Transcript The value of flexibility - Department of Engineering

Risk Management & Real Options
Interlude
The Link to Financial Options and
Black-Scholes
Stefan Scholtes
Judge Institute of Management
University of Cambridge
MPhil Course 2004-05
A financial option is…

A right but not an obligation

To buy (“call”) or sell (“put”)
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A market-valued asset (“underlying asset”)
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At a fixed price (“strike price”)
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At some fixed time in the future (“European”) or during a fixed time
span (“American”)
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Value of a European call option
Value of
the call
at time of
exercise
Stock price
- strike price
Decision: Don’t exercise
Decision: Exercise
Stock price
Strike price
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What is the difference to real options?

FOs are purely financial contracts, i.e., a bet on changing values of the
underlying asset
•

At exercise money changes hands but nothing material (“real”) happens
FOs are traded in markets
•
There exists a market price (law of one price)

FOs have short time horizons

Used to hedge risks
•
•
E.g. a Put on a stock price hedges the owner of the stock against low stock
prices
As stock falls, value of put option rises
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The Black Scholes Model
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Key question: What’s the “correct” market price of a financial option?
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Nobel Prize-winning answer given by Black, Scholes and Merton in 70ies
•

Black-Scholes formula
There are many finance people (in academia) who believe that the
“right” way of valuing a real option is the Black-Scholes valuation model
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How does the B-S model work?
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The B-S model assumes that the underlying asset value follows a
“geometric Brownian motion”
•
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
Another way of saying that the returns have log-normal distributions
Underlying asset value can be modelled in a spreadsheet by a lattice
The B-S model values the option, using the “consistent valuation” of
chance nodes that we had used in the R&D option valuation
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Example
1
Call option
at strike price 4
0
x
1
Bank
account
?
=
1
All moves are triggered by the same
flip of the coin: Price up or Price down
1
5
Stock
price
2
3
“Price up”
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“Price down”
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Example
1
Call option
at strike price 4
x
1
Bank
account
5
0
1/3 *
=
5
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-
2/3 *
1
1
All moves are triggered by the same
flip of the coin
2
Investing $ 1 in stock and borrowing
$ 2/3 from the bank fully REPLICATES
the call payoffs
To buy this REPLICATING PORTFOLIO
I need £ 1/3 – that’s the price of the call
3
“Price up”
3
1
1
1
Stock
price
2
“Price down”
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The general case: Binomial lattice model
All chance nodes follow THE SAME underlying uncertainty:
The price of the asset moves up or down
uS
S
Price of asset moves
up or down
dS
(1+r)
Risk-free investment
r=one-period risk-free rate
1
(1+r)
Cu
C=?
Cd
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Value of the option
on the stock price
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Computing the one-period B-S value
The consistent value for C can be computed as
C
q  Cu  (1  q )  Cd
1 r
where
q
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1 r  d
ud
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Example
1
Call option
at strike price 4
5
0
x
1
1/3 *
=
2
3
-
1
2/3 *
1
1
1
r  0% (in thiscase)
Bank
account
1
5
Stock
price
2
1
2
1   0
q  Cu  (1  q)Cd 3
3 1
x

1 r
1
3
3
“Price up”
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u  5 ,d  2
3
3
1 r  d 1
q

ud
3
“Price down”
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Multi-period models


Financial option is valued by
•
Dividing the time to maturity into a number of periods
•
Spanning out the lattice for the underlying asset value
•
Applying backwards induction, as discussed before, to value the option
The B-S price is the theoretical price one obtains as the number of
periods goes to infinity
•

There is a closed form solution for European options, called the BlackScholes formula
•

10-15 periods is normally sufficient for good accuracy
It also applies to American call options without dividend payments
Spreadsheet example of a lattice valuation of an American call can be
found in “BlackScholesOptionsPricing.xls”
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Hedging and the Non-Arbitrage argument
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The key to financial options valuations is hedging
•
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If the option was cheaper than the replicating portfolio, one could make
risk-less profits (“arbitrage profits”) by buying the option and selling the
replicating portfolio
•


Buying the option and selling the replicating portfolio (or vice versa) has zero
future cash flows, no matter what, because they have the same payoffs in
every state of nature (at least in the model)
and vice versa if the replicating portfolio was cheaper
Only price that would make both, the replicating portfolio and the option
tradable is the price of the replicating portfolio, which is the BlackScholes price
This is called the “non-arbitrage” argument for the options price
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Hedging in a real options situation
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What’s the value of a 10-year lease on a mine?
Extraction rate 10,000 ounces / year
Extraction cost £250 / ounce
Risk-free interest 5%
Company discount rate 10%
Current gold price £260 / ounce
Growth rate of gold price 2.5%
For those who are interested: This is worked out in the spreadsheet
GoldMine.xls
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Summary
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Financial options analysis has been instrumental in raising
awareness in the value of real options analysis
•
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Financial options techniques are valuable to deal with market
uncertainties
•
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Largely responsible for real options lingo
Equivalent to consistent chance node valuation
Blind-folded application of financial options techniques is
dangerous
•
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Hybrid approach to deal with technical and market risk separately
is preferable and can give hugely different results
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Real versus financial options

Most important difference between financial and real options:
Financial options are “priced”
Real options are “valued”

Typically, real options analysis needs to help us make a decision, not to
find the correct price!
•

But: there are situations where we will have to name a “price” – e.g. bidding
Biggest drawback of Black-Scholes: It is often “sold” as a black-box
“…give me the volatility and I give you the correct value of your real
option…”
•
•
People don’t focus enough on the need to tell a good story with the model
B-S is like telling a story in a foreign language; it may well be a great story
but what good is it if no-one is willing to listen?
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