© the mcgraw-hill companies, inc., 2000 irwin/mcgraw hill 20- 1 b40.2302 class #4 bm6 chapters 20,...
TRANSCRIPT
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 1
B40.2302 Class #4
BM6 chapters 20, 21 Based on slides created by Matthew Will Modified 04/18/23 by Jeffrey Wurgler
Spotting and Valuing Options
Principles of Corporate FinanceBrealey and Myers Sixth Edition
Slides by
Matthew Will, Jeffrey Wurgler
Chapter 20
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 3
Topics Covered
Calls, Puts and Shares Financial Alchemy with Options Option Valuation
Constructing equivalent portfolios Risk-neutral valuation Black-Scholes
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 4
Option Terminology
Put Option
Right to sell an asset at a specified exercise price on or before a specified exercise date.
Call Option
Right to buy an asset at a specified exercise price on or before a specified exercise date.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 5
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 6
Option Value
Payoff on a riskless bond/loan at maturity … is fixed (lender’s perspective).
Share Price
Bon
d va
lue
0
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 7
Option Value
Payoff to a share when you want to sell it … depends on share price (share buyer’s perspective).
Share Price
Sha
re v
alue
50
50
0
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 8
Option Value
Call 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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 9
Option Value
Put option value at expiration given a $85 exercise price (put buyer’s perspective).
Share Price
Put
opt
ion
valu
e
80 85
$5
0
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 10
Option Obligations
Buyer Seller
Call option Right to buy asset Obligation to sell asset
Put option Right to sell asset Obligation to buy asset
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 11
Option Value
Call option value at expiration given a $85 exercise price (call seller’s perspective).
Share Price
Cal
l opt
ion
$ pa
yoff
85
0
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 12
Option Value
Put option value at expiration given a $85 exercise price (put seller’s perspective).
Share Price
Put
opt
ion
$ pa
yoff
85
0
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 13
Financial Alchemy
Protective Put = Buy stock and buy put
Share Price
Pos
itio
n V
alue “Protective Put”
Long Put
Long Stock
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 14
Financial AlchemyStraddle = Long call and long put
- Profits from high volatility
Share Price
Pos
itio
n V
alue
Straddle
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 15
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 16
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 17
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 18
Safe versus risky debt
Shareholder value at maturity given $D borrowing (shareholder’s perspective).
Firm asset value
Sha
reho
lder
pay
off
D0
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 19
Safe versus risky debt
Lender value at maturity given $D lending to a risky firm (lender’s perspective).
Firm asset value
Deb
thol
der
payo
ff
D
0
D
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 20
Option Value
Upper Limit
Stock Price
Lower Limit
{Stock price - exercise price, 0}whichever is higher
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 21
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 22
Option Value
Option Price
Stock Price
ACTUAL VALUE
Exercise Price
Notice the shape of an unexpired option’s value
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 23
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 (+)
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 24
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!
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 25
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.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 26
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.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 27
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.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 28
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”
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 29
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!
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 30
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 31
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)
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 32
(d1) = - .3070
N(d1) = .3794
Example
What is the price of a call option given the following?
P = 36 r = 10% = .40
S = 40 t = 90 days / 365
(d2) = - .5056
N(d2) = .3065
Black-Scholes
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 33
Black-Scholes
VCall = N(d1)*P - N(d2)*S*e-rt
= [.3794]*36 - [.3065]*40*e - (.10)(.2466)
= $ 1.70
Example
What is the price of a call option given the following?
P = 36 r = 10% = .40
S = 40 t = 90 days / 365
Real Options
Principles of Corporate FinanceBrealey and Myers Sixth Edition
Slides by
Matthew Will, Jeffrey Wurgler
Chapter 21
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 35
Topics Covered
Real Options Follow-on investments Abandon Wait (and learn) Vary output or production methods
Valuation examples mixed in
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 36
Real option value
Real option value = Value with option - Value without option
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 37
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 38
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)
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 39
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?
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 40
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 41
Case #2: Option to abandon
Opposite of expansion option (a put not a call)
Can bail out (cut your losses) if things look bad
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 42
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?
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 43
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 44
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)
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 45
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
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 46
Basic option value principle:
More time to expiration, more time to gather information = More value (all else equal)
Case #3: Option to wait
Option Value
Underlying asset value
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 47
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?
Case #3: Option to wait
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 48
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
Case #3: Option to wait
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 49
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
Case #3: Option to wait
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
20- 50
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