Volatility Estimation for Stochastic Project Value Models

VOLATILITY ESTIMATION FOR STOCHASTIC

PROJECT VALUE MODELSLuiz E. Brandão

IAG Business School, Pontifícia Universidade Católica do Rio de [email protected]

James S. DyerMcCombs School of Business, University of Texas at Austin

[email protected]

Warren J. HahnGraziadio School of Business and Management, Pepperdine University

[email protected]

15th International Conference on Real Options, Turku, Finland, 2011

mailto:[email protected]



Volatility Estimation for Stochastic Project Value Models - Brandão, Dyer and Hahn


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Ten Sample Paths for a GBM Process

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0 1 2 3 4 5 6 7 8 9Time Period

Val

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($)

90% C.I.

10% C.I.

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ist'n

• A common model of how an asset’s value (V) changes over time is a Geometric Brownian Motion (GBM): dV = Sdt + SSdz

• GBM’s, as well as some other stochastic processes, are used to model the underlying uncertainty in option valuation problems

Stochastic Processes for Asset Value



• Cox, Ross, and Rubinstein (1979) showed that a discrete binomial approximation could be made of a GBM

• A key parameter for this is the volatility (denoted as ), which is defined as the standard deviation of returns

• Volatility determines how widely possible outcomes will vary, and in a binomial model it specifies the up and down moves, e.g., if =18.23%:

which could then be

used to build this value lattice:

Binomial Approximation of a GBM Process

V 0 d

V 0

V 0 u

V 0 u 2 V 0 u 3

V 0 ud

V 0 d 2V 0 d 3

V 0 u 2 d

V 0 ud 2

2.111823.0 eeu t 833.01 ud

Lognormal D

istribution



• For Real Option valuation problems, there may be several relevant uncertainties

• One approach is to model each uncertainty explicitly using a multivariate lattice or simulation approach

• As an alternative, Copeland & Antikarov (2003) suggested consolidating uncertainties into a univariate stochastic process for Project Value

– Advantage of this approach is it simplifies the numerical analysis of the options

• Need to determine the starting value for the underlying asset

– Copeland and Antikarov proposed using the Present Value of the project without options

• The one remaining parameter to determine is the project volatility.

• How do we get the volatility for this asset, given that it is not a market traded asset and there are no historical price series?

Stochastic Processes for Real Options



• As Copeland and Antikarov showed, that we can get the variation of the returns by simulating the uncertainties that affect the project’s value,

– e.g., a project with uncertain revenue and variable operating costs

• Start with a pro forma worksheet for the project, in which we calculate the remaining PV in each time period (current cash flow, plus PV of cash flows from remaining periods)

Volatility of Project Value Process

V0 V1 V2 V3 …. ….



• By running a Monte Carlo simulation of this spreadsheet, we generate a new set of cash flows with each iteration

• The project rate of return (denote as ) is the desired simulation output:

• Simulation yields the standard deviation of .

• This is the volatility () we need to calculate the up and down moves in project value

• Unfortunately, this process needs to be modified in order to obtain an unbiased estimate of the volatility

Using Simulation to Find the Volatility

11 0

0

lnV

V V eV



• In the approach proposed by Copeland and Antikarov, V1 is recalculated with the cash flow realizations in each period (e.g., F1-F5)

• Thus is written as:

• However, the volatility of returns from period 0 to period 1 only depends on the outcome of uncertainty at period 1

– Uncertain cash flows in periods 2-5 have yet to be resolved

– At the end of Period 1, the best unbiased estimates of F2-F5 are the expected values, conditional of the outcome for F1

• Using the above specification of in the simulation produces volatility estimates that are too large, because the uncertainty is overstated

Bias in the Estimate of Volatility

( 1)

11

0

1

ln ln

nt

ttn

tt

t

F eV

V E F e



• In order to verify the results of the simulation, we derive an analytical expression for the volatility.

• Let V (t) be a project subject to a single source of uncertainty S(t) and only variable costs where S follows a GBM diffusion process

• The project cash flows are F(t) = S(t), and by Itô

• Since , then

• Applying Ito we get which shows that the volatility of the project is the same as of the single uncertainty in this case.

Analytical Determination of Project Volatility

SdS Sdt Sdz

SdF Fdt Fdz

( )( ) t

t

V E F t e dt

( )( )

FV

SdV Vdt Vdz



• To correct this bias Brandão, Dyer, and Hahn (2005) proposed an approach the utilizes the conditional expectations for periods beyond F1 .(F2 - F5 in our case).

• Their approach states that the volatility of the project from period 0 to period 1 depends only on the outcome of uncertainty at period 1, as the uncertainty in future periods are yet to be resolved.

• The random variable is then computed as:

• An analytical proof of this intuitive approach can be arrive at by deriving the the variance of expression for and comparing to the expected volatility.

An Unbiased Estimate of Volatility

( 1)1

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0

1

[ ]ln ln

[ ]

nt

ttn

tt

t

F E F eV

V E F e



• Consider a project with fixed costs where the cash flows F(t) are defined as by where is a fixed cost.

• Then the process for F(t) becomes which shows that the volatility of the cash flows F increases by a constant equivalent to due to the operating leverage.

• Integrating over an infinite life project provides

• Applying Itô we arrive at

• This implies that with fixed costs, the project volatility is factored by

where

Project with Fixed Costs (Operating Leverage)

SdV V dt V dz

*V V

*V

V

.

, 0 1F S

( ) ( )SdF F dt F dz

S 0F e

V



• The analytical solution can be used as a benchmark to verify the bias of each simulation method.

• We now analyze a five period project and compare the results for each method.

• Since we are assuming that the project value follows a GBM, the volatility in each period is the same, so only one period volatility need to be determined.

• We will see that this method also allows for easy computation of period by period volatility when this assumption no longer holds.

Project with Fixed Costs (Operating Leverage)



• For the example below, the analytical vol is σ = (19,990+14,372)/19,990 × 25% = 42.97%

• The simulated BDH and C&A vols with different

nº of periods and errors are shown.

An Example

$ 1000 0 1 2 3 4 5

Revenue 10,000 10,600 11,236 11,910 12,625 13,382Variable Costs 30% (3,180) (3,371) (3,573) (3,787) (4,015)

Fixed Costs (3,000) (3,000) (3,000) (3,000) (3,000)Investment (20,000)

Free Cash Flow (20,000) 4,420 4,865 5,337 5,837 6,368

V0 V1

19,990 21,989Invest. = (20,000)

NPV0 = (10) µ = 10%

Revenue Parameters Growth () = 6.0% = 25%

Number of Periods

Vol Analytic

Vol BDH

Vol C&A

5 43,0% 42,5% 66,8%

10 38,7% 39,0% 82,7%

20 35,0% 35,2% 95,8%

50 31,5% 31,5% 108,2%

100 30,5% 30,5% 111,7%

200 30,3% 30,3% 112,9%

Infinite 30,3% 30,3% -



• This modification can also be important for estimating the volatility of returns in each specific period (e.g., ln(V2 /V1), ln(V3 /V2),…)

• Unlike the case for a standard GBM, the volatility may indeed be non-constant for consolidated stochastic processes for Project Value

– Differing effects from the individual processes (e.g., if one or more of the consolidated processes is Mean-Reverting, or has Jumps)

• We can accommodate the non-constant volatility case with non-recombining trees, which gives us greater flexibility for modeling the consolidated process in discrete time

Non-Constant Volatility

a

d

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b

b

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c

c

c

b

d

a



• Modeling the value of a project with:– Mean-Reverting Stochastic process for Revenue

– Variable Cost with a Jump (due to anticipated technology breakthrough)

An Example



• Comparison of Copeland and Antikarov (C&A) results by period with our modified approach in this case

• Note that the results are the same in the last time period (no subsequent period uncertainty to overstate with C&A approach)

Computational Results

Vol Vol

Period C&A BDH

1 27.5% 15.6%

2 27.8% 18.0%

3 28.2% 20.9%

4 28.7% 24.7%

5 30.4% 30.4%

• Due to this the error in the C&A approach is greater for the initial periods.

• How closely do the binomial trees with volatility estimates from the C&A approach and our revised approach, respectively, approximate the actual cash flows?

• As a benchmark, we use the cash flow data from the simulation to determine the volatility (period 5 shown in next slide)



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Cu

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Year 5 Cash Flow

Simulated vs. Modeled CDF'sYear 5 Cash Flow

Simulated

GCE

C&A

Comparing Cash Flow Distributions: Yr 5

Root MSE:

C&A 10.5%

BDH 2.25%



• Volatility is a critical input variable for stochastic process models in real option valuation

• Values derived via existing simulation-based estimation methods, however, are biased high.

• BDH (2005) proposed an approach that models the temporal resolution of uncertainty and updating of conditional expectations in each discrete time increment.

• We analytically verify that this approach is unbiased when the cash flows can be closely approximated by a GBM stochastic process.

• We also show how this approach can be extended to cases with leveraged cash flows and non-constant volatility, which is important when the GBM assumption is not a reasonable one.

• The use of this approach resolves a critical error in the estimation of project volatility using Monte Carlo simulation methods, and ensures more accurate valuation in real option applications.

•

Conclusions

VOLATILITY ESTIMATION FOR STOCHASTIC

PROJECT VALUE MODELSLuiz E. Brandão

IAG Business School, Pontifícia Universidade Católica do Rio de [email protected]

James S. DyerMcCombs School of Business, University of Texas at Austin

[email protected]

Warren J. HahnGraziadio School of Business and Management, Pepperdine University

[email protected]





Volatility Estimation for Stochastic Project Value Models

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