b40.2302 class #4

©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill

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B40.2302 Class #4

BM6 chapters 20, 21 Based on slides created by Matthew Will Modified 04/22/23 by Jeffrey Wurgler

Spotting and Valuing Options

Principles of Corporate FinanceBrealey and Myers Sixth Edition

Slides by

Matthew Will, Jeffrey Wurgler

Chapter 20



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Topics Covered

Calls, Puts and Shares Financial Alchemy with Options Option Valuation

Constructing equivalent portfolios Risk-neutral valuation Black-Scholes


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Option Terminology

Put Option

Right to sell an asset at a specified exercise price on or before a specified exercise date.

Call OptionRight to buy an asset at a specified exercise price on or before a specified exercise date.


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Option Value

The value of an option at expiration depends on the difference between the stock price and the exercise price.

Example - Value at expiration given $85 exercise price

00051525ValuePut 25155000Value Call

110100908070$60eStock Pric


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Option ValuePayoff on a riskless bond/loan at maturity … is fixed (lender’s perspective).

Share Price

Bon

d va

lue

0


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Option ValuePayoff to a share when you want to sell it … depends on share price (share buyer’s perspective).

Share Price

Shar

e va

lue

50

50

0


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Option ValueCall option value at expiration given a $85 exercise price (call buyer’s perspective).

Share Price

Cal

l opt

ion

valu

e

85 105

$20

0


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Option ValuePut option value at expiration given a $85 exercise price (put buyer’s perspective).

Share Price

Put o

ptio

n va

lue

80 85

$50


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Option Obligations

Buyer SellerCall option Right to buy asset Obligation to sell assetPut option Right to sell asset Obligation to buy asset


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Option ValueCall option value at expiration given a $85 exercise price (call seller’s perspective).

Share Price

Cal

l opt

ion

$ pa

yoff

85

0


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Option ValuePut option value at expiration given a $85 exercise price (put seller’s perspective).

Share Price

Put o

ptio

n $

payo

ff

85

0


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Financial Alchemy

Protective Put = Buy stock and buy put

Share Price

Posi

tion

Val

ue “Protective Put”

Long Put

Long Stock


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Financial AlchemyStraddle = Long call and long put - Profits from high volatility

Share Price

Posi

tion

Val

ue

Straddle


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Put-Call Parity

The following two strategies give exactly the same payoff (a “protective put” payoff)… Buy share and buy put Lend money and buy call

… so they must sell at exactly the same price

This leads to the “put-call parity” formula


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Put-Call Parity

Value of a call + PV(Exercise price)= Value of put + Current share price

Holds only for European options Requires put and call with same exercise price If stock pays dividend, need to make adjustment


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Safe versus risky debt

An application of option logic to capital structure:

When a firm borrows, the lender acquires the company and the shareholders obtain the option to buy it back by paying off the debt

Shhs have thus purchased a call option on the firm

The “strike price” is the amount of debt D that must be repaid


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Safe versus risky debtShareholder value at maturity given $D borrowing (shareholder’s perspective).

Firm asset value

Shar

ehol

der p

ayof

f

D0


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Safe versus risky debtLender value at maturity given $D lending to a risky firm (lender’s perspective).

Firm asset value

Deb

thol

der p

ayof

f

D

0

D


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Option Value

Upper Limit

Stock Price

Lower Limit

{Stock price - exercise price, 0}whichever is higher


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Option Value

Option Price

Stock Price

Upper limit: share price

Lower limit: payoff if exercised immediatelyACTUAL VALUE

Exercise Price

Upper and lower limits to call option value


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Option Value

Option Price

Stock Price

ACTUAL VALUE

Exercise Price

Notice the shape of an unexpired option’s value


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Option Value

Determinants of Call Option Price1 - Underlying stock price (+)2 - Exercise (“strike”) price (-)3 - Standard deviation of stock returns (+)4 - Time to option expiration (+)5 - Interest rate (+)


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Why can’t do DCF for options?

Can in principle forecast cash flows

But discount rate is changing over time! Risk of an option changes every time the stock

price moves! E.g. when price goes up, option payoff becomes

more certain, option’s risk & beta go down… A huge nightmare!


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Constructing Option Equivalents

Trick to valuing options is to set up an “equivalent” or “replicating” portfolio that we can already value.

Equivalent portfolio involves both buying a certain fraction of a share (called “option delta” or “hedge ratio”) and borrowing.


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Constructing Option EquivalentsIntel call option

• Strike = $85, six months to exercise, 2.5% interest for six months

• Intel is right now at $85 and can either rise to $106.25 or fall to $68 over next six months (keep it simple)

• Payoffs to call option are therefore:$0 if price falls$21.25 if price rises

• Notice this is same payoff structure you would get from an equivalent portfolio that is long 5/9 of one share and borrows $36.86 from the bank! So must have same value.


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Constructing Option Equivalents If stock goes down,

• 5/9 of share is worth 5/9*68=$37.38• And have to repay $36.86*1.025= -$37.78• Total = $0, just like option

If stock goes up,• 5/9 of share is worth 5/9*106.25=$59.03• And have to repay $36.86*1.025= -$37.78• Total = $21.25, just like option

Price of option must be the same as price of equivalent portfolio. • Equiv. portf. has a value today of 5/9*(85) -36.86 = $10.36. • So option is worth $10.36.


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Risk-neutral valuation

Value of that option was $10.36, independent of investor risk attitudes

• It was based on an arbitrage argument• Even risk-averse investors like arbitrages!

Suggests another way to value options• Pretend people are risk-neutral• Work out expected future value of option in that case• Discount it back at the risk-free rate to get value today

The option-equivalent and RN methods are two different ways to implement “the binomial method”


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Risk-neutral valuation

Intel call option redux• Risk-neutral investors would set the expected return on

the stock equal to interest rate: 2.5% per six months• Know that Intel can either rise 25% or fall 20%. We

can calculate “RN probabilities” of a price rise:2.5%=RNProb(rise)*25%+(1-RNProb(rise))*(-20%)RNProb(rise)=0.50

• Value of call if (rise) is $21.25, if not is $0• Take expected value with Rnprobs and discount at rf

(0.50*21.25+0.50*0)/(1.025) = $10.36• Same answer as replicating portfolio technique!


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Black-Scholes

VCall = N(d1)*P- N(d2)*PV(S)

• Our examples have just been simple up-or-down movements• In these cases, the binomial method is perfect

• In reality, there may be a continuum of outcomes• Black-Scholes formula uses a replicating portfolio argument to derive option value under these circumstances


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VCall - Call option price

N(d1) - Cumulative normal density function at (d1)

P - Current stock price

N(d2) - Cumulative normal density function at (d2)

S - Strike price (take PV using risk-free rate)

t - time to maturity of option (as fraction of year)

- standard deviation of annual returns

Black-ScholesVCall = N(d1)*P- N(d2)*PV(S)


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(d1) = - .3070N(d1) = .3794

ExampleWhat is the price of a call option given the following? P = 36 r = 10% = .40S = 40 t = 90 days / 365

(d2) = - .5056N(d2) = .3065

Black-Scholes


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Black-Scholes

VCall = N(d1)*P - N(d2)*S*e-rt

= [.3794]*36 - [.3065]*40*e - (.10)(.2466)

= $ 1.70

ExampleWhat is the price of a call option given the following? P = 36 r = 10% = .40S = 40 t = 90 days / 365

Real Options

Principles of Corporate FinanceBrealey and Myers Sixth Edition

Slides by

Matthew Will, Jeffrey Wurgler

Chapter 21



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Topics Covered

Real Options Follow-on investments Abandon Wait (and learn) Vary output or production methods

Valuation examples mixed in


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Real option value

Real option value = Value with option - Value without option


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Key questions

When is there a real option?- Clearly defined underlying asset whose value changes

unpredictably over time- Payoffs to asset are contingent on a decision or event

When does the real option have significant value?- Usually when only you can take advantage of it- As barriers to competition fall, options often worth less

Can that value be estimated using an option pricing model?- If underlying asset is traded, and exercise price is known- Usually not as precise as DCF


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Case 1: Follow-on investments

Option to undertake expansion or follow-on investments if tide turns in future

May want to undertake project that is NPV<0 (before considering option value)


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Example: Building Mark I computer gives option to build Mark II computer if platform catches on

NPV of Mark I computer (itself) = - $46 million But gives option to go ahead with Mark II:

Decision arises 3 years from now Required investment in Mark II is $900 million Forecasted cash flows of Mark II are $463 (PV as of today) Mark II cash flows are uncertain: an annual SD of 35 percent Annual interest rate is 10%

Proceed with Mark I? How valuable is the follow-on option?


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Example: Building Mark I computer gives option to build Mark II computer if platform catches on

Option to invest in Mark II is just a 3-year call option on an asset worth $463 million with a $900 million exercise price!

Black-Scholes call value = +$53.59 million

This makes up for the -$46 NPV of the Mark I on its own

Go ahead with Mark I


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Case #2: Option to abandon

Opposite of expansion option (a put not a call)

Can bail out (cut your losses) if things look bad


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Case #2: Option to abandonExample: Choice between two production

technologies. A is specialized: low unit cost, low salvage value. B is general: high unit cost, decent salvage value.

A has cash flows of 18.5 if high demand, 8.5 if low demand B has cash flows of 18 if high demand, 8 if low demand. If can’t ever abandon, want A. But suppose, one year into project know what demand will be.

Can abandon and get 10 out of B (0 for A). If low demand, B is better. What is value of the put option associated with B?


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Example (A vs. B continued)

• If can’t be abandoned, suppose B is worth $12 million– If high demand, B value rises 50% to $18 million– If low demand, B value falls 33% to $8 million

• If can be abandoned, B’s put option is worth $0 if demand is high, $2 million if demand is low

• Say abandonment possible 1 year from now• Say 1 year interest rate is 5%

• Perfect setup for binomial method – implement with RN


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Example (A vs. B continued)5%= RNProb(hi. dem.)*(50%)+ (1-RNProb(hi. dem.))*(-33%)

RNProb(high demand) = .46

Expected put option payoff = .46*0+(1-.46)*2 = $1.08 million

Discount at 5% put value is $1.03 million.

In total, B is worth $12 + $1.03 = $13.03 million(Compare this to the NPV of A, which has no option)


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What if have decent project (NPV>0 today) but may get even better? Not a now-or-never DCF calculation.

When to pull trigger? What is the value of the option to wait?

Case #3: Option to wait


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Basic option value principle:More time to expiration, more time to gather

information = More value (all else equal)


Option Value

Underlying asset value


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Example: Build factory today (NPV>0 already) or delay a year? If delay, factory may be more or less valuable, depending on demand.

Tradeoff: Building today gets cash flowing. But waiting may help avoid a costly mistake.

What is value of option to wait? Build today or wait a year?



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Example: Build today or delay for 1 year?

Today: If invest $180 million, PV = $200 million If low demand, CF1 =$16 and PV going forward = $160

• So return would be (16+160)/(200) = -12% If high demand, CF1 =$25 and PV going forward = $250

• So return (25+250)/(200) = 37.5%

Suppose riskless rate is 5%.

Another binomial problem. Can solve with RN method



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Example: Build today or delay for 1 year?

5%= RNProb(hi. dem.)*(37.5%)+ (1-RNProb(hi. dem.))*(-12%) RNProb(high demand) = .343

Expected call option payoff = .343*(250-180) + (1-.343)*0 = $24.01 million

Discount at 5% call value is $22.87 million.

So “delay for 1 year” value is $22.87 millionvs. “build today” value is $200 - $180 = $20 million



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Case #4: Flexible production

Flexible production facilities give option to:

Vary product mix as demand changes• Computer-controlled knitting machines

Vary production technology as costs change• Utilities with “cofiring equipment” that can use coal or

natural gas• Auto manufacturers with production facilities in

different countries

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