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B40.2302 Class #4
 BM6 chapters 20, 21
 Based on slides created by Matthew Will
 Modified 3/14/2016 by Jeffrey Wurgler
Irwin/McGraw Hill
©The McGraw-Hill Companies, Inc., 2000
Principles of Corporate Finance
Brealey and Myers

Sixth Edition
Spotting and Valuing Options
Slides by
Matthew Will,
Jeffrey Wurgler
Irwin/McGraw Hill
Chapter 20
©The McGraw-Hill Companies, Inc., 2000
20- 3
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
Call Option
Right to buy an asset at a specified exercise
price on or before a specified exercise date.
Put Option
Right to sell 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
Stock Pric e $60
70
80
90
100
110
Call Value
Put Value
0
15
0
5
5
0
15
0
25
0
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0
25
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Option Value
Bond value
Payoff on a riskless bond/loan at maturity … is fixed
(lender’s perspective).
0
Share Price
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Option Value
Share value
Payoff to a share when you want to sell it … depends on
share price (share buyer’s perspective).
50
0
50
Share Price
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Option Value
Call option value
Call option value at expiration given a $85 exercise price
(call buyer’s perspective).
$20
0
85
105
Share Price
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Option Value
Put option value
Put option value at expiration given a $85 exercise price
(put buyer’s perspective).
$5
0
80 85
Share Price
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Option Obligations
Buyer
Call option Right to buy asset
Put option Right to sell asset
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Seller
Obligation to sell asset
Obligation to buy asset
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Option Value
Call option $ payoff
Call option value at expiration given a $85 exercise price
(call seller’s perspective).
0
85
Share Price
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Option Value
Put option $ payoff
Put option value at expiration given a $85 exercise price
(put seller’s perspective).
0
85
Share Price
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Financial Alchemy
Protective Put = Buy stock and buy put
Long Stock
Position Value
“Protective Put”
Long Put
Share Price
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Financial Alchemy
Position Value
Straddle = Long call and long put
- Profits from high volatility
Straddle
Share Price
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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 debt
Shareholder payoff
Shareholder value at maturity given $D borrowing
(shareholder’s perspective).
0
D
Firm asset value
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Safe versus risky debt
Debtholder payoff
Lender value at maturity given $D lending to a risky firm
(lender’s perspective).
D
0
D
Firm asset value
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Option Value
Stock Price
Upper Limit
Lower Limit
{Stock price - exercise price, 0}
whichever is higher
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Option Value
Upper and lower limits to call option value
Upper limit: share price
Option
Price
ACTUAL VALUE
Lower limit: payoff if
exercised immediately
Stock Price
Exercise Price
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Option Value
Notice the shape of an unexpired option’s value
Option
Price
ACTUAL VALUE
Stock Price
Exercise Price
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Option Value
Determinants of Call Option Price
1 - 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 Equivalents
Intel 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
• 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
VCall = N(d1)*P- N(d2)*PV(S)
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Black-Scholes
VCall = N(d1)*P- N(d2)*PV(S)
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
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Black-Scholes
Example
What is the price of a call option given the following?
P = 36
r = 10%
 = .40
S = 40
t = 90 days / 365
(d1) = - .3070
N(d1) = .3794
(d2) = - .5056
N(d2) = .3065
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Black-Scholes
Example
What is the price of a call option given the following?
P = 36
r = 10%
 = .40
S = 40
t = 90 days / 365
VCall = N(d1)*P - N(d2)*S*e-rt
= [.3794]*36 - [.3065]*40*e - (.10)(.2466)
= $ 1.70
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Principles of Corporate Finance
Brealey and Myers

Sixth Edition
Real Options
Slides by
Matthew Will,
Jeffrey Wurgler
Irwin/McGraw Hill
Chapter 21
©The McGraw-Hill Companies, Inc., 2000
20- 35
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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Case 1: Follow-on investments
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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Case 1: Follow-on investments
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 abandon
Example: 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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Case #2: Option to abandon
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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Case #2: Option to abandon
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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Case #3: Option to wait
 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?
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Case #3: Option to wait
 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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Case #3: Option to wait
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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Case #3: Option to wait
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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Case #3: Option to wait
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 million
vs. “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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