Risk Management & Real Options Interlude The Link to Financial Options and Black-Scholes Stefan Scholtes Judge Institute of Management University of Cambridge

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”)


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 pricestrike price


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


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


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


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


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 * --






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


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


Example

xx

11

33

11 00

1111

2255

== 33

55 22

11

11 11

2/3 *2/3 *1/3 *1/3 * --


3

1

1

032

131

1

)1(

3

113

2 ,35

case) (in this %0

r

CqCqx

du

drq

du

r

du





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”


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


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


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


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?

Documents

Risk Management & Real Options Interlude The Link to Financial Options and Black-Scholes Stefan Scholtes Judge Institute of Management University of Cambridge