REAL OPTIONS Alain Bensoussan International Center for Decision and - - PowerPoint PPT Presentation

REAL OPTIONS

Alain Bensoussan International Center for Decision and Risk analysis School of Management University of Texas at Dallas

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 1/81

INTRODUCTION

Real Options theory is an approach to mitigate risks

• f investment projects which is based on two ideas.

The first one is hedging, borrowed from financial

• ptions, when market considerations can be
• introduced. The project risk must be correlated to the

market risk ,in which case tradable assets can be used to hedge. The second idea is flexibility. There is flexibility in the process of decision making. In particular, one may scale down or up the project, one may stop it, one may change orientation. This flexibility allows to react properly when information is obtained on the uncertainties of the evolution.

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EXAMPLES

• ption to defer

natural-resource extraction, real estate staged investment long-development capital-intensive projects

• ption to alter
• perating scale

facility planning and construction in cyclical industries

• ption to abandon

new product introductions in uncertain markets

• ption to switch

goods subject to volatile demand

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TRADITIONAL NPV

Consider a project which will bring a cash flow P(t) per unit of time. Its value is V = T EP(t) exp −rtdt According to the NPV rule, the project can be decided if its net present value V − I ≥ 0, where I is the investment cost. The flexibility is not valued. The discount factor is the risk-free interest rate.

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REFERENCES

A.K. DIXIT, R.S. PINDYCK Investment under Uncertainty, Princeton University Press, Princeton,New Jersey, 1994.

• L. TRIGEORGIS, Real Options ,MIT Press, 1996.
• T. COPELAND, V. ANTIKAROV, Real Options ,

Texere, N.Y. 2003

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COMPLETE MARKET MODEL

We assume continuous time. The randomness is characterized by n standard independent Wiener processes wj(t). We denote by Ft = 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 dYi(t) = Yi(t)(αi(t)dt + σij(t)dwj) where αi(t), σij(t) are processes adapted to Ft.

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The market is complete when the matrix σ(t) is

• invertible. In this case, the information obtained by
• bserving the evolution of the prices of assets is

sufficient to recover the underlying source of noise modelled by the Wiener processes. We assume that these basic assets do not carry coupons.

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MARKET RISK INDICATOR

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) − r1 I) whose definition makes direct use of the invertibility

• f the matrix σ(t).

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We next define Z(t) by the relation dZ(t) = −Z(t)θ(t).dw(t), Z(0) = 1 This process ( a martingale ) is an indicator of the market risk

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RISK PREMIUM AND CAPM

This is justified by the formula αidt = rdt − cov(dYi(t) Yi(t) , dZ(t) Z(t) |Ft) (1) The expected return of each individual asset is the risk-free return plus a premium linked to the specific risk of the asset. Other Approaches: Risk-neutral probability, martingale, CCAPM

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TRADABLE ASSET

We shall consider now an asset whose value P(t) evolves according to dP = P(α(t)dt +

• j

σjdwj) This asset, different from the basic assets of the market, gets its random component from the randomness of the market, but with specific volatility, hence specific risk.

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We consider an expression similar to (1) , namely µdt = rdt − cov(dP(t) P(t) , dZ(t) Z(t) |Ft) (2) which is called the risk-adjusted expected rate of return of P

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BASIC ASSUMPTION

We assume that µ(t) − α(t) = δ(t) ≥ 0 (3) We shall then say that the asset is tradable. The quantity δ(t) will be interpreted as a dividend associated to the asset P(t). The logic is that we can build a risk-free portfolio made of the asset P(t) and a short position on the basic assets of the market. Its value is P(t) −

i πi(t)Yi(t).

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For this portfolio to be risk-free, we must choose πi(t) such that P(t)σj(t) =

• i

πi(t)Yi(t)σij(t) which defines uniquely πi(t), since the matrix σ is

• invertible. We then claim that its expected return at

any time per unit of time should be equal to the risk-free return.

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Recalling that P(t) carries a dividend δ(t) we then have P(t)(α(t) + δ(t)) −

• i

πi(t)αi(t)Yi(t) = r(P(t) −

• i

πi(t)Yi(t)) Using (1) and recalling the definition of µ(t) (2), we

• btain the relation (3).

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COUPONS

In the case the basic assets Yi(t) carry a coupon δi(t) per unit value and unit of time, then one must replace r with r +

ij((σ∗)−1)ijσj(t)δi(t). It is then useful to

consider the case when P(t) is simply one of the basic assets Yk(t). Formula (3) reduces to the relation defining θk.

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VALUE OF A PROJECT

We consider now that P(t) represents the output flow

• f a project, or the revenue of a company. We will

assume α,σij deterministic constants. We also consider that P carries a dividend δ per unit-value and unit of time also constant to simplify. We have the relation α + δ = µ where µ, the risk-adjusted expected rate of return of P is given by (2).

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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). We consider the value of ownership of the project as a tradable asset and write its differential dV = (∂V ∂t + ∂V ∂P Pα + 1 2 ∂2V ∂P 2P 2σ2)dt+ ∂V ∂P P

• j

σjdwj σ2 =

• j

σ2

j

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The risk-adjusted rate of return of the ownership of the project is then ˜ µ = r+∂V ∂P P V [

• ij

((σ∗)−1)ijσjδi−

• j

cov(σjdwj, dZ Z )] Therefore V ˜ µ − rV = ∂V ∂P P(µ − r) We then write that the expected capital return on the

• wnership of the project plus the profit flow per unit

value is equal to ˜ µ.

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This means V ˜ µ = π + ∂V ∂t + ∂V ∂P Pα + 1 2 ∂2V ∂P 2P 2σ2 We see that α can be eliminated. There remains ∂V ∂t + ∂V ∂P P(r − δ) + 1 2 ∂2V ∂P 2P 2σ2 − rV + π = 0 (4)

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SOLUTION OF THE VALUATION EQUATION

Equation (4) is a partial differential equation whose space variable P lies in (0, ∞). We need boundary

• conditions. As far time is concerned, we will in most

cases look for stationary solutions of equation (4) namely ∂V ∂P P(r − δ) + 1 2 ∂2V ∂P 2P 2σ2 − rV + π = 0 (5) which is possible when coefficients are independent

• f time ( as we assumed), the profit flow is also

independent of time and the horizon is infinite. For finite horizon, V (P, T) = 0 whose interpretation is clear.

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Concerning the variable P we need conditions at 0 and ∞. For P = 0 we see formally on equation (5) 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. 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

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β is a solution of 1 2σ2β(β − 1) + (r − δ)β − r = 0 The roots of this quadratic expression are β1, β2 with β1 = 1 2 − r − δ σ2 +

• [1

2 − r − δ σ2 ]2 + 2r σ2 β2 = 1 2 − r − δ σ2 −

• [1

2 − r − δ σ2 ]2 + 2r σ2 and β1 > 1, β2 < 0

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PARTICULAR SOLUTIONS

If π(P) = P then V (P) = P δ . If π(P) = (P − C)+ then one has V (P) =

• V1P β1 if P ≤ C

V2P β2 + P δ − C r if P ≥ C

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where V1 = C1−β1 β1 − β2 (β2 r − β2 − 1 δ ) V2 = C1−β2 β1 − β2 (β1 r − β1 − 1 δ )

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PROBABILISTIC INTERPRETA- TION

We can interpret V (P, t) as follows.The output evolution is described by dP = P((r −δ)ds+

• j

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

t

exp −r(s − t)π(P(s), s)ds Note that the discount applicable is the risk-less discount r.

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VALUATION OF AN OPTION TO IN- VEST

Consider a stationary situation.The problem we face now is that of investing in the project an amount I. In

• ther words, we pay a price I to get a a project of

value V (P). Under NPV ( Net present value) approach we will invest if P ≥ P0 where I = V (P0) 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.

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THE OPTION APPROACH

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 we 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 one of the branches of the alternative is to invest immediately, we must have F(P) ≥ V (P) − I, ∀P

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The other branch is to keep the option. To proceed, we must use valuation concepts. We can form a portfolio made of the option and a short position in the output itself P. A portfolio F(P(t)) − π(t)P(t) will be riskless if π(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.

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The expected increase in capital is F ′(P)Pα + 1 2F”(P)σ2 − F ′(P)P(α + δ) Since it is risk-free it cannot be larger than r(F − PF ′(P)). We get the inequality F ′(P)P(r − δ) + 1 2F”(P)σ2 − rF ≤ 0 Since the alternative has only two branches, at any time one of the inequality must become an equality

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VARIATIONAL INEQUALITY

To simplify the notation define the differential

• perator on smooth functions φ(P) by

Aφ(P) = φ′(P)P(r − δ) + 1 2σ2P 2φ”(P) − rφ(P) then F(P) is the solution of the following problem. F(P) ≥ V (P) − I AF(P) ≤ 0 (F(P) − V (P) + I)AF(P) = 0 (6)

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This problem is called a Variational Inequality (V.I.). 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.

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PROBABILISTIC INTERPRETA- TION

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 dP = P((r − δ)ds +

• j

σjdwj(s)); P(0) = P Consider now stopping times τ adapted to the σ-field generated by the process P(t) then we have F(P) = max

τ

E[(V (P(τ)) − I) exp −rτ] (7)

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THRESHOLD

To solve the V.I. we look for a value P ∗ such that AF = 0, P < P ∗ F(P) = V (P) − I, P ≥ P ∗ F ′(P ∗) = V ′(P ∗) Define Γ(P) = F(P) − V (P) + I

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Then we have Γ(P) = 0, P ≥ P ∗ AΓ = π(P) − rI, P < P ∗ Γ′(P ∗) = 0 (8)

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THEOREM 1. Assume π(P) increasing and that a solution of the system (8) exists. Then F(P) = Γ(P) + V (P) − I is solution of the V.I. (6).

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PARABOLIC VARIATIONAL IN- EQUALITY

We consider now the non-stationary case: The problem (6) becomes F(P, t) ≥ V (P, t) − I ∂F ∂t + AF ≤ 0 (F(P, t) − V (P, t) + I)(∂F ∂t + AF) = 0 F(P, T) = 0 (9) This problem is called a parabolic variational inequality.

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FREE BOUNDARY

We are interested in the same type of solution, namely find a free boundary P ∗(t) with F satisfying ∂F ∂t + AF = 0, P < P ∗(t) F(P, t) = V (P, t) − I, P ≥ P ∗(t) ∂F ∂P (P ∗(t), t) = 0, F(P, T) = 0 (10)

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THEOREM 2. Assume ∂π

∂t ≤ 0, ∂π ∂P ≥ 0, and

π(0, t) = 0. Then there exists a continuous differentiable function F solution of the parabolic variational inequality (9) which is of the form (10) We also have the property dP ∗(t) dt > 0 (11)

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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) (12) where P(t) is the value of the output at time t representing the present time (more precisely the time

• f decision). On the other hand the traditional NPV

(Net Present Value) 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.

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We may define ˆ P(t) such that V ( ˆ P(t), t) = I The NPV rule is equivalent to we invest in the project if P(t) ≥ ˆ P(t) (13) We have the following result THEOREM 3. Under the assumptions of Theorem 2 we have ˆ P(t) < P ∗(t)

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So the NPV rule is always wrong. One invests much too early with this rule. The quantity F(P ∗(t), t) = V (P ∗(t), t) − I > 0 represents the value of flexibility in the decision of investment at time t.

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UNCERTAINTIES ON INVEST- MENT

We consider here a model in which not only the value

• f the output flow is random but also the investment

needed can vary with time. More specifically we have dP = P(αPdt +

• j

σPjdwj) and a similar relation for the investment dI = I(αIdt +

• j

σIjdwj)

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Considering 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.

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We will write σ2

P =

• j

σ2

Pj, σ2 I =

• j

σ2

Ij

and ρσPσI =

• j

σPjσIj

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We suppose to simplify that the profit flow of the project is given by π(P) = P We will then consider a stationary model. We first note that the value of the project does not depend on the investment and therefore in view of the profit flow we have V (P) = P δP

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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 V.I.

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∂F ∂P P(r − δP) + ∂F ∂I I(r − δI) + 1 2 ∂2F ∂P 2P 2σ2

P+

1 2 ∂2F ∂I2 I2σ2

I + ∂2F

∂P∂I PIρσPσI − rF ≤ 0 F − P δP + I ≥ 0 product = 0 (14) where "product " means the product of the two quantities at the left-hand side of the inequalities.

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SCALING

It is convenient to introduce Γ(P, I) = F(P, I) − P δP + I which can be expressed by Γ(P, I) = Iz(P I ) and z(x) is the solution of the V.I.

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∂z ∂xx(δI − δP) + 1 2 ∂2z ∂x2x2(σ2

P + σ2 I − 2ρσPσI) − zδI

−x + δI ≤ 0 z ≥ 0 product = 0 (15)

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SOLUTION

This V.I. can be easily solved. Considering the quadratic form β(δI − δP) + 1 2β(β − 1)(σ2

P + σ2 I − 2ρσPσI) − δI = 0

and its root β1 > 1, we deduce a threshold value x∗ = β1 β1 − 1δP

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and the solution is given by z = f1xβ1 − x δP + 1, x ≤ x∗ and z = 0, x ≥ x∗ The constant f1 is such that there is continuity for x = x∗.

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UNCERTAINTIES DUE TO INCEN- TIVES

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).

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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 − λ0dt Prob(η(t + dt) = 0|η(t) = 0) = 1 − λ1dt

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The output flow is governed by dP = P(αdt +

• j

σjdwj) 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

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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

• nly two values, it is convenient to write

F(P, 0) = F 0(P), F(P, 1) = F 1(P)

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SYSTEM OF V.I.

By the standard Dynamic Programming arguments

• ne obtains easily the system

AF 0(P) + λ1(F 1(P) − F 0(P)) ≤ 0 F 0(P) − P δ + I ≥ 0 product = 0 (16) AF 1(P) + λ0(F 0(P) − F 1(P)) ≤ 0 F 1(P) − P δ + I(1 − θ) ≥ 0 product = 0 (17)

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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

• ne will not invest even when the incentive is in place.

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

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AF 0 + λ1(F 1 − F 0) = 0 F 1(P) = P δ − I(1 − θ) ∀P 1 < P < P 0 and finally F 0(P) = P δ − I F 1(P) = P δ − I(1 − θ) ∀P 0 < P

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THE OPTION OF ABANDONMENT

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

• f temporary suspension whenever P < C and

resuming the activity when P > C with no penalty in both cases. That 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.

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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 (18)

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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 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 (19)

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TWO-SIDED VARIATIONAL IN- EQUALITY

Note first the compatibility condition I + E ≥ 0 which is obvious whenever the two quantities I, 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 a realistic situation. We must have I + E > 0 to avoid situations in which one could get cash but continuously investing and disinvesting.

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The nice feature is that the function Γ(P) = F(P) − V (P) + I is still the solution of a single problem ( no coupling). However it is a two-sided variational inequality and not a one-side variational inequality.

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The problem is expressed as follows 0 ≤ Γ(P) ≤ I + E if 0 < Γ(P) < I + E, then AΓ(P) = P − C − rI if Γ(P) = 0, then AΓ(P) ≤ P − C − rI if Γ(P) = I + E then AΓ(P) ≥ P − C − rI (20) Since AC = −rC the two last conditions reduce to Γ(P) = 0 ⇒ P − C − rI > 0 Γ(P) = I + E ⇒ P − C + rE < 0

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SOLUTION OF THE TWO-SIDED V.I.

The two last conditions guide the intuition. For a sufficiently low output P not only one should not invest, but one should cut the investment. For a sufficiently large output, not only one should continue the project, but one should invest if the project has not yet started. In between, one should continue a project already started but one should not invest. Therefore

• ne looks for two thresholds 0 < PL < PH such that

AΓ(P) = P − C − rI, ∀PL < P < PH and Γ(P) = 0, ∀P > PH Γ(P) = I + E, ∀P < PL

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Clearly we must have PH ≥ C + rI, PL ≤ C − rE We will have in the interval (PL, PH) Γ(P) = Γ1P β1 + Γ2P β2 − P δ + C r + I and we have four constants to obtain, Γ1, Γ2 and PH,PL. There are also four conditions to express the continuity of Γ(P) and its derivative at points PL, PH.

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We obtain the following system Γ1P β1

L + Γ2P β2 L − PL

δ + C r = E Γ1P β1

H + Γ2P β2 H − PH

δ + C r + I = 0 Γ1β1P β1−1

L

+ Γ2β2P β2−1

L

− 1 δ = 0 Γ1β1P β1−1

H

+ Γ2β2P β2−1

H

− 1 δ = 0

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We derive the following system for PH PL

1 δ(1 − β2) + β2 PL(C r − E)

P β1−1

L

=

1 δ(1 − β2) + β2 PH (C r + I)

P β1−1

H 1 δ(β1 − 1) − β1 PL(C r − E)

P β2−1

L

=

1 δ(β1 − 1) + β2 PH (C r + I)

P β1−1

H

It can be shown that this system has a unique solution with PL < PH.

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 69/81

THE OPTION OF MOTHBALLING

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

• r 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 mothballing incurs a fixed cost EM. To reactive from mothballing implies a fixed cost ER.

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 70/81

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 V.I. for V we will need a new

function H(P) which represents the value of the

• ption of mothballing.So in fact we will have a

coupled system for three functions F, V, H.

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 71/81

VARIATIONAL INEQUALITIES

We consider the stationary case. The value of the

• ption to invest is governed by the V.I.

AF(P) ≤ 0 F(P) − V (P) + I ≥ 0 AF(P)(F(P) − V (P) + I) = 0 (21)

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 72/81

The problem of which V (P) is a solution is now given by AV (P) + P − C ≤ 0 V (P) ≥ H(P) − EM (AV (P) + P − C)(V (P) − H(P) + EM) = 0 (22) In (22) the function H(P) represents the value of mothballing.

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 73/81

It is itself governed by the following V.I. AH(P) − M ≤ 0 H(P) ≥ max(V (P) − ER, F − ES) (AH(P) − M)[H(P) − max(V (P) − ER, F − ES)] = 0 (23) 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.

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 74/81

SOLUTION OF THE V.I

To define the solution we need four thresholds 0 < PS < PM < PR < PH 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

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 75/81

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, PH correspond to the thresholds PL, PH defined when there was no option

• f mothballing.

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 76/81

We can next define completely the three functions by solving the differential equations in the respective

• intervals. We can state

F(P) = F1P β1 for P < PH F(P) = V (P) − I for P > PH

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 77/81

V (P) = V2P β2 + P δ − C r for P > PM V (P) = H(P) − EM for P < PM H(P) = H1P β1 + H2P β2 − M r for PS < P < PR H(P) = F(P) − ES for P < PS H(P) = V (P) − ER for P > PR

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 78/81

The solution depends on 8 constants, the four thresholds and the values of constants F1,G2,H1,H2. We write 8 matching conditions (2 per threshold). There is some decoupling. The values H1,G2 − H2,PR,PM are solutions of the system

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 79/81

−H1P β1

R + (G2 − H2)P β2 R + PR

δ − C − M r − ER = 0 −H1β1P β1−1

R

+ (G2 − H2)β2P β2−1

R

+ 1 δ = 0 −H1P β1

M + (G2 − H2)P β2 M + PM

δ − C − M r + EM = 0 −H1β1P β1−1

M

+ (G2 − H2)β2P β2−1

M

+ 1 δ = 0

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 80/81

Similarly, we define F1 − H1,H2,PS, PH by the system (F1 − H1)P β1

S − H2P β2 S + M

r − ES = 0 (F1 − H1)β1P β1−1

S

− H2β2P β2−1

S

= 0 F1P β1

H − G2P β2 H − PH

δ + C r + I = 0 F1β1P β1−1

H

− G2β2P β2−1

H

− 1 δ = 0

In Honor to Professor Wolfgang Runggaldier, July 17 2007 – p. 81/81