What is an Option Worth?• At expiration, an option is worth its intrinsic value.• Before expiration, put-call parity allows us to price

options. But,• To calculate the price of a call, we need to know the put

price.• To calculate the price of a put, we need to know the call

price.

• So, we want to know the value of a call option:• Before expiration, and• Without knowing the price of the put

16-3

A Simple Model to Value Options Before Expiration, I.

• Suppose we want to know the price of a call option with• One year to maturity (T = 1.0)• A $110 exercise price (K = 110)• The current stock price is $108 (S0 =108)• The one-year risk-free rate is 10 percent (r = 10%)

16-4

A Simple Model to Value Options Before Expiration, I.• We know what the stock price will be in one year.

• S1 = $130 or $115 (but no other values).• S1 is still uncertain. • We do not know the probabilities of these two values.

• Therefore, the call option value at expiration will be:• $130 – $110 = $20 or• $115 - $110 = $5

• This call option is certain to finish in the money.• A similar put option is certain to finish out of the money.

16-5

A Simple Model to Value Options Before Expiration, II.

• If you know the price of a similar put, you can use put-call parity to price a call option before it expires.

• The chosen pair of stock prices guarantees that the call option finishes in the money.

• Suppose, however, we want to allow the call option to expire in the money OR out of the money.

$8. $100 - $108C

)$110/(1.10 $108 0-C

r)K/(1-S P - C T0

16-6

The One-Period Binomial Option Pricing Model—The Assumptions • S = the stock price today; the stock pays no

dividends.• Assume that the stock price in one period is either S

× u or S × d, where: • u (for “up” factor) is > 1• and d (for “down” factor) < 1• Suppose S = $100, u = 1.1 and d = .95

• S1 = $100 × 1.1 = $110 or S1 = $100 × .95 = $95

• What is the call price today, if:• K = 100• R = 3%

16-7

The One-Period Binomial Option Pricing Model—The Setup

• Consider the following portfolio:• Buy a fractional share of the underlying asset-

• The Greek letterDelta) = this fraction• Sell one call option• Finance the difference by borrowing $(S – C)

• Key Question:

What is the value of this portfolio, today and at option expiration?

16-8

The Value of this Portfolio(long Shares and short one call) is:

• Important: S is NOT the change in S.

Rather, it is a dollar amount, S = Δ * S.

S - C

S×u - Cu

S×d - Cd

Portfolio Value At ExpirationPortfolio Value Today

Cu and Cd : The intrinsic value of the call if the stock price increases to S×u or decreases to S×d.

16-9

To Calculate Today’s Call Price, C:• There is one combination of a fractional share and

one call that makes this portfolio risk-less.• The portfolio will have the same value when the

underlying asset increases as it does when the underlying asset decreases in value.

• The portfolio is riskless if: Su – Cu = Sd – Cd

• We know all values in this equation, except • S = $100; Su = $110; Sd = $95• Cu = MAX(Su – K, 0) = MAX($110 – 100,0) = $10

• Cd = MAX(Sd – K, 0) = MAX($95 – 100,0) = $0

16-10

First Step: Calculate

To make the portfolio riskless:Su – Cu = Sd – Cd

Su – Sd = Cu – Cd

Su – Sd) = Cu – Cd,

We can calculate := (Cu – Cd) / Su – Sd)

= (10 – 0) / 110 – 95

= 10 / 15

= 2 / 3

16-11

Sidebar: What is ?

• , delta, is the riskless hedge ratio.

• , delta, is the fractional share amount needed to hedge one call.

• Therefore, the number of calls to hedge one share is 1/

16-12

The One-Period Binomial Option Pricing Model—The Formula

• A riskless portfolio today should be worth (S – C)(1+r) in one period.

• So:

(S – C)(1+r) = Su – Cu

(which equals Sd – Cd because we chose the

“correct” ).

• Solve the equation above for C.

16-13

The One-Period Binomial Option Pricing Model—The Formula

• Solving the equation above for C:

r1

Cu)rΔS(1C

r)C(1Cu)rΔS(1

Cr)C(1u)rΔS(1

Cr)C(1ΔSur)ΔS(1

CΔSur)C(1r)ΔS(1

C - Sur)C)(1-S(

u

u

u

u

u

u

16-14

Calculate the Call Price, C.

$5.181.03

$5.33C

.031

10$.07)($200/3)(C

.031

10$1.10).03)(1(2/3)($100C

r1

Cu)rΔS(1C u

16-15

Now We Can Calculate the Call Price, C.

If C= $5.18, what is the price of a similar put?

Using Put-Call Parity:

$2.27P

100$100/1.03$5.18P

$100/1.03$5.18$100P

r)K/(1CSP

16-16

The Two-Period Binomial Option Pricing Model• Suppose there are two periods to expiration instead

of one. • Repeat much of the process used in the one-period

binomial option pricing model.

• This method can be used to price:• European call options. • European put options.• American calls and puts (with a modification to allow for

early exercise).• An exotic array of options (with the appropriate

modifications).

16-17

The Method We can find binomial option prices for two (or more)

periods by using the following five steps:

1. Build a price “tree” for stock prices through time.

2. Use the intrinsic value formula to calculate the possible option values at expiration.

3. Calculate the fractional share needed to form each riskless portfolio at the next-to-last date.

4. Calculate all possible option prices at the next-to-last date.

5. Repeat this process by working back to today.

16-18

The Binomial Option Pricing Model with Many Periods

16-19

What Happens When the Number of Periods Gets Really, Really Big?

• For European options on non-dividend paying stocks, the binomial method converges to the Black-Scholes option pricing formula.

• To calculate the prices of many other types of options, however, we still need to use a computer (and methods similar in spirit to the binomial method).

16-20

The Black-Scholes Option Pricing Model

• The BSOPM calculates the price of a call option before maturity• Dates from the early 1970s• Professors Fischer Black and Myron Scholes• Facilitated option pricing • CBOE was launched soon after BSOPM appeared• 1997 Nobel Prize in Economics

• Important contributions by professor Robert Merton • The Black-Scholes-Merton option pricing model

16-21

C = Current call option valueP = Current put option value

S = Current stock priceK = Option strike or exercise pricer = Risk-free interest rate Stock price volatilityT = time to maturity of the option in yearsln = Natural log function; ln(x)e = 2.71828, the base of the natural log; exp()N(d) = probability that a random draw from a

normal distribution will be less than d

Black-Scholes Option ValuationVariables

Solution Variables

Input Variables

Functions

16-22

Black-Scholes Option Valuation

)()( 21 dNKedSNC rT

Tdd

T

TrKSd

12

2

1

)2/()/ln(

)()( 12 dSNdNKeP rT

16-23

Formula Functions• e-rt = exp(-rt) = natural exponent of the value

of –rt (a discount factor)• ln(S/K) = natural log of the "moneyness"

term, S/K• N(d1) and N(d2) denotes the standard

normal probability for the values of d1 and d2.

• Formula makes use of the fact that:

N(-d1) = 1 - N(d1)

16-24

Example: Computing Pricesfor Call and Put Options

Suppose you are given the following inputs:

S = $50

K = $45

T = 3 months (or 0.25 years)

= 25% (stock volatility)

r = 6%

16-25

Step 1: Calculating d1 and d2

0.900380.250.251.02538Tσdd

1.02538 0.125

0.25 0.09125 0.10536

0.250.25

0.2520.250.064550ln

Tσ

T2σrKSlnd

12

2

2

1

16-26

Step 2: Excel’s “=NORMSDIST(x)” Function

=NORMSDIST(1.02538) = 0.84741 = N(d1)

=NORMSDIST(0.90038) = 0.81604 = N(d2)

N(-d1) = 1 - N(d1):

N(-1.02538) = 1 – N(1.02538)

= 1 – 0.84741 = 0.15259 = N(-d1)

N(-0.90038) = 1 – N(0.90038)

= 1 – 0.81604 = 0.18396 = N(-d2)

16-27

Step 3a: The Call Price

C = SN(d1) – Ke–rTN(d2)

= $50(0.84741) – 45(e-(.06)(.25))(0.81604)

= 50(0.84741) – 45(0.98511)(0.81604)

= $6.195

16-28

Step 3b: The Put Price:

P = Ke–rTN(–d2) – SN(–d1)

= $45(e-(.06)(.25))(0.18396) – 50(0.15259)

= 45(0.98511)(0.18396) – 50(0.15259)

= $0.525

16-29

Verify Our Results Using Put-Call Parity

$5.42$44.33$50.00$5.42

45e-50$0.565$5.985

KeSPC

0.25)(0.06

rT

Note: Options must be European-style

16-30

Valuing the Options Using ExcelStock Price: 50.00 Stock: 50.00Strike Price: 45.00 Discounted Strike: 44.33

Volatility (%): 25.00Time (in years): 0.2500

Riskless Rate (%): 6.00

d(1): 1.02538N(d1): 0.84741 N(-d1): 0.15259

d(2): 0.90038N(d2): 0.81604 N(-d2): 0.18396

Call Price: 6.195$

Put Price: 0.525$

16-31

Using a Web-based Option Calculator• www.numa.com.

16-32

Varying the Option Price Input Values• An important goal of this chapter is to show how an

option price changes when only one of the five inputs changes.

• The table below summarizes these effects.

16-33

Factors Influencing Option Values

Table 16.3Call Put

Underlying stock price S + - Delta

Strike price of option contract K - +Time remaining to expiration T + +Volatility of the underlying stock price σ + +Risk-free interest rate r + -

Effect on Option Value

Input Factor

Common Name

16-34

Varying the Underlying Stock Price

• Changes in the stock price have a big effect on option prices.

16-35

Varying the Time Remaining Until Option Expiration

16-36

Varying the Volatility of the Stock Price

16-37

Varying the Interest Rate

16-38

Calculating Delta• Delta measures the dollar impact of a change in

the underlying stock price on the value of a stock option.

Call option delta = N(d1) > 0

Put option delta = –N(–d1) < 0

• A $1 change in the stock price causes an option price to change by approximately delta dollars.

16-39

Example: Calculating DeltaStock Price: 50.00 Stock: 50.00Strike Price: 45.00 Discounted Strike: 44.33

Volatility (%): 25.00Time (in years): 0.2500

Riskless Rate (%): 6.00

d(1): 1.02538N(d1): 0.84741 N(-d1): 0.15259

d(2): 0.90038N(d2): 0.81604 N(-d2): 0.18396

Call option delta = 0.84741Call Price: 6.195$ Put option delta = -0.15259

Put Price: 0.525$

16-40

The Call "Delta" Prediction:• Call delta = 0.84741

if the stock price increases by $1, the call option price will increase by about $0.85

• From the previous example:• Stock price = $50• Call option price = $6.195

• If the stock price is $51:• Call option value = $7.060• Increase of about $0.868

16-41

The Put "Delta" Prediction:• Put delta value = -0.15259

if the stock price increases by $1, the put option price will decrease by $0.15.

• From the previous example:• Stock price = $50• Put option price = $0.525

• If the stock price is $51:• Put option value = $0.390 • Decrease of about $0.14

16-42

Hedging with Stock Options• You own 1,000 shares of XYZ stock and you want

protection from a price decline.• Let’s use stock and option information from before—

in particular, the “delta prediction” to help us hedge.• You want changes in the value of your XYZ shares

to be offset by the value of your options position. That is:

options # Delta Option shares price stock in Change

options # price option in Change shares price stock in Change

16-43

Hedging Using Call Options—The Prediction

• Delta = 0.8474; stock price declines by $1:

12.- 100 / 1,180.08-

1,180.08- 0.8474 / 1,000- options #

options # 0.8474 1,000 1-

options # Delta shares price stock in Change

Write 12 call options with a $45 strike to hedge your stock.

16-44

Hedging with Calls - Results• Call option gain nearly offsets your loss of $1,000.• Why is it not exact?

• Call Delta falls when the stock price falls.• Therefore, you did not sell quite enough call options.

Stock Price Portfolio Value Call PriceCall Position

Value

$50 $50,000 $6.200 -$7,440

$49 $49,000 $5.370 -$6,444

-$1 ($1,000) -$0.830 $996

16-45

Hedging Using Put Options—The Prediction

• Delta = -0.1526; stock price declines by $1:

66. 100 / 6,553.08

6,553.08 0.1526- / 1,000- options #

options # 0.1526- 1,000 1-

options # Delta shares price stock in Change

Buy 66 put options with a strike of $45 to hedge your stock.

16-46

Hedging Using Put Options: Results

• Put option gain more than offsets $1,000 loss • Why is it not exact?

• Put Delta also falls (gets more negative) when the stock price falls.

• Therefore, you bought too many put options—this error is more severe the lower the value of the put delta.

• To get closer: Use a put with a strike closer to at-the-money.

Stock Price Portfolio Value Put PricePut Position

Value$50 $50,000 $0.530 $3,498

$49 $49,000 $0.700 $4,620

-$1 ($1,000) $0.170 $1,122

16-47

Hedging a Portfolio with Index Options

• Many institutional money managers use stock index options to hedge equity portfolios

• Regular rebalancing needed to maintain an effective hedge • Underlying Value Changes • Option Delta Changes• Portfolio Value Changes• Portfolio Beta Changes

valuecontract Option delta Option

value Portfolio beta Portfolio contracts option of Number

16-48

Hedging with Stock Index Options

valuecontract Option delta Option

value Portfolio beta Portfolio contracts option of Number

You manage a $10 million stock portfolio.

You attempt to maintain a portfolio beta of 1.00

You decide to hedge your position by buying index put options with a contract value of $125,000 and a delta of 0.579.

How many contracts do you need to buy?

16-49

Hedging with Stock Index Options

Portfolio value = $10 million Portfolio beta = 1.00

Index contract value = $150,800 Option delta = 0.579

valuecontract Option delta Option

value Portfolio beta Portfolio contracts option of Number

115 10015080.579

10,000,0001.00

100 ValueUnderlyingDelta Option

ValuePortfolioBeta Portfolio contracts option of Number

Sell 115 call options

16-50

Implied Standard Deviations

• Implied standard deviation (ISD) • = Implied volatility (IVOL) • Stock price volatility estimated from an option

price • Of the six BSOPM input factors, only stock price

volatility is not directly observable

• Calculating an implied volatility requires:• All 5 other input factors• Either a call or put option price

16-51

CBOE Published Implied Volatilities for Stock Indexes

• CBOE publishes 3 volatility indexes:• S&P 500 Index Option Volatility (VIX)• S&P 100 Index Option Volatility (VXO)• Nasdaq 100 Index Option Volatility (VXN)

• Each volatility index calculated using ISDs from eight options:• 4 calls and 4 puts, each with two maturity dates:

• 2 slightly out of the money • 2 slightly in the money

• Provides investors with information about market volatility in the coming months.

16-5252

Employee Stock Options, ESOs

• ESO = call option that a firm grants (gives) to employees.

• ESO life generally 10 years• Cannot be sold• “Vesting” period of about 3 years

• If employee leaves the company before “vested," the ESOs are lost

• Once vested, ESOs can be exercised any time over its remaining life

16-5353

Why are ESOs Granted?• Motivates employees to make decisions

that help the stock price increase• Powerful motivator

• Payoffs can be large.• High stock prices: ESO holders gain and

shareholders gain.• No upfront costs to the company

• Can act as a substitute for ordinary wages• Helpful in recruiting employees

16-5454

ESO Repricing• Generally issued exactly “at the money”

• Intrinsic value = zero.• No value from immediate exercise.• Still valuable

• “Underwater” Options• Stock price falls after the ESO is granted• “Restriking” or “Repricing.”

• Companies lower the strike prices • Controversial practice

16-5555

ESO Repricing Controversy• PRO: Once an ESO is “underwater,” it loses

its ability to motivate employees.• Small chance for payoff • Employees may leave to get “fresh” options.

• CON: Lowering strike price = reward for failing• Decisions by employees made the stock price fall.• If employees know that ESOs will be repriced, the

ESOs loose their ability to motivate employees.

16-56

ESOs Today• Most companies award ESOs on a regular

basis.• Quarterly• Annually

• Therefore, employees will always have some “at the money” options.

• Regular grants of ESOs mean that employees always have some “unvested” ESOs—giving them the added incentive to remain with the company.

16-57

Valuing Employee Stock Options• Companies must report estimates of ESO

values (FASB 123)• BSOPM modified by Merton widely used for

this purpose• For example:

• In December 2002, the Coca-Cola Company granted ESOs with a stated life of 15 years.

• ESOs often exercised before maturity, so Coca-Cola also used a life of 6 years to value these ESOs.

16-58

)d(NKe)d(NSeC rTyT21

Employee Stock Option Valuation

Tdd

T

T)/yr()K/Sln(d

12

2

1

2

“S” equals the stock’s price at the grant date

The Black-Scholes-Merton Option Pricing Model incorporates the possibility that the underlying stock will pay a dividend during the life of the option:

“y” = the annual dividend yield for the underlying stock

16-59

Example: Valuing Coca-ColaESOs Using Excel

Stock Price: 44.55 Stock Price: 44.55Discounted Stock: 35.10 Discounted Stock: 40.23

Strike Price: 44.66 Strike Price: 44.66Discounted Strike: 19.13 Discounted Strike: 36.41

Volatility (%): 25.53 Volatility (%): 30.20Time (in years): 15 Time (in years): 6

Riskless Rate (%): 5.65 Riskless Rate (%): 3.40Dividend Yield (%): 1.59 Dividend Yield (%): 1.70

d(1): 1.10792 d(1): 0.50458N(d1): 0.86605 N(d1): 0.69307

d(2): 0.11915 d(2): -0.23517N(d2): 0.54742 N(d2): 0.40704

Call Price: 19.92$ Call Price: 13.06$

16-60

Summary: Coca-Cola Employee Stock Options

16-61

Useful Websites

• www.jeresearch.com (information on option formulas)• www.option-price.com (for a free option price calculator)• www.numa.com (for “everything about options”)• www.wsj.com/free (option price quotes)• www.ino.com (Web Center for Futures and Options)• www.optionetics.com (Optionetics)• www.pmpublishing.com (free daily volatility summaries)• www.ivolatility.com (for applications of implied volatility)