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Page 1: Professor Robert B.H. Hauswald Kogod School of · PDF fileProfessor Robert B.H. Hauswald Kogod School of Business, AU ... – financial markets: spot or futures – corporate finance:

FIN 673 Pricing Real Options

Professor Robert B.H. Hauswald

Kogod School of Business, AU

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 2

From Financial to Real Options

• Option pricing: a reminder– messy and intuitive: lattices (trees)– elegant and mysterious: Black-Scholes-Merton

• Option theory in corporate finance?– managerial flexibility and projects as options– strategy as a collection of options

• Key concepts: arbitrage ideas– risk-adjusted probabilities– risk-neutral pricing

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Having the Cake and Eat It, too

• Options confer contractual rights on holder:– a right to buy (sell) a fixed amount of currency at (over)

a specified time (period) in the future at a price specified today

• Insurance vs. fixed commitment:– right to buy or sell at discretion of holder

– “wait and see” security: even over time

– have an opinion while cutting off catastrophes

• Right means choice: choice means value

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 4

A Short Options Menu: Review

• Style: European or American exercisable at maturity only (e) or any time (a)

• Type: the right to buy (call) or to sell (put)– corporate: growth = call, retrenchment = put

• Underlying: – financial markets: spot or futures

– corporate finance: real asset, firm value

• Parties: buyer (holder), seller (writer)

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Pricing Terminology: Review

• Three price elements:– current price of underlying asset: spot, futures, forward– strike (exercise): price at which transaction occurs– (option) premium: the option’s price itself

• Price location: at/in/out-of-the-money options– at: current spot = strike– in: option profitable if exercised immediately– out: option could not be profitably exercised

• Intrinsic value: extent to which an option is in-the-money (profit of immediate exercise)

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 6

Pricing Real Options

• How do I take into account the risk of the real options?

• Does it matter if the underlying asset is traded in financial markets?

• How do I go about implementing a real options model?

• What are the limitations of real options analysis?

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Compare Projects

• The insight: all projects or assets with similar risk should have similar returns

• The challenge: find a “twin” security or project -and use its rate of return

• The alternative: use a standard equilibrium theory that relates risk to return (economics)

• The strategy: In the case of options, need to find a “replicating portfolio” that has the same risk

• The concept: no need to appeal to “no-arbitrage”

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 8

Futures Market?

• If there is a futures market for the underlying product (e.g. oil), then PV is readily computed– it is simply today’s price of the product (adjusted for a

“convenience yield”) times the volume.

• If we don’t have a futures market, we need to find the appropriate rate of return for the underlying project (the firm’s cost of capital perhaps)

– risk-neutral valuation

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Implementing Real Options Analysis

• There are three different approaches to value real options:– formulaic approach (e.g. Black-Scholes)– lattice model (e.g. binomial model)– Monte-Carlo Simulation

• Formulas are easy to implement, – but they have limited applicability and are very

much black boxes – the other two approaches are more viable in

general

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 10

Price Determinants

• Current spot price, (dividend and) interest rate– futures or forward: by C&C from spot price and interest rates– foregone revenue in real option: dividend yield

• Exercise price• Time to maturity: length of period to expiration• Underlying price process: volatility• Type: European or American• A right: use probability theory to evaluate contingencies• Prerequisite: a model of the underlying asset value• Distributional assumption: the spot (forward) price’s

logarithmic change is normally distributed

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1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 11

Pricing European Options: BSM

– Apply the classics: modify the seminal work of Black, Scholes and Merton to calculate theoretically fair prices

– Pricing formula: call

– Put:– Interpretation: payoffs S - K and K - S weighted by

• discount factor: future strike and spot• probability of prices realizations: expected values• PCP - put-call-parity: fundamental arbitrage equation

( ) ( ){ } ( )[ ]( ) ( )( )[ ] tTdd

tTtTrKSd

dKNtTrdNSc

t

tt

−−=−

−++=

−−−=

σσ

σ 122

1

21

,1

2log

exp

( ) ( ){ } ( )[ ]21 exp dKNtTrdNSp tt −−−+−−=

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 12

Black-Scholes-Merton Example

• Assumes option is European - exercise only at the option’s maturity date– e.g. residual value guarantee on a machine - a put

option which can be only exercised at T.

Value at T (forward price) 48 d1= 0.098Guarantee level 50 d2= -0.302Maturity (years) 5 N(-d1)= 0.461Volatility of Value at T 0.4 N(-d2)= 0.619Annual interest rate 0.05 P = (X N(-d2) - F N(-d1)) exp(-rt) 6.859

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1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 13

Black-Scholes with Dividends

• Dividends are a form of “asset leakage” – if dividend are paid repeatedly, adjust B-S-M to

allow for constant proportional dividends:

yielddividendconstanta is and and

2ln where

2

3

33

33

δσ

σ

σ

δ

δ

t

rt

rtt-t

SeS

t

]t/σ[r/K)(S=d

)tN(dKe)N(dS

)tN(dKe)N(dSec

δ

δ

δ

=

++

−−=

−−=

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 14

Perpetual Options: Infinite-Horizon

• Consider an opportunity to develop a piece of land:

),,(,1

*,1 *

δσγγ

γγ

γ

rfunctionXPX

VPP

=−

=−

=

Value of developed land 100 Gamma 1.862Cost of development 100 P* 216.1Annual Volatility 0.2

Annual interest rate 0.06 Option Value 27.7Annual "Dividend Yield" 0.045

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1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 15

Lattice Methods: Trees

• Most common is the binomial model– one up or down movement at a time– workhorse of the financial industry: pricing American options

• Solve by starting at the end and working backwards– time honored principle: dynamic programming (engineering),

backward induction

• Probabilities in the lattice have been adjusted – to reflect risk of underlying variable; discount at risk-free rate– pricing theory

• For example, an option to invest in a project

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 16

Three Period Binomial Option Pricing Example: Review

• There is no reason to stop with just two periods: generalize to three, four, …periods

• The principles are the same:– find q– construct the underlying asset value lattice working

forward– construct the option value working backward

• Find the value of a three-period at-the-money call option written on a $25 stock that can go up or down 15 percent each period when the risk-free rate is 5%

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1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 17

Stock Price Lattice

$25

28.75

21.25

2/3

1/3

)15.1(00.25$ ×

2)15.1(00.25$ ×

)15.1)(15.1(00.25$ −×

2)15.1(00.25$ −×

)15.1(00.25$ −×

3)15.1(00.25$ ×

)15.1()15.1(00.25$ 2 −×

2)15.1()15.1(00.25$ −××

3)15.1(00.25$ −×

33.06

24.44

2/3

1/3

18.06

2/3

1/3

15.35

2/3

1/3

38.02

2/3

1/3

20.77

2/3

1/3

28.10

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 18

Risk-Neutral Probabilities: Review

• The key to finding q is to note that it is already impounded into an observable security price: the value of S(0)

S(0), V(0)

S(U), V(U)

S(D), V(D)

q

1- q)1(

)()1()()0(

fr

DVqUVqV

+×−+×=

)1(

)()1()()0(

fr

DSqUSqS

+×−+×=

A minor bit of algebra yields:)()(

)()0()1(

DSUS

DSSrq f

−−×+

=

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1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 19

$25

28.75

21.25

2/3

1/3

15.35

2/3

1/3

38.02

28.10

2/3

1/3

20.77

2/3

1/3

33.06

24.44

2/3

1/3

18.06

2/3

1/3

]0,25$02.38max[$),,(3 −=UUUC

13.02

]0,25$10.28max[$

),,(),,(

),,(

33

3

−===

DUUCUDUC

UUDC

3.10

]0,25$77.20max[$

),,(),,(

),,(

33

3

−===

UDDCDUDC

DDUC

0

]0,25$35.15max[$

),,(3

−=DDDC

0

)05.1(

10.3$)31(02.13$32),(2

×+×=UUC

9.25

)05.1(

0$)31(10.3$32

),(),( 22

×+×== UDCDUC

1.97

)05.1(

0$)31(0$32

),(2

×+×=DDC

0

)05.1(

97.1$)31(25.9$32

)(1

×+×=UC

6.50

)05.1(

0$)31(97.1$32

)(1

×+×=DC

1.25

4.52

)05.1(

25.1$)31(50.6$320

×+×=C

Call Option Lattices

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 20

Risk-Neutral Valuation in Practice

• Use observed volatility to determine size of up and down steps and generate value lattice: fit model to observed uncertainty!

• Some more algebra yields (note continuous compounding!):

S(0), V(0)

S(U), V(U)

S(D), V(D)

q

1- q

du

de

du

dr

dSuS

dSSr

DSUS

DSSrq

frf

ff

−−≅

−−+

=

−−×+

=−

−×+=

)1(

)0()0(

)0()0()1(

)()(

)()0()1(

?;1

, =⋅===⇒ ∆−∆ dueu

deu tt σσσ

( )0)( uSUS =

)1(

)()1()()0(

fr

DSqUSqS

+×−+×=

( ) ( )0dSDS =

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1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 21

Binomial Real Options

1. Calculate PV of project’s value (net cash flows) taking future financial strategy (WACC) as given: V = S

2. Find appropriate risk-free interest rate: r3. Determine the required investment amount(s): K4. Model current asset value (cash flow) uncertainty

5. Build cash flow and associated option value lattices6. Recover object of interest (c, V, K); extend model

du

drq

udeuerr ttr

−−=⇒=== ∆∆

~1,,~:, 2 σσ

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 22

Parameter InputsProject value 100Exercise Price 100Maturity (years) 2Annual Volatility 0.3Annual interest rate 0.07Number of Periods 4Step Size (T/N) 0.5Annual lost revenues 0.04Exercise Price at Maturity 100

Risk-Neutral Probabilitiesu 1.236311d 0.808858rhat 1.03562dhat 1.020201q = ((rhat/dhat) - d)/(u-d) 0.482521

Binomial Model: Example

Lattice for the Underlying Project ValueDate Jun-97 Jun-98 Jun-99Downs/Period 0 1 2 3 4

0 100.00 123.63 152.85 188.97 233.621 80.89 100.00 123.63 152.852 65.43 80.89 100.003 52.92 65.434 42.80

Lattice for the Option ValueDate Jun-97 Jun-98 Jun-99Downs/Period 0 1 2 3 4

0 17.01 30.78 53.75 88.97 133.621 5.35 11.47 24.62 52.852 0.00 0.00 0.003 0.00 0.004 0.00

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1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 23

From Option to Project ValuationProject Variable Call Option

Required expenditure X, K Strike, exercise price

Operating value of assets S, F Price of underlying asset (spot, futures, forward)

Length of time to final decision

t, T-t Time to expiration

Riskiness of operating CFs Variance of underlying asset’s return, price, etc.

Time value of money r Default risk-free rate of return

du

drqdu

−−=⇒⇒

~,2σ

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 24

Monte-Carlo Simulation

• Some applications involve options that are “path-dependent”– their values depend on the particular path of cash flows

(not just the lattice node at some point in time)

• Compound options: options on options– feasibility study to build prototype with new technology

• Monte-Carlo simulation: somewhat similar to scenario analysis – can properly account for path probabilities and risk– works in a forward, rather than backward, fashion

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Implementation Challenges

• Multiple sources of uncertainty– model the interaction of risks

• Modeling resolution of uncertainty– what is learned when: from decision to value trees

• Estimating inputs– volatility– distribution of underlying– cost of capital on underlying– “dividend yields:” lost revenue (in %)

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 26

Benefits of Option Analysis

• Plausibility check: initial outlay, PV of project, later investment(s)– given any two, back out the third

• Pricing business and financial strategy– warm and fuzzy becomes cold and hard

• Corporate finance:– why overpay? paying a premium now amounts to what?

– applications: resource extraction, growth, synergy, R&D, governance (abandonment, cash out ) options

π+= PNPVANPV

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Who is Using Real Options?

• A survey of 4000 CFOs reported that“Twenty seven percent of CFOs said that they always use real options to analyze large projects.” (Graham and Harvey, “The theory and practice of corporate finance,” Journal of Financial Economics 60, May/June 2001: 187-243 )

• Industries applying real-options analysis (no order): pharmaceuticals, petrochemicals, aerospace, power generation, mineral extraction, finance, real estate, electronics, forest products, telecommunications, metallurgy, oil and gas, etc.

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 28

Capital Budgeting TechniquesHow freqently does your firm use the following techniques when deciding which project or

acquisition to pursue?Source: Graham Harvey JFE 2001 n =392

0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00%

APV

Profitability index

Simulation analysis

Book rate of return

Real options

Discounted payback

P/E multiple

Sensitivity analysis

Payback

Hurdle rate

NPV

IRR

Eva

luat

ion

tec

hn

iqu

e

% always or almost always

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Binomial Real Options: Appendix

• The four most common types of real options1. The opportunity to make follow-up investments.2. The opportunity to abandon a project3. The opportunity to “wait” and invest later.4. The opportunity to vary the firm’s output or

production methods.

• Recall the relationship between active and passive NPV:Value “Real Option” = NPV with option

- NPV without option

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 30

Intrinsic Value identifies it as what type of option?

Option to Wait

Option Price

Asset Price

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Intrinsic Value + Speculative Value = Option Value

Speculative (time) Value = Value of being able to wait

Option to Wait

Option Price

Asset Price

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 32

More time = More value

Option to Wait

Option Price

Asset Price

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• Dalby Airways Ltd is considering the purchase of a turboprop aeroplane for its business. – If the business fails, an option exists to sell the aeroplane for

$500,000; current value of plane is $553,000– Risk-free rate: 5%

• Given the following decision tree of possible outcomes– what is the value of the offer (i.e. the put option) and – what is the most Dalby Airways should pay for the option?

• Difference between decision tree and valuation lattice?

Option to Abandon: Put Option

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 34

Decision Tree: Not a Valuation Lattice, yet

Year 0 Month 6 Month 12

832 (22.6%)

679 (22.6%)

(18.4%)

PV = 553 553

(22.6%)

451 (18.4%)

368 (18.4%)

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After 12 months

Option value = exercise price - asset value

Example: 500 - 368 = 132 (or $132 000)

Intrinsic Value

Year 0 Month 6 Month 12

832 (0)

679

PV = 553 553 (0)

451

368 (132)

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 36

Valuation Lattice: Extract RN QAfter 6 months: Probability of an up-movement

Note: movements expressed in rates of change, not our usual up, down and interest factors (same expression, though)

( )( )change downside - change upside

change downside - rateinterest =q

Example:

( )( )( )( ) 510

418622

41852.

.- - .

.- - . q ==

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Option Value: 6MAfter 6 months

Example: If firm value in month 6 is $451, the option value is:

= (0.51)(0) + (0.49)(132) = $65

Value at month 6: discount back

= 65/1.025 = $63

Year 0 Month 6 Month 12

832 (0)

679 (0)

NPV = 553 553 (0)

451(63)

368 (132)

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 38

Option Value: 12MNow

Expected return = (0.51)(0) + (0.49)(63) = $31

Value today: discount back

= 31/1.025 = $30

Year 0 Month 6 Month 12

832 (0)

679 (0)

NPV = 553 (30) 553 (0)

451(63)

368 (132)

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• Decision trees for valuing “real options” in a corporate setting cannot be practically done by hand. – Introduce binomial & B-S-M models

• Calibrate parameters to observed quantities– investment projects

– corporate strategies

– synergies from M&A or corporate cooperation

Corporate Options

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 40

u1 d change downside 1

e u change upside 1 h

==+

==+ σ

Binomial Pricing

( )tyear a offraction a as timeh

asseton returns annual ofdeviation standard

∆==σ

Where:

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Investment Project

Price = 36 σ = 0.40 ∆t = 30 days

Exercise Price = 40 r = 10%

Maturity = 90 days

Binomial Example

( )

0.8917 1.12151 d

1.1215 e u 365300.4

==

==

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 42

40.37

32.10

36

37.401215.13610

=×=× uVuV

Binomial Pricing

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40.37

32.10

36

37.401215.13610

=×=× uVuV

10.328917.3610

=×=× dVdV

Binomial Pricing

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 44

50.78 = price

40.37

32.10

25.52

45.28

36

28.62

40.37

32.10

36

1+=× tt VuV

Binomial Pricing

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50.78 = price

10.78 = intrinsic value

40.37

.37

32.10

0

25.52

0

45.28

36

28.62

36

40.37

32.10

Binomial Pricing

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 46

50.78 = price

10.78 = intrinsic value

40.37

.37

32.10

0

25.52

0

45.28

5.60

36

28.62

40.37

32.1036

( ) ( )

36530

1.01

0.370.4925 10.780.5075

with 1

11

+

⋅+⋅=

=+

⋅+⋅= ++uTuT

dtdutu IVV r

V q Vq

The greater of

Binomial Pricing

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50.78 = price

10.78 = intrinsic value

40.37

.37

32.10

0

25.52

0

45.28

5.60

36

.19

28.62

0

40.37

2.91

32.10

.10

36

1.51

Binomial Pricing

1

11

r

V q Vq V sdtdsutu

st +⋅+⋅= ++

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 48

Expanding the binomial model to allow more possible price changes

1 step 2 steps 4 steps

(2 outcomes) (3 outcomes) (5 outcomes)

Binomial vs. Black-Scholes

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How estimated call price changes as number of binomial steps increases No. of steps Estimated value

1 48.1

2 41.0

3 42.1

5 41.8

10 41.4

50 40.3

100 40.6

Black-Scholes 40.5

Binomial vs. Black-Scholes