FIN 673 Pricing Real Options

Professor Robert B.H. Hauswald

Kogod School of Business, AU

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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

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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 options 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)

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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

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Futures Market?

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

convenience yield) times the volume.

If we dont have a futures market, we need to find the appropriate rate of return for the underlying project (the firms 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

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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) prices

logarithmic change is normally distributed

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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

=

++=

=

122

1

21

,1

2log

exp

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

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Black-Scholes-Merton Example

Assumes option is European - exercise only at the options 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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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

t

rt

rtt-t

SeS

t

]t/[r/K)(S=d

)tN(dKe)N(dS

)tN(dKe)N(dSec

=

++

=

=

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Perpetual Options: Infinite-Horizon

Consider an opportunity to develop a piece of land:

),,(,1

*,1 *

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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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

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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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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

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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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$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

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Risk-Neutral Valuation in Practice

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