real options: the binomial model - american … · this example illustrates how binomial option...

American University

FIN-02-002Revised January 14, 2003

Real Options:

The Binomial Model

Parameter Inputs Risk-Neutral ValuationProject value V 100 Annual interest rate r 0.07 u = eσ∆t 1.236311Exercise Price K 100 Number of Periods N 4 d = u−1 0.808858Maturity (years) T 2 Step Size ∆t = T/N 0.5 r̃ = er∆t 1.03562Annual Volatility σ 0.3 Annual lost revenues δ 0.04 δ̃ = eδ∆t 1.020201

q = r̃/δ̃−du−d 0.482521

Lattice for the Underlying Project ValueDate 06-97 06-98 06-99Downs/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 42.80Lattice for the Option Value

Date 06-97 06-98 06-99Downs/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

Figure 1: Class Example for Binomial Real Option Valuation (Aircraft Delivery Option)

This example illustrates how binomial option pricing techniques are to be adapted for the valuationof an investment project by real options methods. The project data and corresponding optionparameters are as follows:

Investment ProjectProject Parameter Option Variable Value

Initial PV of net operating cash flows V0 V0 100mInvestment amount I K 100mInvestment decision horizon T 2 yearsAnnual volatility σ 30.00%Annual interest rate r 7.00%Number of periods (6M) N 4Step size ∆t = T

N24 = 0.5

Annual lost revenue (dividend yield) δ 4.00%

Professor Robert Hauswald prepared this technical note as the basis for class discussion rather than to illustrateeither effective or ineffective handling of an administrative situation.

c© Robert B.H. Hauswald, Kogod School of Business, American University, Washington, DC 20016-8044. To ordercopies or request permission for reproduction, please send email to [email protected]. No part of this publica-tion may be reproduced in any form or by any means, or used without prior consent.

Real Options FIN-02-002

The first step in any valuation exercise under uncertain is to describe the nature of the ran-domness and which variables it affects. Hence, we construct the binomial lattice (tree) for theproject’s value which we take to be a random variable. However, we wish to choose the up anddown step sizes in such a manner as to conserve the given volatility of the underlying assets’ value,be they physical plant, intellectual property, the present value of net operating cash flows (or, if youprefer, free cash flows, EBITDA, etc.). Currently the present value of net cash flows (asset value)is V0 = 100m; however, we know that the annual volatility of V is σ = 0.3. We need to convertthis annual volatility to a period (6 months) volatility of σ6M = σ

√∆t = 0.3

√0.5 = 0.2121.1 To

conserve this volatility in the lattice, we now choose the lattice’s up-factor u and down-factor d sothat

u = exp{

σ√

∆t}

= e0.3√

0.5 = 1.2363

d =1u

= exp{−σ√

∆t}

= e−0.3√

0.5 = 0.8088

Note that by a theoretical argument for financial options the up and down factors need to be chosenin such a way that d < r̃ < u where r̃ = er∆t is the interest rate factor.2 Otherwise, the risk-neutralprobabilities that we need to calculate the value of the real option are ill defined. These two factorsgenerate the project’s random (present value) of cash flows so that the original cash flow volatility ispreserved. You should verify that the next period project cash flow values conform to V1 (u) = u ·V0

and V1 (d) = d·V0. In t = 2, we have V2 (uu) = u·V1 (u) = u2V0, V2 (ud) = uV1 (d) = udV0 = V2 (du) ,etc. for all time periods and states. Figure 2 shows the result of the recombining lattice that wehad in class (Figure 1). For analytic purposes, we continue with the truncated values of the latticereproduced in Figure 2.

An investment opportunity is the right but not the obligation to invest in any given project.So, after waiting for two years the project’s managers will presumably have learned enough todetermine in which state the project will end up. Imagine a situation where the net operating cashflow uncertainty stems from three risk factors: production costs, output prices and demand. Lowproduction costs, high demand, high output prices result in of the lattice’s upper end nodes; highproduction costs, low demand because, e.g., intense competition leads to a lower one.

Managerial flexibility means that the firm can observe the economic environment and thendecide to go ahead or pull the plug on the project.3 We, therefore, evaluate the payoff to investmentsin each of the states in 2 year’s time (at the end notes) and select the ones for which our payoffs (NPVof project in this particular state: V4− I) are positive (why? what type of option do we have?). Asan example, in the lowest note going forward with the investment would net 42.80− 100 = −57.20so we should refrain from investing and replace the end node value with 0. The general rule foran option to invest (a call option) is c4 = [V4 − I]+ = max {V4 − I, 0} which is nothing but thepositive part of the project’s NPV at t = 4.

The preceding investment decision rule brings out a second feature (real) options analysis:lattice models of (real or financial) asset prices are always solved going backward, i.e., you start atthe last node and calculate the instrument’s value folding the lattice back toward t = 0. Havingreplaced the cash flow process’ value at the last node with the value of the option at exercise (t = 4)we, therefore, calculate the investment option’s value in the period before (t = 3) and record thevalue at each corresponding node.

1Note the sqare root: since volatility is the square root of variance, you need to take the square root of anymultiplicative factor of the volatility, too.

2We need to define interest rate and discount factors in terms of continuous compounding so that they areconsistent with the up and down factors. Also, despite their intimidating use of e, the exponential formula, they areactually much easier to work with.

3In NPV analysis we first go ahead (or not) and then observe the resolution of uncertainty.

2

FIN-02-002 Real Options

100.00

123.63

80.88

152.84

100.00

188.96

123.63

80.88

233.62

152.84

100

65.4265.42

52.91

42.80

t = 0 : Inception t = 1 : 6M t = 2 : 12M t = 3 : 18M t = 4 : 24M

Figure 2: Investment Project Value Process

c0

c1(u)

c1(d)

c2(u2)

c2(ud)

c3(u3)

c3(u2d)

c3(ud2)= 0

[233.62− 100]+ = 133.62

[152.84− 100]+ = 52.84

[100− 100]+ = 0

[65.42− 100]+ = 0c2(d2)= 0

c3(d3)= 0

[42.80− 100]+ = 0


Figure 3: Investment Option Value Process

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The last step involves taking into account the riskiness of the underlying process and to makesure that there are no arbitrage opportunities. Incredibly, both requirements can be satisfied bythe same operation. Absence of arbitrage is an equilibrium condition. It insures that (1) our modelis well specified (one should not be able to make unlimited profits without taking on risk) and(2) that we correctly price the inherent risks (equilibrium means that we have found the marketprice of risk). Hence, we need to extract our risk-neutral probabilities q from the underlyingvalue process. A further advantage of this approach is that we then can price and contingent(i.e., state dependent) claim (e.g., futures contract, financial option) or managerial decision (e.g.,investment, abandonment, technology switch decision) using these very same probabilities as longas the underlying value process remains the same.4

By analogy with their financial counterparts, real options also rely on risk-neutral pricing.Hence, we adjust the probabilities for the project’s underlying riskiness to make them risk-neutral.It is one of the celebrated results in finance that for asset valuation you should always use risk-neutral probabilities.5 To find these magical probabilities, we use their securities analog and definethe probability of a risk-adjusted up-movement as

q =r̃ − d

u− d

where r̃ = er∆t is the interest rate factor (in the lecture, it was just called r). The value of anyfinancial instrument or asset can now be recursively calculated as

Xt−1 = d (t− 1, t)EQt−1 [d (t− 1, t) Xt]

where EQt−1 is the expectation (weighted average) under the risk-neutral probabilities at time t− 1

of the asset’s value in the next period t and d (t− 1, t) the one period discount factor (discrete:1

r̃t−1; continuous: e−rt−1∆t). However, for our case we need to make one last adjustment because

postponing the project incurs a loss in terms of forgone incremental revenue (for the rest of the firm)at the rate δ (analogous to a stock paying dividends at rate δ). Hence, the risk-neutral probabilitiesof our revenue generating process have to be adjusted by δ̃ = eδ∆t to become

q =r̃δ̃− d

u− d(1)

Let us consider an investment opportunity. In the last period t = T, you know what the financialinstrument’s (or asset’s) payoffs CT are, e.g., for a call option it would be CT = max {ST −K, 0} .In our case, you set X4 = c4 = max {VT − I, 0} and replace the terminal values with the investmentproject’s payoff in this state (see Figure 3). Next, we calculate the value of the investment optionfor all states in the previous period and continue backward through the lattice until we reach itsbeginning (t = 0) . The general formula for any state s (combination of previous up and downmovements, e.g., s = udu = u2d) is

ct−1 (s) =1r̃

[qct (su) + (1− q) ct (sd)] (2)

4Replacing the primitive probabilities p with the risk-neutral ones q is precisely the step that converts a decisiontree specified with respect to p into a valuation lattice.

5Aside: we know that we need to take the volatility into account somewhere. We have three candidates: discountfactors, probabilities and step sizes. So, we could have adjusted the discount factor had we known the risk premium(market price of risk) or a market-determined discount rate. Alternatively, we could have kept arbitrary (given)probabilities of up and down movements and used the T-bill rate for discounting but then adjusted the step size tomake the process risk-neutral. Here, we follow the usual procedure of pricing with risk-neutral probabilities.

4


For our investment project, we start with t = 3 and calculate the expected discounted investmentoption value for all four states taking into consideration that by waiting and then investing we loserevenue to the tune of 4% p.a. We will, therefore, have to deduct this as a cost from our investmentoption value by adjusting the discount factor. The per period parameter values are

r̃ = er∆t = e0.07· 12 = 1.0356 ⇒ d (t− 1, t) =

11.0356

δ̃ = eδ∆t = e0.04· 12 = 1.0202

q =r̃δ̃− d

u− d=

1.03561.0202 − 0.80881.2363− 0.8088

= 0.4825, 1− q = 0.5175

Start at the bottom of the tree (for a sellout option, a put, you would start at the top) in t = 3where the option expires worthlessly next period:

c3

(d3

)=

11.0356

[0.4825 · 0 + (1− 0.4825) · 0] = 0

c3

(ud2

)=

11.0356

[0.4825 · 0 + (1− 0.4825) · 0] = 0

c3

(u2d

)=

11.035636

[0.4825 · 52.84 + (1− 0.4825) · 0] = 24.62

c3

(u3

)=

11.0356

[0.4825 · 133.62 + (1− 0.4825) · 52.84] = 88.66

Moving on to the previous period (t = 2), we find by the same method:

c2

(d2

)= 0

c2 (ud) =1

1.0356[0.4825 · 24.62 + (1− 0.4825) · 0] = 11.47

c2

(u2

)=

11.0356

[0.4825 · 88.66 + (1− 0.4825) · 24.63] = 53.62

so that for t = 1 we have

c1 (d) =1

1.0356[0.4825 · 11.47 + (1− 0.4825) · 0] = 5.34

c1 (u) =1

1.0356[0.4825 · 53.62 + (1− 0.4825) · 11.48] = 30.72

which, finally, yields the option value as

c0 =1

1.0356[0.4825 · 30.72 + (1− 0.4825) · 5.34] = 16.98

Figure 4 shows the evolution of the investment opportunity’s value (call option premium).By expected NPV criteria, the project would not have been undertaken (just breaking even is

not good enough in the face of uncertainty):

E [NPV ] = E [PV0 (CF )]− C0 = 100− 100 = 0E [NPVMF ] = c0 − E [NPV ] = 16.98

Managerial flexibility (MF ) in form of ”wait and see” is worth USD 16.98m: by postponing invest-ment for two years to see how our environment evolves adds USD 16.98m to our project throughthe resolution of uncertainty. Managers come in twice: first, they postpone, learn and observe

5


16.98

30.72

5.34

53.62

11.47

88.66

24.62

0

133.62

52.84

0

00

0

0


Figure 4: The Value of Flexibility in Project Value

the environment that is reflected in the net cash flow lattice. Second, they decide in which cir-cumstances to go forward and when to abandon the investment idea altogether. Incidentally, youshould ask yourselves whether it follows that we always should postpone an investment and, then,indefinitely so. Would this strategy somehow enhance value even more?

After you understand this example, you might want to calculate the investment’s value processby using the Black-Scholes-Merton formula for dividend paying stock where N is the normal dis-tribution and log the natural logarithm (I refuse to write the ancient ln for logarithmus naturalis):

ct = e−δ(T−t)VtN (d1)− e−r(T−t)IN (d2)

d1 =log Vt

I +(r − δ + σ2

2

)(T − t)

σ√

T − t(3)

d2 = d1 − σ√

T − t =log Vt

I +(r − δ − σ2

2

)(T − t)

σ√

T − t

Evaluating the option formula at t = 0 gives you the initial value of the investment project c0 (withV0) according to BSM inclusive of the embedded optionality. Finally, use the binomial tree andBSM formulae for puts to calculate the value of a buyout offer (salvage value guarantee, option toshut down and dispose of assets, etc.) good for two years. Assume that in two years time you havethe option to sell your investment project for USD 100m. How much would this option be worth?You proceed in complete analogy with the investment opportunity analysis.

By now, you have probably noticed that the investment project (real option) valuation latticedoes not quite match the one given in class and reproduced at the beginning of this note. As thefinal value of the investment opportunity does not seem to come out correctly, you might wonderabout rounding errors. However, you should compare the upper branch of the lattice. Before youread on, you might want to pause and ask yourselves for which option type we just calculatedthe value. What did we implicit assume about its exercise behavior? If you answered European

6


100.00

23.63

0

52.84

0

88.96

23.63

0

133.62

52.84

0

00

0

0


Figure 5: The Investment’s Intrinsic Value Process

option you were right on the money: we either undertake the project in t = 0 or two years laterin t = 4. In fact the investment opportunity value calculated in class (see spreadsheet Real OptionClass Example.xls) is an American option valuation. As soon as the time is right, the firm will goforward with the investment. Here is how you would proceed in this case.

First, you replace all values in the original cash flow lattice with the intrinsic value of the optionat this node: you take the bigger of Vt− I or 0 at each node, i.e., Ct (s) = max {Vt (s)− I, 0}. Youobtain Figure 5.

Next, you calculate the American option values c̃3

(d3

)for all t = 3 nodes in the intrinsic value

lattice that connect to at least one non-zero end node. As in the preceding European case, youshould find:

c̃3

(d3

)= c3

(d3

)=

11.0356

[0.4825 · 0 + (1− 0.4825) · 0] = 0

c̃3

(ud2

)= c3

(ud2

)=

11.0356

[0.4825 · 0 + (1− 0.4825) · 0] = 0

c̃3

(u2d

)= c3

(u2d

)=

11.0356

[0.4825 · 52.84 + (1− 0.4825) · 0] = 24.62

c̃3

(u3

)= c3

(u3

)=

11.0356

[0.4825 · 133.62 + (1− 0.4825) · 52.84] = 88.66

However, instead of recording these option values in the option value lattice you have to evaluate themerits of waiting (premium c3 (s)) against immediate exercise (intrinsic value C3 (s)) for each states. After all, this is precisely what an American option is all about. So, in state s = u3 you wouldcompare waiting vs. investing decision as c3

(u3

)= 88.66 < C3

(u3

)= 88.96 and immediately

go forward with the project. However, in state s = u2d it pays to be patient and wait anotherperiod: the value of waiting c3

(u2d

)= 24.62 is higher than the value of immediate investment

7


C3

(u2d

)= 23.63. The general decision and, hence, rule is

c̃n (s) = max {cn (s) , Cn (s)} , n = 1, ..., N (4)

where n denotes the time period, s the node, c̃ the American option value, c the one-period Europeanoption value and C the intrinsic value (same for both types). Figure 6 illustrates this step.

Finally, we calculate recursively the investment option values in each state using the valuationrule (4) for the preceding time periods (note how I substitute 88.96 into the expression for c̃2

(u2

)as compared to the European case):

c̃2

(d2

)= c2

(d2

)= 0

c̃2 (ud) = c2 (ud) = 11.48

c̃2

(u2

)=

11.0356

[0.4825 · 88.96 + (1− 0.4825) · 24.62] = 53.75

c̃1 (d) = c1 (d) = 5.35

c̃1 (u) =1

1.0356[0.4825 · 53.75 + (1− 0.4825) · 11.48] = 30.78

c̃0 =1

1.0356[0.4825 · 30.78 + (1− 0.4825) · 5.35] = 17.01

Figure 7 summarizes the value process. As you can see, having more flexibility in that you canimmediately exercise the investment option instead of waiting until t = 4 adds another USD 30,000to the project: c̃AM

0 − cEU0 = 17.01− 16.98 = 0.03.

23.63 ∨ 24.62= 24.62

88.96 ∨ 88.66= 88.96

133.62

52.84

0

0

0

t = 4 : 24M

0 ∨ c2(ud)= c2(ud)c̃0

23.63 ∨ c1(u)

0 ∨ c1(d)= c1(d)

t = 0 : Inception

52.84 ∨ c2(u2)

t = 1 : 6M

0

0

0

t = 2 : 12M t = 3 : 18M

88.96 ∨ 88.66 := max{88.96, 88.66} = 88.9652.84 ∨ c2(u2) := max{52.84, c2(u2)}

Figure 6: American Option Valuation

8


17.01

30.78

5.35

53.75

11.48

88.96

24.62

0

133.62

52.84

0

00

0

0


Figure 7: Project Valuation with Immediate Investment Decision

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real options: the binomial model - american … · this example illustrates how binomial option...

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