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Abstract We present here the theory of real options using extensively the techniques of variational inequalities. This powerful technique is the right tool to capture the various possibilities that express the ﬂexibility, the main background of real options.

1. Introduction A key reference for the theory of real options is the study by Dixit and Pindyck [1994], although the title of the book is different. Variational inequalities developed in a different context were initially used for stochastic control in the study by Bensoussan and Lions [1982]. We show here that this technique is the appropriate one to handle the ﬂexibility, which is the hallmark of real options. Real options theory is an approach to mitigate risks of investment projects that stems from two ideas. The ﬁrst idea is hedging, borrowed from ﬁnancial options, when market considerations can be introduced. We refer to situations in which the project risk is correlated to the market risk, and one can use tradable assets to hedge. The second idea is ﬂexibility. The option concept pertains to the range of decisions. In particular, one may scale down or up the project, one may stop it, or one may change orientation. This ﬂexibility allows to react properly when information is obtained on the uncertainties of the evolution. We shall ﬁrst review what can be obtained from ﬁnancial theory and adapted to investment decisions in the continuous time and in the discrete time. We will then treat the ﬂexibility in the decision making.

Mathematical Modeling and Numerical Methods in Finance Copyright © 2008 Elsevier B.V. Special Volume (Alain Bensoussan and Qiang Zhang, Guest Editors) of All rights reserved HANDBOOK OF NUMERICAL ANALYSIS, VOL. XV ISSN 1570-8659 P.G. Ciarlet (Editor) DOI 10.1016/S1570-8659(08)00013-6 531

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2. Tradable assets 2.1. Complete market model We assume continuous time. The randomness is characterized by n standard independent Wiener processes wj (t). We denote F t = σ(wj (s), j = 1, . . . , n; s ≤ t). There are n basic assets on the market whose prices are denoted by Yi (t) whose evolution is governed by dY i (t) = Yi (t)(αi (t)dt + σij (t)dwj ), where αi (t) and σij (t) are processes adapted to F t . The market is complete when the matrix σ(t) is invertible. In this case, the information obtained by observing the evolution of the prices of assets is sufﬁcient to recover the underlying source of noise modeled by the Wiener processes. In addition to the random assets, there is a riskless asset whose evolution is characterized by Y0 (t) = exp rt. Denoting by α(t) the vector with components αi (t), we consider the process θ(t) = σ −1 (t)(α(t) − r1I), whose deﬁnition makes direct use of the invertibility of the matrix σ(t). We next deﬁne Z(t) by the relation dZ(t) = −Z(t)θ(t) · dw(t), Z(0) = 1. A key property is the following Proposition 2.1. The processes Z(t)Yi (t) exp −rt is a F t martingale. From the martingale property, we can write E[Z(T)Yi (T) exp −rT |F t ] = Z(t)Yi (t) exp −rt, ∀T > t hence, Yi (t) = E

Z(T ) Yi (T) exp −r(T − t)|F t , Z(t)

which can also be written as follows Yi (t) = E[Yi (T )|F t ] exp −r(T − t) + cov

Yi (T ) Z(T ) t , |F . exp r(T − t) Z(t)

(2.1)

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This formulation is similar to that of the pricing of assets in the Markowitz model. The ﬁrst term in the right-hand side is the expected value at time t of the value Yi (T ) discounted with the free-risk discount rate r. The second term is a premium linked to the risk. This risk premium is expressed by the covariance between the value at T discounted at rate r and a ﬁxed market indicator, which does not depend on i. This indicator is similar to the Market portfolio and the covariance is similar to the β of the asset. Let us now take T = t + δ with δ small. We can write Z(T ) t t |F , Yi (t) exp rδ = E[Yi (t + δ)|F ] + cov Yi (t + δ), Z(t) and approximating exp rδ by 1 + rδ, we get the formulation δZ(t) t t |F , σij (t)cov δwj (t), E[δYi (t)|F ] = rYi (t)δ − Yi (t) Z(t)

(2.2)

j

where δf(t) = f(t + δ) − f(t). The risk premium can be expressed in terms of the covariance between the noises affecting the price values Yi (t) and the market indicator. We can also express Eq. (2.2) as follows dZ(t) t σij (t)cov dwj (t), (2.3) |F , αi dt = rdt − Z(t) j

which can be seen as a consequence of Ito’s calculus and the deﬁnition of θ(t). Note also the equivalence with dY i (t) dZ(t) t αi dt = rdt − cov (2.4) , |F . Yi (t) Z(t) 2.2. Risk premium and CCAPM We can recover these formulas by the Consumer Capital Asset Pricing Model (CCAPM) approach. In this approach, consider a generic “investor” who invests in the market. He should be representative of the behavior of investors on the market. Suppose that at time t his portfolio is made of an amount πi (t) of asset i and an amount πf (t) of riskless asset. His wealth at time t is then given by πi (t)Yi (t) + πf (t) exp rt. X(t) = i

The evolution of X(t) is only governed by the evolution of the portfolio. This implies πi (t)dY i (t) + πf (t)r exp rt. dX(t) = i

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We can eliminate πf (t) between these two relations. Besides, we use the amounts relative to the wealth, namely, i (t) =

πi (t)Yi (t) . X(t)

With these elements, we can derive the evolution of the wealth as follows dX(t) = rX(t)dt + X(t)(t).σ(t)(dw(t) + θ(t)dt), and thus one observes that X(t)Z(t) exp −rt is also a F t -martingale and more precisely d(X(t)Z(t) exp −rt) = X(t)Z(t) exp −rt(σ ∗ (t)(t) − θ(t)).dw(t).

(2.5)

This martingale property has an important consequence. First, E(X(T )Z(T ) exp −rT ) = X(0), where the initial wealth is known and not random. To a given portfolio, evolution corresponds a given wealth proﬁle. It is remarkable that one can invert the statement. If one considers an arbitrary wealth at time T , which is a F T-measurable random variable satisfying the previous constraint, called the budget equation, then the wealth at any time is determined by the martingale property, and it is realized by a unique portfolio, thanks to the representation of martingales with respect to a ﬁltration generated by Wiener processes. The generic investor will choose his portfolio according to some utility function U depending on his ﬁnal wealth. In other words, he seeks to maximize EU(X(T )) exp −βT, where β is a subjective discount linked to the investor, as the utility function is. From this consideration, he can ﬁnd his optimal ﬁnal wealth, provided the budget equation constraint is respected. So, his problem reduces to ﬁnding X(T ) such that maximize EU(X(T )) exp −βT under the constraint E(X(T )Z(T ) exp −rT ) = X(0). We may introduce a Lagrange multiplier λ to deal with the constraint, and the optimal ˆ ) must satisfy ﬁnal wealth X(T ˆ )) exp −βT = λZ(T ) exp −rT. U (X(T Taking the conditional expectation with respect to F t , we deduce ˆ )) U (X(T Z(T ) . = ˆ ))|F t ] Z(t) E[U (X(T

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We plug this value in the pricing relation of assets to get ˆ )) U (X(T Yi (T ) t t , |F , Yi (t) = E[Yi (T )|F ] exp −r(T − t) + cov ˆ ))|F t ] exp r(T − t) E[U (X(T (2.6) which is a CCAPM pricing formula. 2.3. Deﬁnition of a tradable asset A tradable asset is an asset whose value Q(t) evolves according to the relation dQ = Q(a(t)dt + b(t) · dw(t)) with a drift such that dQ dZ , |F t . a(t) = r − E cov Q Z The logic is that one could constitute a portfolio of the basic assets and risk less asset, which is equal to Q(t), carries identical risk as the tradable asset, and consequently provides the same expected return, in view of the absence of arbitrage. This portfolio is equal to X(t) =

n

πi (t)Yi (t) + πf (t) exp rt = Q(t),

i=1

and its evolution is dX(t) =

n

πi (t)dY i (t) + rπf (t) exp rt dt.

i=1

So, dX(t) = rQ(t)dt +

n

πi (t)Yi (t)(αi (t) − r)dt +

i=1

i,j=1

Equaling the risk term between dQ and dX yields bj (t)Q(t) =

n

πi (t)Yi (t)σij (t),

i=1

and by equaling the expected return, we obtain a(t)Q(t) = rQ(t) +

n i=1

n

πi (t)Yi (t)(αi (t) − r).

πi (t)Yi (t)σij (t)dwj .

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We use Eq. (2.3) to obtain a(t)Q(t)dt = rQ(t)dt −

n

πi (t)Yi (t)

i=1

n j=1

dZ t |F , σij (t)E cov dwj , Z

hence, a(t)Q(t)dt = rQ(t)dt − Q(t)

n j=1

dZ bj (t)E cov dwj , |F t . Z

Finally, we obtain

dQ(t) dZ(t) t a(t)dt = rdt − E cov , |F . Q(t) Z(t)

(2.7)

It is easy to derive the portfolio, which replicates the tradable asset Q(t), Indeed, we have πi (t)Yi (t) = Q(t)

((σ ∗ )−1 )ij (t)bj (t) j

and thus πf (t) exp rt = Q(t) − Q(t)

((σ ∗ )−1 )ij (t)bj (t). ij

An equivalent way to obtain the relation (2.7) is to constitute a portfolio made of the tradable asset Q(t) and a short position on the market assets Yi (t). Such a portfolio leads to a wealth x(t) = Q(t) −

πi (t)Yi (t).

i

We recall that a short position on an asset means that one sells immediately the asset and receives the corresponding cash. If Q(t) is tradable, one should ﬁnd the values of πi (t) so that the preceding portfolio is risk free, hence satisﬁes dx(t) = rx(t)dt. The fact that it is risk free results from the idea that by taking a short position on the market assets we eliminate the risk. The advantage of this approach is also that one does not introduce cash explicitly.

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2.4. Case of dividends Let us assume now that the market assets bear a coupon. More precisely, the asset Yi (t) yields a coupon δi (t) per unit value and per unit of time, hence Yi (t) yields Yi (t)δi (t)dt during the period t, t + dt. Consider a tradable asset that carries a dividend denoted by q(t) per unit value and per unit of time. The processes δi (t) and q(t) are adapted to F t . The formulas change slightly. Indeed, during dt the increase of wealth due to the possession of Yi (t) is dY i (t) + Yi (t)δi (t)dt, arising from the increase of price of the asset and the from the payment of the dividend. If we consider the riskless portfolio πi (t)Yi (t), x(t) = Q(t) − i

then we can write dx(t) = Q(a + q)dt −

πi Yi (αi + δi )dt +

i

Qbj −

j

πi Yi σij dwj .

i

Proceeding as above, we obtain the formula (a(t) + q(t))dt = rdt + δi (t)((σ ∗ )−1 )ij (t)bj (t)dt

ij

dQ(t) dZ(t) t − E cov , |F . Q(t) Z(t)

(2.8)

We can naturally apply formula (2.8) to Yk (t). We must use the values a = αk ;

bj = σkj ;

q = δk

Plugging these values in (2.8) yields immediately (2.3). Indeed, this formula is not affected directly by the coupons on Yi (t) since they express uniquely the risk part of the asset. 3. Valuation of contingent claims 3.1. Claims on market assets A claim is a right on the underlying assets to be exercised in a contract. We will consider ﬁrst a claim on the market assets Yi (t) components of a vector Y(t). Since the approach is based on Ito’s calculus, we will assume that αi (t) = αi (Y(t), t), σij (t) = σij (Y(t), t), δi (t) = δi (Y(t), t) in which the functions on the right-hand side are deterministic (the randomness is captured through the process Y(t)).

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A claim on the market assets has a value at time t, which is a function of the values Y(t), expressed as F(Y(t), t). It is a capital asset and is fully paid. We assume that the possession of this claim generates a proﬁt per unit of time q(F, t), which is equivalent to a dividend q(t) =

q(F, t) F

in which the argument F has to be replaced with F(Y(t), t) to obtain the precise value. We consider Q(t) = F(Y(t), t) as a tradable asset, which carries the above dividend. We can write dQ(t) = Q adt + bj dwj , j

where the values of a and bj can be obtained, thanks to Ito’s calculus and the Ito differentials of Yi (t). We get easily Fa =

1 ∂2 F ∂F ∂F + αi Yi + σij Yi Yj ∂t ∂Yi 2 ∂Yi ∂Yj i

ij

∂F Fbj = Yi σij . ∂Yi i

We can, then, apply the formula (2.8) and cancel identical terms on both sides. We can state the following. Theorem 3.1. A tradable contingent claim on the market assets F(Y, t) must be a solution of the partial differential equation (PDE) 1 ∂2 F ∂F ∂F (r − δi )Yi + σij Yi Yj − rF + q = 0. + ∂t ∂Yi 2 ∂Yi ∂Yj i

(3.1)

ij

The boundary conditions are part of the description of the contract related to the contingent claim. When we will consider real options as contingent claims, we will make precise these boundary conditions. 3.2. Claims on a tradable asset Suppose now that we have a tradable asset Q(t), which is not one of the market assets and we have a claim on this asset. The tradable asset itself is described by the Ito differential bj (Q, t)dwj (t)). dQ = Q(a(Q, t)dt + j

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We shall assume that it carries a dividend per unit value and unit of time denoted by δ(Q, t). We do not use the notation q(t) for the dividend since it will be used for the contingent claim. Since the asset is tradable, we have the relation see Eq. (2.8) (δ(Q, t) + a(Q, t))dt = r +

ij

δi (t)((σ ∗ )−1 )ij (t)bj (Q, t)dt

⎡

⎛

− E ⎣cov ⎝

j

⎤ ⎞ dZ(t) ⎠ t ⎦ bj (Q, t)dwj (t), |F . Z(t)

The contingent claim is valued by a function F(Q, t), and it brings a proﬁt q(Q, t). Proceeding as in the case of contingent claims on market assets, we derive easily the following equation for the valuation function ∂F ∂F 1 ∂2 F + (r − δ(Q, t))Q + |b(Q, t)|2 − rF + q(Q, t) = 0. ∂t ∂Q 2 ∂Q2

(3.2)

3.3. Claims on a nontradable asset Some limited extensions can be obtained for claims on nontradable assets.We assume that the nontradable asset obeys the stochastic differential dQ = Q(a(Q, t)dt +

bj (Q, t)dwj (t)),

j

but since this asset is nontradable, the drift a(Q, t) does not satisfy the relation (2.8). This asset carries a dividend δ(Q, t) per unit value and per unit of time. We follow the model of Dixit and Pindyck (see [1994]). We assume there are tradable assets, the value of which is denoted by Qi (t), i = 1, . . . , m verifying dQi = Qi (Ai (Q, t)dt +

Bij (Q, t)dwj (t)),

j

which provides a dividend Di (Q, t) per unit value and per unit of time. We note that in the above model, the functions Ai (Q, t), Bij (Q, t), and Di (Q, t) depend on Q and not on Qi . The main assumption is that the vector b can be written as a linear combination of the vectors Bi , whose components are Bij . Namely, b=

m

γi Bi .

i=1

This assumption is satisﬁed when m = n and the matrix Bij (Q, t) is invertible.

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The contingent claim is a function F(Q, t). We consider a portfolio made of the contingent claim and a short position on the assets Qi (t), whose wealth is given by πi (t)Qi (t), x(t) = F(Q, t) − i

and we want this portfolio to be risk free. A routine calculation shows that, to ensure risk free, we must have the relations ∂F Qbj (Q, t) = πi (t)Qi (t)Bij (Q, t). ∂Q i

By the assumption, we introduce the vector γ (Q, t) of components γi (Q, t), which in the case of invertibility of the matrix B is given by (B∗ (Q, t))−1 b(Q, t). The risk-free condition is equivalent to the portfolio πi (t)Qi (t) = γi (Q, t)Q

∂F . ∂Q

Expressing that this portfolio satisﬁes dx = rx(t)dt, we obtain the valuation equation ∂F 1 ∂2 F ∂F (γi (Ai + Di − r))Q |b(Q, t)|2 − rF + q(Q, t) = 0. + (a − + ∂t ∂Q 2 ∂Q2 i

(3.3) We can also recover the relation (3.3) by expressing directly that the assets Qi (t) and the contingent claim F(Q, t) are tradable. We must have dZ(t) ((σ ∗ )−1 )kj Bij δk − E cov Bij dwj (t), |F t . (3.4) Ai + Di = r + Z(t) j

kj

Exercise 3.1. Obtain Eq. (3.3) by using (3.4), computing the Ito differential of F and expressing that the claim F(Q, t) is tradable. 3.4. Valuation of futures A future is a contingent claim, which is not payable at the time of agreement. It is not a capital asset. It is an agreement to perform an exchange at some future date. Therefore, it does not carry dividends or any interest rate. If F(Y(t), t) represents the value of a future based on market assets, then we can form a portfolio with the future and a short position on the market assets, which is risk free. However, setting πi (t)Yi (t), x(t) = F(Y(t), t) − i

Real Options

we have dx(t) = −r

541

πi (t)Yi (t)dt

i

since F cannot yield any interest rate. Developing the above conditions, we obtain 1 ∂2 F ∂F ∂F (r − δi )Yi + σij Yi Yj = 0 + ∂t ∂Yi 2 ∂Yi ∂Yj i

(3.5)

ij

4. Valuation of a project 4.1. Description of the model A project is characterized by an output called P(t). This is the basic “asset,” which is the source of value. It is measured in dollars. It is a stochastic process governed by the equation dP = αPdt + P σj dwj , (4.1) j

where w(t) is a the n-dimensional Wiener process w1 , . . . , wn modeling the randomness of the economy. So the output carries the same randomness as the economy in general. So we can consider P(t) as an asset, which may be traded or not. To simplify a little bit, we shall assume that α and σ are constant and deterministic, as are the noise correlations σij and the coupons δi of the basic assets of the economy Yi (t). Suppose P carries a dividend δ per unit value and per unit of time. It is constant to simplify. If P is tradable, we have the relation α + δ = μ, where μ called the risk-adjusted expected rate of return of P is given by (see Eq. (2.8) dZ μ=r+ ((σ ∗ )−1 )ij σj δi − cov σj dwj , . Z ij

j

Even when there is no dividend, it is economically meaningful to assume μ > α since α is the expected capital gain. This assumption implies that the randomness involved in P brings more opportunities than risks. P should generate more wealth than its expected capital gain. The difference μ − α can be denoted by δ > 0 and acts as a dividend. If P is not tradable, we shall assume that P is spanned by ﬁnancial markets, which means that there exist traded assets Qi (t) perfectly correlated with P as follows ⎛ ⎞ dQi = Qi ⎝Ai dt + Bij dwj ⎠ . j

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A. Bensoussan

We assume that γi Bij . σj = i

In addition, we require that γi = 1. i

With these assumptions we are in the same situation as above by deﬁning γi (Ai + Di ) μ= i

and δ = μ − α. 4.2. The valuation equation The project carries a ﬂow of proﬁts given by π(P, t) when the output is P at time t. Denote by V(P, t) the value of owning the project (we can also think of a ﬁrm instead of a project and speak about the value of the ﬁrm). If the output itself is tradable, then we have α = μ − δ, where μ is the risk-adjusted rate of return of the output dZ ((σ ∗ )−1 )ij σj δi − cov σj dwj , . μ=r+ Z ij

j

Considering the value of ownership of the project as a tradable asset and writing its differential, we have ∂V ∂V ∂V 1 ∂2 V 2 2 dV = dt + + Pα + P P σ σj dwj , 2 ∂t ∂P 2 ∂P ∂P j

where we have used the notation σ2 = σj2 . j

The risk-adjusted rate of return of the ownership of the project is then ∂V P ∗ −1 dZ μ ˜ =r+ . ((σ ) )ij σj δi − cov σj dwj , ∂P V Z ij

j

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Therefore, μ ˜ −r =

1 ∂V P(μ − r). V ∂P

We then write that the expected capital return on the ownership of the project plus the proﬁt ﬂow per unit value is equal to μ. ˜ This means Vμ ˜ =π+

∂V ∂V 1 ∂2 V 2 2 + Pα + P σ . ∂t ∂P 2 ∂P 2

Replacing α and eliminating common terms yield ∂V ∂V 1 ∂2 V 2 2 P σ − rV + π = 0. + P(r − δ) + ∂t ∂P 2 ∂P 2

(4.2)

4.3. Solution of the valuation equation Eq. (4.2) is a partial differential equation whose space variable P lies in (0, ∞). We need boundary conditions. As far as time is concerned, we will, in most cases, look for stationary solutions of Eq. (4.2), namely, 1 ∂2 V 2 2 ∂V P(r − δ) + P σ − rV + π = 0, ∂P 2 ∂P 2

(4.3)

which is possible when coefﬁcients are independent of time (as we assumed), the proﬁt ﬂow is also independent of time, and the horizon is inﬁnite. If we cannot consider the horizon as inﬁnite, we may consider the following terminal condition V(P, T ) = 0, whose interpretation is clear. Concerning the variable P we need conditions at 0 and ∞. For P = 0, we see formally on Eq. (4.3) that V(0) =

π(0) . r

It is natural to assume that π(0) = 0, so V(0) = 0. This assumes implicitly that V does not have a singularity at 0. In fact, the condition V(0), makes perfect economic sense. Concerning the condition at inﬁnity, we need to specify a growth condition. We shall require a growth condition similar to that of π(P). Leaving aside a particular solution, which will be valid for P large, the general solution is an exponential V(P) = exp βP with β to be a solution of 1 2 σ β(β − 1) + (r − δ)β − r = 0. 2

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The roots of this quadratic expression are β1 , and β2 , with 1 r−δ 1 r−δ 2r β1 = − 2 + [ − 2 ]2 + 2 , 2 2 σ σ σ 1 r−δ 1 r−δ 2r β2 = − 2 − [ − 2 ]2 + 2 , 2 2 σ σ σ and β1 > 1, β2 < 0 Exercise 4.1. Show that if π(P) = P, then V(P) =

P . δ

Exercise 4.2. Assume π(P) = (P − C)+ , then one has

V P β1 if P ≤ C 1 V(P) = P C β2 V2 P + δ − r if P ≥ C,

where V1 =

C1−β1 β1 − β 2

C1−β2 V2 = β1 − β 2

β2 β2 − 1 − r δ

β1 β1 − 1 − . r δ

Note that V1 , V2 > 0. The previous proﬁt ﬂow function is interpreted as follows: there is an operating cost C and when P < C, then the project can be interrupted, with no cost. Whenever P ≥ C, the project can be resumed with no cost either. Exercise 4.3. Assume π(P) = kP γ , γ > 1, then V(P) =

kP γ ρ

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with 1 ρ = r − γ(r − δ) − σ 2 γ(γ − 1). 2 The preceding proﬁt ﬂow is motivated by the following optimization π(P) = max[Ph(v) − C(v)], v

where h(v) represents a production function and C(v) a variable operating cost. The particular case h(v) = vθ , 0 < θ < 1, C(v) = c yields the function chosen above with γ=

1 > 1. 1−θ

4.4. Probabilistic interpretation We can interpret V(P, t) as follows.The output evolution is described by dP = P((r − δ)ds + σj dwj (s)), s ≥ t; P(t) = P. j

Note that the drift has been changed from α to r − δ. Then, we have ∞ exp −r(s − t)π(P(s), s)ds. V(P, t) = E t

Note that the discount applicable is the riskless discount r. Exercise 4.4. Obtain by the probabilistic interpretation V(P) in the cases π(P) = P, π(P) = (P − C)+ , and π(P) = kP γ . 4.5. Possibility of death Let us suppose that the project can be stopped accidentally by an external event, which arrives at a random time τ distributed with an exponential density with rate λ. Using the probabilistic interpretation, we can write τ V(P, t) = E exp −r(s − t)π(P(s), s)ds|τ > t . t

The conditional density of τ given τ > t is λ exp −λ(θ − t)dθ on the interval t, ∞. Exercise 4.5. Check that ∞ V(P, t) = E exp −(r + λ)(s − t)π(P(s), s)ds . t

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5. Valuation of an option to invest 5.1. Motivation In the previous section, we have discussed the value V(P, t) of a project depending on a random output whose value at time t is P. The value stream of this project is a proﬁt ﬂow π(P, t). To simplify the discussion, we will consider the stationary case V(P) corresponding to a proﬁt ﬂow π(P). Note that whenever π(P) is an increasing function of P (a natural assumption), V(P) is also increasing in P. This can be seen by using the probabilistic formula and recognizing that P(s) being log normal with initial condition P is increasing in P. The problem we face now is that of investing in the project an amount I. In other words, we pay a price I to get a a project of value V(P). Under net present value (NPV) approach, we will invest if P ≥ P0 , where I = V(P0 ). 5.2. The option approach Although natural and used constantly, the NPV approach has a serious ﬂaw. It rules out one possibility, that of postponing the decision to invest to wait for more favorable values of P. It does not take into consideration the ﬂexibility, which is key in decision making. The option approach aims at introducing a ﬂexibility in the time of decision. At any time, we can either invest immediately, in which case we get V(P) − I, or postpone a little bit of time and consider that we have a contingent claim to be valued according to valuation techniques already discussed. We denote by F(P) the value of the option. What is the problem to be solved to ﬁnd this function. Since on the branches of the alternative is to invest immediately, we must have F(P) ≥ V(P) − I, ∀P. The other branch is to keep the option. To proceed, we must use valuation concepts. Let us assume, for instance, that the output P is not directly tradable, but there exists a tradable asset Q governed by σj dwj (t)). dQ = Q(Adt + j

We can form a portfolio made of the option and a short position in the tradable asset Q. To achieve a riskless portfolio, we must assume F(P) to be smooth, so we can apply Ito’s calculus. A portfolio F(P(t)) − π(t)Q(t) will be riskless, if Q(t)π(t) = P(t)F (P(t)). If we keep the option, we get just the increase in capital, since the option by itself does not carry any dividend. The expected increase in capital is 1 F (P)Pα + F (P)σ 2 − F (P)PA. 2

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Since it is risk free, it cannot be larger than r(F − PF (P)). As usual, we set δ = A − α > 0 and we get the inequality 1 F (P)P(r − δ) + F (P)σ 2 − rF ≤ 0. 2 Since the alternative has only two branches, at any time, one of the inequality must become an equality. 5.3. Variational inequality We can summarize by stating that F(P) is solution of the following set of differential inequalities and complementarity slackness condition F(P) ≥ V(P) − I 1 F (P)P(r − δ) + F (P)σ 2 − rF ≤ 0 2 1 (F(P) − V(P) + I)(F (P)P(r − δ) + F (P)σ 2 − rF) = 0. 2

(5.1)

This problem is called a variational inequality (VI). It must be completed by boundary conditions and smoothness conditions. We take F(0) = 0, and we assume a growth condition similar to that of V(P), hence of π(P). In addition, we require F to be continuously differentiable. We can give a probabilistic interpretation for F(P) as a problem of optimal stopping. Recall that we must consider that P(t) has the differential σj dwj (s)); P(0) = P. dP = P((r − δ)ds + j

Consider now stopping times τ adapted to the σ-ﬁeld generated by the process P(t), then we have F(P) = max E[(V(P(τ)) − I) exp −rτ]. τ

To solve the VI, we look for a value P ∗ , such that 1 F (P)P(r − δ) + F (P)σ 2 − rF = 0, P < P ∗ 2 F(P) = V(P) − I, P ≥ P ∗ F (P ∗ ) = V (P ∗ ). Deﬁne (P) = F(P) − V(P) + I.

(5.2)

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Then, we have (P) = 0, P ≥ P ∗ 1 (P)P(r − δ) + (P)σ 2 − r = π(P) − rI, P < P ∗ 2 ∗ (P ) = 0.

(5.3)

Theorem 5.1. Assume that π(P) increases and that a solution of the system (5.3) exists. Then, F(P) = (P) + V(P) − I is solution of the VI (5.1). Proof. We have to check (P) > 0, P < P ∗ 1 (P)P(r − δ) + (P)σ 2 − r ≤ π(P) − rI, P > P ∗ . 2

(5.4)

First, we must have π(P ∗ ) − rI > 0. Indeed, consider γ(P) = (P). It is the solution of 1 γ P(r − δ + σ 2 ) + σ 2 P 2 γ − rγ = π (P) 2 and γ(P ∗ ) = 0. Since π (P) > 0, it follows that γ(P) < 0, P < P ∗ . This implies also γ (P ∗ ) = (P ∗ ) > 0, hence necessarily from the equation letting P ↑ P ∗ , we obtain the desired property. Hence, the second part of condition (5.4) is satisﬁed, since π(P) is increasing. To prove the ﬁrst part of assertion (5.4), we ﬁrst notice that (P) > 0 for P sufﬁciently close to P ∗ . Indeed, we may write the formula 1 1 ∗ 2 (P) = (P − P ) λ (P ∗ + λμ(P − P ∗ ))dλdμ, 0

0

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and the integrand is positive for P sufﬁciently close to P ∗ . This, of course, postulates that (P) is a continuous function. Introduce next P∗ deﬁned by π(P∗ ) = rI, P∗ < P ∗ , then, we ﬁrst prove that (P) > 0, ∀P∗ < P < P ∗ . If this assertion is not true, then there exists P0 = P∗ , with P∗ < P0 < P ∗ and (P0 ) = 0. Let P¯ be this point. One should have ¯ = 0, (P) ¯ < 0, (P) ¯ > 0, (P) ¯ − rI > 0. Hence, which is impossible since π(P) (P) > 0, ∀P∗ ≤ P < P ∗ . On [0, P∗ ], let P be the minimum of (P). We cannot have (P) < 0. If so, P is in the interior of the interval and ( P ) = 0, (P) > 0, ( P ) < 0. ¯ − rI > 0, which is impossible. The proof has been completed. Hence, π(P) 5.4. Parabolic VI We consider now the nonstationary case: the problem (5.1) becomes F(P, t) ≥ V(P, t) − I ∂F ∂F 1 ∂2 F 2 + P(r − δ) + σ − rF ≤ 0 ∂t ∂P 2 ∂P 2 ∂F ∂F 1 ∂2 F 2 (F(P, t) − V(P, t) + I) + P(r − δ) + σ − rF =0 ∂t ∂P 2 ∂P 2

(5.5)

F(P, T) = 0. This problem is called a parabolic VI. We cannot use the same approach as that used for stationary VI (called also elliptic VI). This is because of the extra term, which is the partial derivative with respect to time. We are interested in the same type of solution, namely, ﬁnd a free boundary P ∗ (t) with F satisfying ∂F 1 ∂2 F 2 ∂F + P(r − δ) + σ − rF = 0, P < P ∗ (t) ∂t ∂P 2 ∂P 2 F(P, t) = V(P, t) − I, P ≥ P ∗ (t) ∂F ∗ (P (t), t) = 0, F(P, T) = 0. ∂P

(5.6)

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∂π Theorem 5.2. Assume ∂π ∂t ≤ 0, ∂P ≥ 0, and π(0, t) = 0. Then, there exists a continuous differentiable function F solution of the parabolic VI (5.5), which is of the form (5.6)

Proof. It is convenient to introduce (P, t) = F(P, t) − V(P, t) + I and to write the parabolic VI, as follows: (P, t) ≥ 0 ∂ ∂ 1 ∂2 2 σ − r ≤ π(P, t) − rI + P(r − δ) + ∂t ∂P 2 ∂P 2 ∂ 1 ∂2 2 ∂ + P(r − δ) + σ − r − π(P, t) + rI = 0 (P, t) ∂t ∂P 2 ∂P 2 (P, T) = I

(5.7)

We shall show the existence of a continuously differentiable solution of (5.7), which satisﬁes ∂ ∂ ≥ 0, ≤ 0. ∂t ∂P

(5.8)

If this is true, then note that (0, t) satisﬁes ∂ (0, t) − r(0, t) ≤ −rI ∂t ∂ (0, t) (0, t) − r(0, t) + rI = 0 ∂t (0, t) ≥ 0, (0, T ) = I; therefore, (0, t) = I. We then deﬁne P ∗ (t) = {inf P|(P, t) = 0.} Note that P ∗ (T ) = ∞. By deﬁnition ∂ 1 ∂2 2 ∂ σ − r = π(P, t) − rI, P < P ∗ (t). + P(r − δ) + ∂t ∂P 2 ∂P 2 Next for P > P ∗ (t), we have 0 ≤ (P, t) ≤ (P ∗ (t), t) = 0. Hence, (P, t) = 0 for P > P ∗ (t). Clearly, P ∗ (t) > P∗ (t)

(5.9)

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with π(P∗ (t), t) = rI. The study of the parabolic VI is done using the penalty approximation ∂ ∂ 1 ∂ 2 2 1 − + P(r − δ) + σ + ( ) − r = π(P, t) − rI ∂t ∂P 2 ∂P 2 with the initial condition (P, T ) = I. By classical methods, the solution of the penalty approximation converges toward a solution of the parabolic VI. However, if we consider η (P, t) =

∂ , ∂t

then it is the solution of ∂η ∂π(P, t) ∂η 1 ∂ 2 η 2 η σ − 1I <0 − rη = + P(r − δ) + 2 ∂t ∂P 2 ∂P ∂t with ﬁnal condition η (P, T ) = π(P, T ) from which (and the assumption) it follows that η (P, t) ≥ 0. Similarly, consider γ (P, t) =

∂ , ∂P

which is the solution of ∂γ ∂γ 1 ∂2 γ 2 γ ∂π(P, t) + P(r − δ + σ 2 ) + , σ − 1I <0 − δγ = 2 ∂t ∂P 2 ∂P ∂P with ﬁnal condition γ (P, T ) = 0 from which (and the assumption again) we obtain γ (P, t) ≤ 0. Letting go to 0, we obtain the properties (5.8).

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Note that from the sign conditions (5.8), we deduce dP ∗ (t) > 0, dt

(5.10)

which follows from differentiating the relation (P ∗ (t), t) = 0. 5.5. Comparison with NPV From the preceding characterization of the value of the option to invest in a project F(P, t) it follows that we use the following decision rule. We invest in the project if P(t) ≥ P ∗ (t),

(5.11)

where P(t) is the value of the output at time t representing the present time (more precisely the time of decision). On the other hand, the traditional NPV decision rule tells that one invests at time t provided that V(P(t), t) ≥ I. A natural question is to compare these two decision rules. We begin by stating the property v(P, t) =

∂V(P, t) > 0. ∂P

(5.12)

This is similar to the reverse property for (see (5.8)). Differentiating Eq. (4.2) in P, we obtain ∂v 1 ∂2 v 2 ∂π(P, t) ∂v + P(r − δ + σ 2 ) + = 0, σ − δv + 2 ∂t ∂P 2 ∂P ∂P with ﬁnal condition v(P, T) = 0. ˆ such that It follows easily that v(P, t) is positive. We may deﬁne P(t) ˆ V(P(t), t) = I. The NPV rule is equivalent to the following rule ˆ we invest in the project if P(t) ≥ P(t) We prove the following. Theorem 5.3. Under the assumptions of Theorem 5.2, we have ˆ < P ∗ (t). P(t) So the NPV rule is always wrong. One invests much too early with this rule.

(5.13)

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Proof. The result will follow from the fact that F(P ∗ (t), t) = V(P ∗ (t), t) − I > 0. ˆ From the deﬁnition of P(t) and the monotonicity property of V(P, t) with respect to P the result is obtained. We remark that F(P, t) is the solution of the problem ∂F 1 ∂2 F 2 ∂F + P(r − δ) + σ − rF = 0, P < P ∗ (t) ∂t ∂P 2 ∂P 2 F(0, t) = 0 F(P, T) = 0 ∂F ∗ ∂V ∗ (P (t), t) = (P (t), t) > 0. ∂P ∂P We claim that F(P, t) ≥ 0, ∀P, t such that P ≤ P ∗ (t), t ≤ T. It is sufﬁcient to consider a minimum P0 , t0 of the function F(P, t) and to prove that F(P0 , t0 ) ≥ 0. Suppose we have F(P0 , t0 ) < 0. We cannot have P0 = 0 or t0 = T . We cannot have either P0 = P ∗ (t0 ). Indeed, since P0 , t0 is a minimum of F(P, t), F(P0 − , t0 ) ≥ F(P0 , t0 ) ∗ from which we get ∂F ∂P (P0 , t0 ) ≤ 0, which is not possible if P0 = P (t0 ). So the point P0 , t0 is in the interior of the domain, for which the partial differential equation holds. A look at the equation shows that F(P0 , t0 ) > 0, which is impossible. Hence, the positivity of F . Finally, this positivity combined with the strict positivity of the partial derivative in P at P ∗ (t) implies the strict positivity of F(P ∗ (t), t), hence the result.

The quantity F(P ∗ (t), t) = V(P ∗ (t), t) − I > 0 represents the value of the option at time t, hence the value of ﬂexibility in the decision of investment at time t. Exercise 5.1. Solve the parabolic VI in the case of the proﬁt ﬂow π(P, t) = P exp −λt. Show that V(P, t) =

P exp −λt(1 − exp −(λ + δ)(T − t)) λ+δ

P ∗ (t) =

β1 I(λ + δ) β1 − 1 exp −λt(1 − exp −(λ + δ)(T − t))

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hence, V(P ∗ (t), t) =

β1 I. β1 − 1

6. Extensions The model studied so far has many limitations. There is a single project under consideration (or a single ﬁrm). The price for investment is ﬁxed. Within this framework, the investment will be decided. The only question is when there is no possibility of abandonment or temporary mothballing. Also, the investment is external, there is no policy of building capital. The objective in the following sections is to relax some of these limitations and also to consider how the market mechanism will lead to an equilibrium. 7. Uncertainties on investment 7.1. Assumptions We consider here a model in which not only the value of the output ﬂow is random but also the investment needed can vary with time. More speciﬁcally, we have dP = P αP dt + σPj dwj j

and a similar relation for the investment dI = I αI dt + σIj dwj . j

To ﬁx the ideas that both P and I are tradable assets, we will consider their risk-adjusted expected returns μP , μI and assume as usual μP − αP = δP > 0, μI − αI = δI > 0, and δP ,δI can be interpreted as dividends associated with the output ﬂow or the investment ﬂow. We will write 2 2 σPj , σI2 = σIj , σP2 = j

and ρσP σI =

j

σPj σIj .

j

We suppose to simplify that the proﬁt ﬂow of the project is given by π(P) = P.

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We will then consider a stationary model. We ﬁrst note that the value of the project does not depend on the investment; therefore, in view of the proﬁt ﬂow and Exercise 4.1, we have V(P) =

P . δP

7.2. Valuation of the option The value of the option F(P, I) depends on P and I. It did not appear explicitly when I was just a constant, but now its evolution must be taken into consideration. By standard arguments, one obtains that F(P, I) must be the solution of the VI ∂F ∂F P(r − δP ) + I(r − δI ) ∂P ∂I + F−

1 ∂2 F 2 2 1 ∂2 F 2 2 ∂2 F P σ + I σ + PIρσP σI − rF ≤ 0 P I 2 ∂P 2 2 ∂I 2 ∂P∂I

(7.1)

P +I ≥0 δP

product = 0, where “product” means the product of the two quantities on the left-hand side of the inequalities. It is convenient to introduce (P, I) = F(P, I ) −

P + I, δP

and we get the problem ∂ ∂ 1 ∂2 2 2 P(r − δP ) + I(r − δI ) + P σP ∂P ∂I 2 ∂P 2 +

1 ∂2 2 2 ∂2 PIρσP σI − r ≤ P − IδI I σ + I 2 ∂I 2 ∂P∂I

(7.2)

≥0 product = 0. Fortunately, this two-dimensional problem can be reduced to a one-dimensional problem by scaling considerations. In fact, the solution can be expressed by (P, I) = Iz

P I

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and z(x) is the solution of the VI. ∂z 1 ∂2 z 2 2 x (σP + σI2 − 2ρσP σI ) − zδI ≤ x(δI − δP ) + ∂x 2 ∂x2 x − δI z ≥ 0

(7.3)

product = 0. This VI can be easily solved. Considering the quadratic form 1 β(δI − δP ) + β(β − 1)(σP2 + σI2 − 2ρσP σI ) − δI = 0 2 and its root β1 > 1, we deduce a threshold value x∗ =

β1 δP β1 − 1

and the solution is given by x z = f1 xβ1 − + 1, x ≤ x∗ δP and z = 0, x ≥ x∗ . The constant f1 is such that there is continuity for x = x∗ . 8. Uncertainties due to incentives 8.1. Setting of the model We suppose in this model that the investment cost I can be reduced by an incentive to invest decided by government. However, the decision of the government is random. The incentive can be introduced or withdrawn according to a birth and death process. More precisely, the investment cost is a stochastic process given by the model I(t) = I(1 − θη(t)), where η(t) is a stochastic process independent of P(t). The process η(t) can take two values, 1 or 0. It evolves as a Markov chain with Prob(η(t + dt) = 1|η(t) = 1) = 1 − λ0 dt Prob(η(t + dt) = 0|η(t) = 0) = 1 − λ1 dt. The output ﬂow is governed by dP = P αdt + σj dwj j

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and α = μ − δ, where μ is as usual (assuming that P is tradable) the risk-adjusted rate of return. Since the value of the project does not depend on I, we still have V(P ) =

P . δ

The value of the option depends on the process η(t). So if η(0) = η, it is a function F(P, η). Since η can take only two values, it is convenient to write F(P, 0) = F 0 (P ),

F(P, 1) = F 1 (P ).

8.2. System of VI The functions F 0 (P ) and F 1 (P ) are solutions of a system of coupled VI. To establish it, a convenient way is to use the probabilistic interpretation, recalling that the drift α must be replaced with r − δ. We have P(τ) − I(1 − θη(t)) exp −rτ . F(P, η) = max E τ δ To simplify the notation, deﬁne the differential operator on smooth functions φ(P) by 1 Aφ(P) = φ (P )P(r − δ) + σ 2 P 2 φ (P ) − rφ(P ). 2 By the standard dynamic programming arguments, one obtains easily the system AF 0 (P ) + λ1 (F 1 (P ) − F 0 (P )) ≤ 0 P +I ≥0 δ product = 0 F 0 (P ) −

(8.1)

AF 1 (P ) + λ0 (F 0 (P ) − F 1 (P )) ≤ 0 P + I(1 − θ) ≥ 0 δ product = 0. F 1 (P ) −

(8.2)

8.3. Solution of the system The solution of the system is guided by intuition. There is a threshold of the output ﬂow for which it makes sense to invest whether or not the incentive is in place. Similarly, there is a threshold below which one will not invest even when the incentive is in place.

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In between, one will invest when the incentive is in place and will postpone decisions when the incentive is not in place. So, we look for two numbers 0 < P 1 < P 0 such that AF 0 + λ1 (F 1 − F 0 ) = 0 AF 1 + λ0 (F 0 − F 1 ) = 0 ∀P < P 1 AF 0 + λ1 (F 1 − F 0 ) = 0 F 1 (P) =

P − I(1 − θ) δ

∀P 1 < P < P 0 and ﬁnally P −I δ P F 1 (P) = − I(1 − θ) δ

F 0 (P) =

∀P 0 < P. One can check that three differential operators play a role in the solution. They are denoted as follows: A0 φ(P) = Aφ(P) A1 φ(P) = Aφ(P) − λ1 φ(P) A2 φ(P) = Aφ(P) − (λ1 + λ0 )φ(P). The notation is explained by the presence of zero λ term, one λ term, or two λ terms. We associate to these differential operators second-order algebraic equations 1 2 σ β(β − 1) + (r − δ)β − r = 0 2 1 2 σ β(β − 1) + (r − δ)β − (r + λ1 ) = 0 2 1 2 σ β(β − 1) + (r − δ)β − (r + λ1 + λ0 ) = 0, 2 whose roots are denoted as follows: β(0)1 , β(0)2 ;

β(1)1 , β(1)2 ;

β(2)1 , β(2)2 ,

where the second index is 1 for the root larger than 1, and 2 for the negative root.

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Exercise 8.1. Show that one can write for P < P 1 F1 =

λ0 λ1 F1a P β(0)1 + λ0 F1s P β(2)1 λ0 + λ 1

F0 =

λ0 λ1 F1a P β(0)1 − λ1 F1s P β(2)1 . λ0 + λ 1

Next for P 1 < P < P 0 , we have F1 =

P − I(1 − δ) δ

F 0 = F10 P β(1)1 + F20 P β(1)2 +

I(1 − θ) λ1 P − λ1 . δ δ + λ1 r + λ1

For P > P 0 , the values of F 0 (P) and F 1 (P) are known. We, thus, have six constants entering into the deﬁnition of F 0 F 1 , namely, F1a , F1s , F10 ,F20 , P 0 , and P 1 . We can express six matching conditions. At P 1 we can write two conditions for F 0 and two for F 1 . Next, at P 0 we can write two conditions for F 0 . This deﬁnes completely the solution. It remains to show that this solution solves indeed the system of VI (8.1) and (8.2). 9. The option of abandonment 9.1. Setting of the model In the preceding models, once the investment is decided the project is continued to its end and a value is collected. We have also considered the possibility of temporary suspension whenever P < C and resuming the activity when P > C with no penalty in both cases. This resulted in simply taking a proﬁt ﬂow function given by π(P) = (P − C)+ . We consider now the possibility of abandonment. This means we may decide to stop the project when it is active. This will entail a ﬁxed cost denoted by E. If such a decision occurs, we are put back in the situation before the decision of investment was made. We may, thus, start again, but from scratch, paying the same investment cost I. So, there is no beneﬁt from previous investment. Let us consider a stationary model. The option part (decision to invest) is the same as before, provided, of course, we have the right value function for the project, namely, we have (recalling the deﬁnition of the operator A) AF(P) ≤ 0 F(P) − V(P) + I ≥ 0 AF(P)(F(P) − V(P) + I) = 0.

(9.1)

However, now the value V(P) is not deﬁned independently of F . Indeed, if one stops the project, then one goes back to the situation before investing. In addition, when the project runs, it faces a proﬁt ﬂow P − C, since the possibility of postponement with no

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penalty is no longer present. It follows that V(P) is the solution of AV(P) + P − C ≤ 0 V(P) − F(P) + E ≥ 0 (AV(P) + P − C)(V(P) − F(P) + E) = 0.

(9.2)

9.2. Two-sided VI Note ﬁrst the compatibility condition I + E ≥ 0, which is obvious whenever the two quantities I and E are positive. However, this leaves room for negative E. This possibility is useful whenever there is some cash recovered when the project is dismantled, which is a realistic situation. However, we must have I + E > 0 to avoid situations in which one could get cash but with continuous investment and disinvestment. The nice feature is that the function (P) = F(P) − V(P) + I is still the solution to a single problem (no coupling). However, it is a two-sided VI and not a one-sided VI. The problem is expressed as follows: 0 ≤ (P) ≤ I + E if 0 < (P) < I + E, then A(P) = P − C − rI if (P) = 0, then A(P) ≤ P − C − rI if (P) = I + E, then A(P) ≥ P − C − rI.

(9.3)

Since AC = −rC, the two last conditions reduce to (P) = 0 ⇒ P − C − rI > 0 (P) = I + E ⇒ P − C + rE < 0. 9.3. Solution of the two-sided VI The two last conditions guide the intuition. For a sufﬁciently low output P, one should not only not invest but also cut the investment. For a sufﬁciently large output one should not only continue the project but also invest if the project has not yet started. In between, one should continue a project already started but one should not invest. Therefore, one looks for two thresholds 0 < PL < PH such that A(P) = P − C − rI, ∀PL < P < PH

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and (P) = 0, ∀P > PH (P) = I + E, ∀P < PL . Clearly, we must have PH ≥ C + rI,

PL ≤ C − rE.

We will have in the interval (PL , PH ) (P) = 1 P β1 + 2 P β2 −

P C + + I, δ r

and we have four constants to obtain, 1 , 2 and PH , PL . There are also four conditions to express the continuity of (P) and its derivative at points PL and PH . We obtain the following system C PL + =E δ r P C H β β 1 PH1 + 2 PH2 − + +I =0 δ r 1 β1 −1 β2 −1 1 β1 PL + 2 β2 P L − =0 δ 1 β −1 β −1 1 β1 PH1 + 2 β2 PH2 − = 0. δ β

β

1 PL1 + 2 PL2 −

We derive the following system for PH and PL β2 C β2 C 1 1 δ (1 − β2 ) + PL r − E δ (1 − β2 ) + PH r + I = β −1 β −1 PL1 PH1 1 δ (β1

β1 PL β −1 PL2

− 1) −

C r

−E

=

1 δ (β1

β2 PH β −1 PH1

− 1) +

C r

+I

It can be shown that this system has a unique solution with PL < PH (see Dixit [1989]).

10. The option of mothballing 10.1. Description of the model We introduce a new possibility for an active project, that of mothballing instead of abandoning. From a situation of mothballing, a project can be reactivated or abandoned. In a situation of mothballing, the proﬁt ﬂow is lost and in addition a maintenance cost M must be paid. It is smaller than the operating cost C. To put a project in a situation of

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mothballing incurs a ﬁxed cost EM . To reactive from mothballing implies a ﬁxed cost ER . Finally, to abandon a project from mothballing represents a cost ES . The quantity E = ES + EM represents the ﬁxed cost of abandoning an active project. In writing the VI for V , we will need a new function H(P), which represents the value of the option of mothballing. So, in fact, we will have a coupled system for three functions F, V , and H. 10.2. Variational inequalities We consider the stationary case. The value of the option to invest is governed by the VI. AF(P) ≤ 0 F(P) − V(P) + I ≥ 0

(10.1)

AF(P)(F(P) − V(P) + I ) = 0. The problem of which V(P) is a solution is now given by AV(P) + P − C ≤ 0 V(P) ≥ H(P) − EM

(10.2)

(AV(P) + P − C)(V(P) − H(P) + EM ) = 0. In (10.2), the function H(P) represents the value of mothballing. It is itself governed by the following VI AH(P) − M ≤ 0 H(P) ≥ max(V(P) − ER , F − ES )

(10.3)

(AH(P) − M)[H(P) − max(V(P) − ER , F − ES )] = 0. The second inequality expresses the fact that if mothballing is stopped, it is to go back to a state of active project or abandon, in which case one is back in the situation preceding investing. 10.3. Solution of the VI To deﬁne the solution, we need four thresholds 0 < PS < PM < PR < PH ,

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which trigger the following situations for P > PH , F(P) = V(P) − I for P > PR , H(P) = V(P) − ER for P < PM , V(P) = H(P) − EM for P < PS , H(P) = F(P) − ES . As a consequence, we have for P < PS , V(P) = F(P) − E. So below PS , an active project abandons without going to the mothballing step. So PS and PH correspond to the thresholds PL and PH deﬁned when there was no option of mothballing. We can next deﬁne completely the three functions by solving the differential equations in the respective intervals. We can state F(P) = F1 P β1 for P < PH F(P) = V(P) − I for P > PH V(P) = V2 P β2 +

C P − for P > PM δ r

V(P) = H(P) − EM for P < PM H(P) = H1 P β1 + H2 P β2 −

M for PS < P < PR r

H(P) = F(P) − ES for P < PS H(P) = V(P) − ER for P > PR . The solution depends on eight constants, the four thresholds and the values of constants F1 , G2 , H1 , and H2 . We write eight matching conditions (two per threshold). There is some decoupling. The values H1 ,G2 − H2 ,PR ,PM are solutions of the system PR C−M − − ER = 0 δ r 1 β −1 β −1 − H1 β1 PR1 + (G2 − H2 )β2 PR2 + = 0 δ PM C−M β β − H1 PM1 + (G2 − H2 )PM2 + − + EM = 0 δ r 1 β −1 β −1 − H1 β1 PM1 + (G2 − H2 )β2 PM2 + = 0. δ β

β

− H1 PR1 + (G2 − H2 )PR2 +

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Similarly, we deﬁne F1 − H1 ,H2 ,PS , PH by the system β

β

(F1 − H1 )PS 1 − H2 PS 2 + β −1

(F1 − H1 )β1 PS 1

M − ES = 0 r β −1

− H 2 β2 P S 2

=0

PH C + +I =0 δ r 1 β −1 β −1 F1 β1 PH1 − G2 β2 PH2 − = 0. δ β

β

F1 PH1 − G2 PH2 −

11. Reﬂecting barriers 11.1. Barrier at high output ¯ To justify this We want to prevent the output P(t) from going beyond a barrier P. model, we refer to the fact that at high values of the output, the project is submitted to competing projects from other ﬁrms. At the barrier, we assume that sufﬁciently many ﬁrms will compete preventing the output value to go beyond the barrier. The stochastic process that models the output is the following σj dwj (t)) − dξ(t). dP = P(αdt + j

We have, in fact, a pair P(t) and ξ(t) uniquely deﬁned by the following properties: ξ(t) is a continuous process adapted to the ﬁltration generated by the Wiener process, and P(t) is continuous adapted and has the Ito differential written above. Moreover, ξ(t) is ¯ nondecreasing and does not increase when P(t) < P. We assume that the proﬁt ﬂow is the output value itself P. We want to deﬁne the value ¯ For P < P, ¯ the evolution of P(t) is the same as the V(P) of an active project, for P ≤ P. nonreﬂected process, some by standard arguments we shall have ¯ AV(P) + P = 0, ∀P < P. ¯ However, at P¯ the process is instantaneously We need a boundary condition at P. reﬂected so ¯ V(P¯ − ). V(P) Since is arbitrarily small, we must have ¯ V (P). Exercise 11.1. Show that the solution of AV(P) + P = 0, ∀P < P¯ V(0) = 0,

¯ =0 V (P)

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is given by V(P) =

1 β1 1−β1 P P P¯ . − δ δβ1

(11.1)

We see, in particular, that ¯ = V(P)

P¯ β1 − 1 . δ β1

(11.2)

¯ it makes no We now turn to the value F(P) of the option to invest in the project. At P, sense to wait since the output will decline certainly in the near future. So, we must have ¯ = V(P) ¯ − I. F(P) For P < P¯ we are in the same situation as without barrier. Therefore, F(P) is the solution to the problem AF(P) ≤ 0, F(P) − V(P) + I ≥ 0 AF(P)(F(P) − V(P) + I ) = 0, ∀P < P¯

(11.3)

¯ = V(P) ¯ − I. F(P) Exercise 11.2. Assume that P∗ =

β1 ¯ δI < P, β1 − 1

then show that F(P) = F1 P β1 , P < P ∗ F(P) = V(P) − I, P ∗ ≤ P ≤ P¯ F1 =

(11.4)

1 (P ∗ 1−β1 − P¯ 1−β1 ). δβ1

We see that F1 > 0, hence the value of the option is positive for P ≤ P ∗ . ¯ We need to have indifference between the interest Can we reach an equilibrium at P? to invest or to stay out. This means that ¯ = I. V(P) ¯ we check easily that Recalling the formula for V(P), P¯ = P ∗ . This implies F1 = 0, and the value of the option is always 0.

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11.2. Case of two barriers ¯ with a process The model with one barrier can be generalized to two barriers, P and P, P(t) governed by dP = P(αdt + σj dwj (t)) − dξ(t) + dη(t), j

where P(t), ξ(t), and η(t) are adapted continuous processes, ξ(t) and η(t) are nondecreasing, and dξ(t) = 0, if P(t) < P¯ dη(t) = 0, if P(t) > P. These conditions determine a unique triple P(t), ξ(t), and η(t). The logic of the lower barrier is the same as the upper barrier. The output P(t) is sufﬁciently low and competing projects will be naturally withdrawn. We adapt the language for the upper barrier to the lower one. We now have a system F(P), V(P) solution of AF(P) ≤ 0, F(P) − V(P) + I ≥ 0 AF(P)(F(P) − V(P) + I ) = 0, ∀P < P¯ ¯ = V(P) ¯ −I F (P) = 0, F(P) AV(P) + P − C ≤ 0

(11.5)

V(P) − F(P) + E ≥ 0 (AV(P) + P − C)(V(P) − F(P) + E) = 0 ¯ = 0, V(P) = F(P) − E, V (P) where E represents the ﬁxed cost of abandon. We need two thresholds with ¯ P < PL < PH < P, and we have the formulas V(P) = V1 P β1 + V2 P β2 +

P C − , for PL < P < P¯ δ r

F(P) = F1 P β1 + F2 P β2 , for P < P < PH . So, we have six constants PL , PH , F1 , F2 , V1 , V2

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to deﬁne. We can write six conditions F (P) = 0, F(PH ) = V(PH ) − I, F (PH ) = V (PH ) ¯ = 0, V(PL ) = F(PL ) − E, V (PL ) = F (PL ). V (P) Exercise 11.3. Show that PH ,PL , V1 − F1 , V2 − F2 can be deﬁned by the system C PH − =I δ r 1 β1 −1 β2 −1 + (V2 − F2 )β2 PH + =0 (V1 − F1 )β1 PH δ C PL β1 β2 − = −E (V1 − F1 )PL + (V2 − F2 )PL + δ r 1 β1 −1 β2 −1 + (V2 − F2 )β2 PL + = 0. (V1 − F1 )β1 PL δ β

β

(V1 − F1 )PH1 + (V2 − F2 )PH2 +

(11.6)

We complete with the relations F1 β1 P β1 −1 + F2 β2 P β2 −1 = 0 V1 β1 P¯ β1 −1 + V2 β2 P¯ β2 −1 = 0.

(11.7)

¯ provided the values are in the interval Note that PH and PL do not depend on P and P, ¯ (P, P) We will deﬁne equilibrium in this setup by the condition ¯ PL = P, PH = P, arguing that at equilibrium the general thresholds and those for an individual project coincide. We, then, have ¯ = V (P) ¯ = 0, F (P) = F (P) from which one obtains F1 = F2 = 0; therefore, F(P) = 0, ∀P. There is no value in the option of abandonment. Moreover, F1 = F2 = 0; therefore, ¯ = I. V(P) = −E, V(P)

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12. Equilibrium model 12.1. General description From a better economic interpretation, we replace projects by ﬁrms. Instead of abandoning a project, we will speak of ﬁrms becoming idle, and instead of investing in a project, we will speak of ﬁrms becoming active. We consider an economy in which N ﬁrms per unit of time enter in the market at any time. In an equilibrium, there will be permanently Q ﬁrms that are active. However, in this model, when a ﬁrm enters in the market, it does not become immediately active. It can wait sometime before becoming active, for which it will have to pay the usual fee I. To explain this fact, we assume that the output value is given by P(t) = X(t)D(Q). Here, D(Q) represents the demand, which depends on the number of active ﬁrms. It is a given function. The process X(t) describes the random component of the output value. The random component is modeled by dX = X(αdt + σdw(t)). There is, in addition, an initial value ξ, which is also random and independent of w(t). We assume that it has a probability density g(x). To enter into the market means that a ﬁrm will learn about the initial value of its random component. It has to pay a price R for this information. After it has observed this initial value, it may decide to become active or to wait till a better value of the output. So, in permanence, there will be Q active ﬁrms and M waiting ﬁrms. The numbers N, Q, M are the unknowns of a possible equilibrium. However, if ﬁrms live forever, there cannot be any equilibrium. So, we assume that there is an exogenous death process. The lifetime is, thus, a random variable T , whose distribution is exponential with rate λ. Moreover, T is assumed to be independent of ξ and w(t). In the following analysis, we will not introduce risk-adjusted expected return to simplify the presentation and ﬁrms are risk neutral. 12.2. Value of options A ﬁrm entering in the market pays the fee R to observe the initial value ξ of its random component. During its lifetime, it starts by waiting for some time before it becomes active, for which an extra I has to be paid. The external death process applies in the same way to idle and active ﬁrms. The value of an active ﬁrm, knowing that its initial ξ = x, is given by T V(x, Q) = E X(t)D(Q) exp −rtdt|X(0) = x , 0

where we have emphasized the dependence in Q. It is easy to check that V(x, Q) =

xD(Q) . r+λ−α

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The value of an option to become active is given by F(x, Q) = max E[V(X(τ))1Iτ

where τ is any stopping time. The probabilistic interpretation is the easiest way to F . However, for computations, we refer to the VI formulation. Let us deﬁne the differential operator 1 Aφ(x) = xαφ (x) + x2 σ 2 φ (x) − (r + λ)φ(x). 2 Then, F(x, Q) is the solution of AF(x, Q) ≤ 0 F(x, Q) − V(x, Q) + I ≥ 0 AF(x, Q)(F(x, Q) − V(x, Q) + I ) = 0. There will be a threshold x∗ (Q) given explicitly by x∗ =

β1 I(r + λ − α) (β1 − 1)D(Q)

and F(x, Q) = F1 (Q)xβ1 , for x ≤ x∗ F(x, Q) = V(x, Q) − I, for x ≥ x∗ . 12.3. Choice of Q We complete the deﬁnition of F(x, Q) by noting that β1 is the root larger than 1 of the second-order equation βα +

σ2 β(β − 1) − (r + α) = 0 2

and F1 (Q) =

D(Q) (r + λ − α)β1

β1

β1 − 1 I

β1 −1 .

Having deﬁned the function F(x, Q), the value of Q is given by the equation EF(ξ, Q) =

∞

g(x)F(x, Q)dx. 0

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A. Bensoussan

12.4. Distribution of ﬁrms For the sequel, it is convenient to work with the stochastic process Y(t) = logX(t), which is the solution of dY = νdt + σdw, with 1 ν = α − σ2. 2 We set y∗ = log x∗ and Y(0) = η = log ξ, where the probability distribution of η is h(x). Let us consider again that the process has a lifetime T with probability density an exponential with rate λ. Deﬁne also τ = {inf t|Y(t) ≥ y∗ .} We are interested in the following probability distribution ψ(y, t)dy = Prob {y < Y(t) < y + dy and t < T ∧ τ}, for y < y∗ . This function is the solution of the Kolmogorov equation ∂ψ ∂ψ 1 ∂2 ψ + λψ = 0 +ν − ∂t ∂y 2 ∂y2 ψ(y∗ , t) = 0, ψ(−∞, t) = 0 ψ(y, 0) = h(y). The boundary conditions are clear. To derive the PDE, one uses the following relation for 0 < y < y∗ , t > 0 ψ(y, t) Eψ(y − νdt − σdw, t − dt)(1 − λdt) and expands the right-hand side. Now, the number of ﬁrms arrived between s and s + ds for which y < Y(t) < y + dy is, thus, Nψ(y, t − s)ds. It follows that t number of ﬁrms for which y < Y(t) < y + dt = Nφt (y) = N ψ(y, s)ds. 0

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At equilibrium, ﬁrms are in ﬁxed positions. For those waiting, we can state that ∞ ψ(y, s)ds. number of ﬁrms waiting at equilibrium in position y = Nφ(y) = N 0

We obtain the differential equation ν

∂φ 1 ∂2 φ − + λφ = h(y) ∂y 2 ∂y2

(12.1)

φ(y∗ ) = φ(−∞) = 0. 12.5. Number of new entrants at equilibrium We want to obtain the value of N, which sustains an equilibrium. Let us ﬁrst check that the rate of activation, which is deﬁned by Number of ﬁrms becoming active in an interval dt N = − φ (y∗ )σ 2 . dt 2 An easy way to obtain this result is to consider the binomial approximation of the √Wiener process. From a position y, the particle moves during the interval dt to y + σ √ √ dt with √ probability 12 (1 + σν dt) and to position y − σ dt with probability 12 (1 − σν dt). It follows that the number of new active ﬁrms during the interval dt is approximately √ ν√ 1 dt), Nφ(y∗ − σ dt)(1 − λdt) (1 + 2 σ which is approximately −

N ∗ 2 φ (y )σ dt, 2

which implies the result. We deduce that the number of new active ﬁrms in the interval dt is −

N ∗ 2 φ (y )σ dt + N(1 − H(y∗ ))dt, 2

where H(y) is the cumulative distribution function corresponding to the density h(y). The additional term above corresponds to the number of ﬁrms, which are immediately active once they enter in the market. In order to keep the number of active ﬁrms ﬁxed equal to Q, the number of new activated ﬁrms during dt must coincide with the number that disappear during dt, which is Qλdt. Therefore, we obtain the relation Qλ = −

N ∗ 2 φ (y )σ + N(1 − H(y∗ )), 2

which deﬁnes the number N of new entrants per unit of time.

References Bensoussan, A., Lions, J.L. (1982). Applications of Variational Inequalities in Stochastic Control (North Holland, Amsterdam, The Netherlands). Dixit, A.K. (1989). Entry and exit decisions under uncertainty. J. Polit. Econ. 97, 620–638. Dixit, A.K., Pindyck, R.S. (1994). Investment Under Uncertainty (Princeton University Press, Princeton, NJ).

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