risk management & real options interlude the link to financial options and black-scholes stefan...
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Risk Management & Real Options
Interlude
The Link to Financial Options and Black-Scholes
Stefan ScholtesJudge Institute of Management
University of Cambridge
MPhil Course 2004-05
2 September 2004 © Scholtes 2004 Page 2
A financial option is…
A right but not an obligation
To buy (“call”) or sell (“put”)
A market-valued asset (“underlying asset”)
At a fixed price (“strike price”)
At some fixed time in the future (“European”) or during a fixed time span (“American”)
2 September 2004 © Scholtes 2004 Page 3
Value of a European call option
Stock priceStock price
Value of Value of the callthe callat time ofat time ofexerciseexercise
Strike priceStrike price
Decision: Don’t exerciseDecision: Don’t exerciseDecision: ExerciseDecision: Exercise
Stock price- strike price
2 September 2004 © Scholtes 2004 Page 4
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
2 September 2004 © Scholtes 2004 Page 5
The Black Scholes Model
Key question: What’s the “correct” market price of a financial option?
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
2 September 2004 © Scholtes 2004 Page 6
How does the B-S model work?
The B-S model assumes that the underlying asset value follows a “geometric Brownian motion”
• 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
2 September 2004 © Scholtes 2004 Page 7
Example
xx
11
33
11 00
1111
2255
Call optionCall optionat strike price 4at strike price 4
BankBankaccountaccount
Stock Stock priceprice
==
“Price up” “Price down”
?
All moves are triggered by the sameAll moves are triggered by the sameflip of the coin: Price up or Price downflip of the coin: Price up or Price down
2 September 2004 © Scholtes 2004 Page 8
Example
xx
11
33
11 00
1111
2255
== 33
55 22
11
11 11
2/3 *2/3 *
All moves are triggered by the sameAll moves are triggered by the sameflip of the coinflip of the coin
Investing $ 1 in stock and borrowing Investing $ 1 in stock and borrowing $ 2/3 from the bank fully$ 2/3 from the bank fully REPLICATES REPLICATES the call payoffsthe call payoffs
To buy thisTo buy this REPLICATING PORTFOLIO REPLICATING PORTFOLIOI need I need £ 1/3 £ 1/3 – that’s the price of the call– that’s the price of the call
1/3 *1/3 * --
“Price up” “Price down”
Call optionCall optionat strike price 4at strike price 4
BankBankaccountaccount
Stock Stock priceprice
2 September 2004 © Scholtes 2004 Page 9
The general case: Binomial lattice model
S
uS
dS
Price of asset movesup or down
1
(1+r)
(1+r)
Risk-free investmentr=one-period risk-free rate
Cu
Cd
Value of the optionon the stock price
C=?
All chance nodes follow THE SAME underlying uncertainty:The price of the asset moves up or down
2 September 2004 © Scholtes 2004 Page 10
Computing the one-period B-S value
The consistent value for C can be computed as
where
r
CqCqC du
1
)1(
du
drq
1
2 September 2004 © Scholtes 2004 Page 11
Example
xx
11
33
11 00
1111
2255
== 33
55 22
11
11 11
2/3 *2/3 *1/3 *1/3 * --
“Price up” “Price down”
3
1
1
032
131
1
)1(
3
113
2 ,35
case) (in this %0
r
CqCqx
du
drq
du
r
du
Call optionCall optionat strike price 4at strike price 4
BankBankaccountaccount
Stock Stock priceprice
2 September 2004 © Scholtes 2004 Page 12
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
• 10-15 periods is normally sufficient for good accuracy
There is a closed form solution for European options, called the Black-Scholes formula
• 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”
2 September 2004 © Scholtes 2004 Page 13
Hedging and the Non-Arbitrage argument
The key to financial options valuations is hedging• 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)
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
• 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 Black-Scholes price
This is called the “non-arbitrage” argument for the options price
2 September 2004 © Scholtes 2004 Page 14
Hedging in a real options situation
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
2 September 2004 © Scholtes 2004 Page 15
Summary
Financial options analysis has been instrumental in raising awareness in the value of real options analysis
• Largely responsible for real options lingo
Financial options techniques are valuable to deal with market uncertainties
• Equivalent to consistent chance node valuation
Blind-folded application of financial options techniques is dangerous
• Hybrid approach to deal with technical and market risk separately is preferable and can give hugely different results
2 September 2004 © Scholtes 2004 Page 16
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?