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  • 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

  • 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

    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)

  • 1/24/2011 Real-Options Pricing Robert B.H. Hauswald 5

    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)

    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?

  • 1/24/2011 Real-Options Pricing Robert B.H. Hauswald 7

    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 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

  • 1/24/2011 Real-Options Pricing Robert B.H. Hauswald 9

    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) prices

    logarithmic change is normally distributed

  • 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

    =

    ++=

    =

    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 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

  • 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

    t

    rt

    rtt-t

    SeS

    t

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

    )tN(dKe)N(dS

    )tN(dKe)N(dSec

    =

    ++

    =

    =

    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 *

    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

  • 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%

  • 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

    +

    =

  • 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 uncertai