How To Value Real Options

(1)

FIN 673

Pricing Real Options

Professor Robert B.H. Hauswald

Kogod School of Business, AU

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

(2)

1/24/2011 Real-Options Pricing © Robert B.H. Hauswald 3

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

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)

(3)

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)

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?

(4)

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”

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

(5)

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

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

(6)

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

( ) { ( )} ( )

[ ]

( )

[ ]

^d ^d ^T ^t

t T t T r

K S d

d KN t T r d

N S c

t t t

−

− =

− +

+

=

−

=

σ σ

σ 2 1

2 1

1 , 2

log

exp

( ) { ( )} ( )

[

S N d1 exp rT t KN d2

]

p_t = − _t − + − − −

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= 0.098

Guarantee level 50 d2= -0.302

Maturity (years) 5 N(-d1)= 0.461

Volatility of Value at T 0.4 N(-d2)= 0.619

Annual interest rate 0.05 P = (X N(-d2) - F N(-d1)) exp(-rt) 6.859

(7)

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:

yield dividend

constant a

is and

and

2 ln

where

2 3

3 3

δ σ

σ σ

δ δ

t rt

rt t

- t

Se S

t

]t / [r σ /K)

= (S d

) t N(d

Ke ) N(d S

) t N(d

Ke ) N(d Se c

δ

δ δ

−

=

+ +

−

=

−

=

Perpetual Options: Infinite-Horizon

• Consider an opportunity to develop a piece of land:

) , , ( 1 ,

*

1

*

, ^γ ^γ ^σ ^δ

γ γ

γ

r function X

X P V

P

=

= −





 





Value of developed land 100 Gamma 1.862

Cost of development 100 P* 216.1

Annual Volatility 0.2

Annual interest rate 0.06 Option Value 27.7

Annual "Dividend Yield" 0.045

(8)

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

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%

(9)

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

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 (

rf

D V q U

V V q

+

×

− +

= ×

) 1 (

) ( ) 1 ( ) ) (

0 (

rf

D S q U

S S q

+

×

− +

= ×

A minor bit of algebra yields:

) ( ) (

) ( ) 0 ( ) 1 (

D S U S

D S S q r^f

−

×

= +

(10)

$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 . 38 max[$

) , ,

3(UU U = −

C

13.02

] 0 , 25

$ 10 . 28 max[$

) , , ( ) , , (

) , , (

3 3

3

−

=

= D U U C U D U C

U U D C

3.10

] 0 , 25

$ 77 . 20 max[$

) , , ( ) , , (

) , , (

3 3

3

−

=

= U D D C D U D C

D D U C

0

] 0 , 25

$ 35 . 15 max[$

) , ,

3(

−

= D D D C

0

) 05 . 1 (

10 . 3

$ ) 3 1 ( 02 . 13

$ 3 ) 2 ,

2(

× +

= × U U C

9.25

) 05 . 1 (

0

$ ) 3 1 ( 10 . 3

$ 3 2

) , ( ) ,

( ₂

2

× +

×

=

=C DU D

U C

1.97

) 05 . 1 (

0

$ ) 3 1 ( 0

$ 3 2

) ,

2(

× +

×

= D D C

0

) 05 . 1 (

97 . 1

$ ) 3 1 ( 25 . 9

$ 3 2

)

1(

× +

×

= U C

6.50

) 05 . 1 (

0

$ ) 3 1 ( 97 . 1

$ 3 2

)

1(

× +

×

= D C

1.25 4.52

) 05 . 1 (

25 . 1

$ ) 3 1 ( 50 . 6

$ 3 2

0

× +

= × C

Call Option Lattices

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

d u

d e d u

d r

dS uS

dS S

r D

S U S

D S S q r

rf

f

f f

−

≅ −

−

= +

−

×

= +

−

×

= +

) 1 (

) 0 ( ) 0 (

) 0 ( ) 0 ( ) 1 ( )

( ) (

) ( ) 0 ( ) 1 (

? 1 ;

, = = ⋅ =

=

⇒ ^∆ e⁻ ^∆ u d d u

e

u ^σ ^t ^σ ^t

σ

( )

⁰

) (U uS

S =

) 1 (

) ( ) 1 ( ) ) (

0 (

rf

D S q U

S S q

+

×

− +

= ×

( )

^D ^dS

( )

⁰

S =

(11)

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: r

3. Determine the required investment amount(s): K 4. Model current asset value (cash flow) uncertainty

5. Build cash flow and associated option value lattices 6. Recover object of interest (c, V, K); extend model

d u

d q r

d u e

u e r

r ^r ^t ^t

−

= −

= ⇒

=

= ^∆ ^∆

1 ~

,

~ ,

:

, σ

² ^σ

Parameter Inputs

Project value 100

Exercise Price 100

Maturity (years) 2

Annual Volatility 0.3

Annual interest rate 0.07

Number of Periods 4

Step Size (T/N) 0.5

Annual lost revenues 0.04

Exercise Price at Maturity 100

Risk-Neutral Probabilities

u 1.236311

d 0.808858

rhat 1.03562

dhat 1.020201

q = ((rhat/dhat) - d)/(u-d) 0.482521

Binomial Model: Example

Lattice for the Underlying Project Value

Date Jun-97 Jun-98 Jun-99

Downs/Period 0 1 2 3 4

0 100.00 123.63 152.85 188.97 233.62

1 80.89 100.00 123.63 152.85

2 65.43 80.89 100.00

3 52.92 65.43

4 42.80

Lattice for the Option Value

Date Jun-97 Jun-98 Jun-99

Downs/Period 0 1 2 3 4

0 17.01 30.78 53.75 88.97 133.62

1 5.35 11.47 24.62 52.85

2 0.00 0.00 0.00

3 0.00 0.00

4 0.00

(12)

From Option to Project Valuation

Project 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 d

u d q r d

u −

= −

⇒

⇒ ~

2 , σ

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

(13)

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 %)

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

+

= PNPV

ANPV

(14)

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.

Capital Budgeting Techniques

How 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

Evaluation technique

(15)

Binomial Real Options: Appendix

• The four most common types of real options

1. The opportunity to make follow-up investments.

2. The opportunity to abandon a project 3. 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

Intrinsic Value identifies it as what type of option?

Option to Wait

Option Price

Asset Price

(16)

Intrinsic Value + Speculative Value = Option Value Speculative (time) Value = Value of being able to wait

Option to Wait

Option Price

Asset Price

More time = More value

Option to Wait

Option Price

Asset Price

(17)

• 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

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%)

(18)

After 12 months

Option value = exercise price - asset value Example: 500 - 368 = 132 (or $132 000)

Intrinsic Value

832 (0) 679

PV = 553 553 (0)

451

368 (132)

Valuation Lattice: Extract RN Q

After 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)

( )

(

^upside^interest^change^rate^-^-^downside^downside^change^change

)

= q

Example:

( )

(

²²²^.⁵^.⁶^-^-^-¹⁸^-¹⁸^.⁴^.⁴

)

⁰^.⁵¹

q = =

(19)

Option Value: 6M

After 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

832 (0) 679 (0)

NPV = 553 553 (0)

451(63)

368 (132)

Option Value: 12M

Now

Expected return = (0.51)(0) + (0.49)(63) = $31 Value today: discount back

= 31/1.025 = $30

832 (0) 679 (0)

NPV = 553 (30) 553 (0)

451(63)

368 (132)

(20)

• 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

1u d change downside

1

e u change upside

1 ^h

=

= +

=

+ ^σ

Binomial Pricing

( )

^t

year a of fraction a

as time h

asset on returns annual

of deviation standard

∆

= σ = Where:

(21)

Investment Project

Price = 36 σ = 0.40 ∆t = 30 days Exercise Price = 40 r = 10%

Maturity = 90 days

Binomial Example

( )

0.8917

1.1215

d 1

1.1215

e

u

^0.4 ³⁰³⁶⁵

=

40.37

32.10 36

37 . 40 1215 . 1 36

1 0

=

×

=

×u V_u V

Binomial Pricing

(22)

40.37

32.10 36

37 . 40 1215 . 1 36

1 0

=

×

=

×u V_u V

10 . 32 8917 . 36

1 0

=

×

=

×d V_d V

Binomial Pricing

50.78 = price

40.37

32.10 45.28

36

28.62 40.37

32.10 36

+1

=

× _t

t u V

V

Binomial Pricing

(23)

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

50.78 = price

40.37 .37

32.10 0

25.52 0 45.28

5.60

36

28.62 40.37

32.10 36

( ) ( )

365 130 . 0 1

0.37 0.4925 10.78

0.5075

with 1

1 1

+

⋅ +

= ⋅

+ =

⋅ +

= ^u⋅ ^ut⁺ ^d ^dt⁺ V_uT IV_uT r

V q V q

The greater of

Binomial Pricing

(24)

50.78 = price

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

1 1

r V q V

V_st q^u ^sut ^d ^sdt +

⋅ +

= ⋅ ⁺ ⁺

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

(25)

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