Real Options with Competition and Incomplete Markets

Transcription

1 Real Options with Competition and Incomplete Markets Alain Bensoussan and SingRu Celine) Hoe Abstract Ever since the first attempts to model capital investment decisions as options, financial economists have sought more accurate, more realistic real options models. Strategic interactions and market incompleteness are significant challenges that may render existing classical models inadequate to the task of managing the firm s capital investments. The purpose of this paper is to address these challenges. The issue of incompleteness comes in for the valuation of payoffs due to absence of a unique martingale measure. One approach is to valuate assets by considering a rational utility-maximizing consumer/investor s joint decisions with respect to portfolio investment strategy and consumption rule. In our situation, we add the stopping time as an additional decision. We employ variational inequalities V.I.s) to solve the optimal stopping problems corresponding to times to invest. The regularity of the obstacle payoffs received at the decision time) is a major element for defining the optimal strategy. Due to the lack of smoothness of the obstacle raised by the game problem, the optimal strategy is a two-interval solution, characterized by three thresholds. Keywords Stackelberg leader-follower game Utility maximization Bellman equation Optimal stopping Mathematics Subject Classification 00) 9G80 9A30 9A5 A. Bensoussan B) International Center for Decision and Risk Analysis, School of Management, University of Texas at Dallas, 800 West Cambell Rd, SM30, Richardson, TX , USA axb04600@utdallas.edu A. Bensoussan City University of Hong Kong, Hong Kong, China S.C.) Hoe Texas A&M University-Commerce, Commerce, TX 7549, USA hoceline0@yahoo.com Y. Kabanov et al. eds.), Inspired by Finance, DOI 0.007/ _, Springer International Publishing Switzerland 04 9

2 30 A. Bensoussan and S.C.) Hoe Investment Game Problems and General Model Assumptions We consider a Stackelberg leader-follower game for exploiting an irreversible investment opportunity with payoffs of a continuous stochastic income stream Yt) for a fixed cost K. We limit the flexibility in the investment decisions to the times when to invest. The roles of leader and follower are predetermined by regulations. Each firm chooses its individual stopping time to invest over an infinite horizon with the constraint that the follower be forbidden to undertake the investment until the leader has already done so. By investing K, the leader receives δ Yt)per unit time till the follower s entry. Once both have entered, each gets a continuous cashflow stream δ Yt)per unit time, with δ <δ. Consider a probability space Ω, F, Q) with W t) = W t), W 0 t)) T a standard Wiener process. The asset S representing the market and the cashflow process Y evolve as follows: dst) = rst)dt + σst)λdt + dwt)), ) dyt) = Yt) αdt + ς ρdwt) + ρ dw 0 t)) ), ) where Wt) and W 0 t) are independent Wiener processes, ρ < is the correlation coefficient between market uncertainty and the cashflow process uncertainty, and r risk-free rate), σ,λ,α,ς are all constants. The market is incomplete since the market asset S can span only the portion of the stochastic cashflow risk driven by the Wiener process Wt), leaving the remaining risk driven by W 0 t) unhedgeable. There is no unique martingale measure, so the risk-neutral pricing is no longer appropriate, and an alternative must be developed in this framework. We adopt utility-based pricing in which a risk averse investor/firm maximizes the expected utility of consumption. We assume that the investor s risk preferences are characterized by a constant absolute risk aversion utility function UC)= γ e γc 3) where the argument C is the investor s consumption, and γ is his/her risk aversion parameter, γ>0. Remark We allow for negative consumption. For C R, U increases from to 0. As C, it leads to huge negative values. We interpret this effect as a penalty to the utility maximization investor. We could of course impose the constraint of non-negative consumption. However, imposing non-negativity on the consumption would rule out the analytical solutions for further developments, a property we would like to retain for the full analysis. Therefore, we choose to accept for negative consumption which could lead huge negative utility values big penalties for our utility maximization investor) instead of imposing the non-negativity constraint on the consumption. We also note that the negative consumption occurs when x becomes very negative and we cannot avoid this situation since x R. Each firm maximizes its expected discounted utility from consumption over an infinite horizon, subject to choice over investment timing, consumption, hedge po-

3 Real Options with Competition and Incomplete Markets 3 sition in the market asset, and allocation in the riskless bond. Thus, each firm considers undertaking the investment as an additional decision besides portfolio investment and consumption decisions. The decision remains a stopping time, for which the right approach is that of variational inequality V.I.) [, 5]. Our duopoly game requires us to solve two V.I.s corresponding to the leader s and the follower s optimal stopping respectively. As such, we will need two obstacles corresponding to each V.I. We obtain the obstacles from solving continuous control problems, i.e., portfolio investment and consumption decisions, and we call this as solutions to postinvestment utility maximization. Employing the obstacles obtained, we then form V.I.s to solve the optimal stopping problems, and we call this as solutions to preinvestment utility maximization. One point to note is that we need to consider an auxiliary problem of which the cashflow process ) hits zero; the problem will then be reduced to classical investment-consumption portfolio decisions. We next summarize the general notations used in the paper to facilitate reading: τ for the follower s stopping time and θ for the leader s stopping time; F x, y) for the follower s obstacle, i.e., solution to follower s postinvestment utility maximization, and Fx,y) for the follower s solution to the V.I., i.e., solution to the follower s preinvestment utility maximization; L x, y) for the leader s obstacle, i.e., solution to leader s postinvestment utility maximization, and Lx,y) for the leader s solution to the V.I., i.e., solution to the leader s preinvestment utility maximization; Fx) for the solution to the classical investment-consumption utility maximization, i.e., no augmented stochastic income stream Yt). We detail follower s problem and solution in Sect. and the leader s in Sect. 3. We conclude in Sect. 4. We omit most of the proofs except the main result. Follower s Problem and Solution We start with the follower s investment problem. Given the initial wealth, x, the follower optimizes his portfolio by dynamically choosing allocations in the market asset S, the riskless bond, and the consumption rate, C. The follower s wealth, X, evolves as follows: dxt) = πt)xt)σλdt + dw t)) + rxt)dt Ct)dt, t < τ, Xτ) = Xτ 0) K, dxt) = πt)xt)σλdt + dw t)) + rxt)dt Ct)dt + δ Yt)dt, αdt + ς ρdwt) + ρ dw 0 t) )), t >τ, dyt) = Yt) X0) = x, Y0) = y, 4) where πt) is the proportion of wealth invested in asset S, Ct) is the consumption rate, and τ is the stopping time to undertake the investment, chosen optimally by the

4 3 A. Bensoussan and S.C.) Hoe follower. The wealth process is discontinuous at τ.from4), we observe that the wealth process has two possible evolution regimes. To facilitate further exposition, we introduce the processes X 0 and X regime 0 and regime, respectively): dx 0 t) = πt)x 0 t)σ λdt + dw t)) + rx 0 t)dt Ct)dt, 5) dx t) = πt)x t)σ λdt + dw t)) + rx t)dt Ct)dt + δ Yt)dt. 6) The follower s problem is to maximize his expected discounted utility from consumption by choosing stopping time τ, consumption rate C, and investment strategy π. We have to solve the problem in two steps, beginning with the utility maximization after τ postinvestment utility maximization) and then solving the complete utility maximization prior to τ preinvestment utility maximization). The rationale behind this two-step procedure is because we need a clearly defined obstacle function when solving the stopping time problem.. Postinvestment Utility Maximization After τ, the follower solves his utility maximization as a control problem of portfolio selections and consumption rules augmented by a stochastic cashflow stream δ Yt)per unit time. To facilitate representation, for F t -adapted processes πt), Ct), we introduce the local integrability conditions and define I i = { E T 0 πt)x i t) ) dt <, T, E T 0 Ct)) dt <, T, τ i N = inf{t : Xi t) < N}, i = 0,. 8) The follower reveals his preference through his expected discounted utility of consumption, and so, to the pair C ), π )), we introduce the objective function J C ) ) = E 0 7) e μt U Ct) ) dt, 9) where μ, a constant, is the discount rate. This function is well-defined, but it may take the value. Since the follower can manage his investment-consumption portfolio, we consider the following control problem: F x, y) = sup {π ),C )} U x,y J C ) ), 0)

5 Real Options with Competition and Incomplete Markets 33 where U x,y {π, = C) : I ; τ as N N ;e μt Ee rγ X T )+fyt)) ) 0, } as T, and fy)is a positive function of linear growth with f0) = 0 which will be made precise later cf. 7), 6)). We associate the value function F x, y) with the Bellman equation: μf F F rx + δy) + y αy + F } y y ς + sup {UC) C F + sup π { πxσ λ F + F yςρ y C ) + F π x σ } = 0. ) The domain is x R, y>0. We note that if y = 0, then Yt)= 0 for all t. The problem reduces to the classical investment-consumption problem with the solution given by: Fx)= { rγx rγ exp + μ + } λ. ) r We thus have: F x, 0) = Fx). 3) We look for a solution of 0) in the form F x, y) = { rγx rγ exp + fy))+ μ + } λ 4) r in which, by 3), f0) = 0. 5) By 4), defining the optimal feedback, and Ĉx,y) = γ ln F πx,y) = λ F + yςρ F y, F σx we reduce the Bellman equation ) to: y ς f + α λςρ)yf rγy ς ρ )f rf + δ y = 0. 6)

6 34 A. Bensoussan and S.C.) Hoe Proposition The value function, fy)= inf E {v ) U y } 0 e δ rt Y y t) + ) v t) dt 7) with dy y t) = Y y t) [ α λςρ + ς rγ ρ )vt) ] dt + Y y t)ςdw t), Y y 0) = y, U y ={v ) : E 0 e rt v t)dt <, e rt EY y T ) 0 as T }, 8) is the unique function in C 0, ) solving 6), 5) on the interval [0,εy+ M ε ], and such that fy) as y. Proposition The function f y) is bounded. We now state the result that the value function given by 0) is indeed of the form 4). Theorem The function F x, y) given by 4) coincides with the value function given by 0).. Preinvestment Utility Maximization We now turn to the problem of optimal stopping with the obstacle defined by F x, y), the solution to the postinvestment utility maximization. Before the stopping time τ, the wealth process is governed by 5) and the cashflow process evolves as ). Set θ 0 = inf{t : Yt)= 0}. At time τ θ 0 the follower stops. If θ 0 τ,the investment never takes place and the follower receives FX 0 θ 0 )), where Fx) is given by ). If τ<θ 0, the follower receives F X 0 τ) K,Yτ)) at the stopping time τ, where F x, y) is given by 4). Therefore, the objective function is: ) J x,y C ), π ), τ [ τ θ 0 = E e μt U Ct) ) dt + F X 0 τ) K,Yτ) ) e μτ τ<θ F X 0 θ 0 ) ) ] e μθ 0 θ 0 τ, 9) δε r + α λςρ ) M ε is defined as: M ε = ς r γ ρ.ifr + λςρ α>0, we can take M ε = 0, hence ) ε =. Note that ε can be arbitrarily small. δ r α+λςρ

7 Real Options with Competition and Incomplete Markets 35 and we define the associated value function: ) Fx,y) = sup J x,y C ), π ), τ, 0) [ + sup π πxσ λ F + yςρ F y Fx,y) F x K,y), {π ),C ),τ} U 0 xy where } Ux,y {C,π,τ): 0 = I 0 ; τ θ 0 < a.s.; τ = lim τ 0 τ θ 0 N a.s.. As a consequence of Dynamic Programming, assuming sufficient smoothness of the function Fx,y), we may write the strong formulation of V.I. that Fx,y) must satisfy as follows: μf + F F rx + y αy + F ς y ) + sup y C UC) C F ) + π x σ ] F 0, Fx,y) F x K,y) )[ μf + F F rx + y αy + F y ς y + sup C UC) C F ) + supπ [ πxσ λ F + yςρ F y ) + π x σ F ]] = 0. ) We have the boundary condition: Fx,0) = Fx). ) We look for a solution of the form: Fx,y) = [ rγ rγ exp x + gy) ) + μ + ] λ. 3) r Using 3) and 4) and defining the optimal feedback and Ĉx,y) = γ ln F πx,y) = λ F + yςρ F y, F σx we transform V.I. ) to the form: y ς g + g yα λςρ) y ς rγ ρ )g rg 0, gy) fy) K, ) [ gy) fy)+ K y ς g + g yα λςρ) y ς rγ ρ )g rg] = 0, g0) = 0. 4)

8 36 A. Bensoussan and S.C.) Hoe This V.I. cannot be interpreted as a control problem because the non-linear operator is connected to a minimization problem, while the inequalities are connected to a maximization problem. So, gy) is more appropriately the value function of a differential game rather than of a control problem. Define uy) = gy) fy)+ K. Then 4) becomes using the equation of fy)cf.6)): y ς u yu α λςρ yf ς rγ ρ ) ) + y ς rγ ρ )u + ru δ y + rk, u 0, u [ y ς u yu α λςρ yf ς rγ ρ ) ) + y ς rγ ρ )u + ru+ δ y rk ] = 0, u0) = K. 5) We study 5) by the threshold approach. Let ŷ be fixed, to be determined below. We consider the Dirichlet problem y ς u yu α λςρ yf ς rγ ρ ) ) + y ς rγ ρ )u + ru = δ y + rk, 0 <y<ŷ, u0) = K, uŷ) = 0. 6) For ŷ fixed, this problem is a classical Bellman equation. Similar to Proposition, equation 6) is a Bellman equation of the following control problem with the controlled diffusion: dy y t) = Y y t) [ α λςρ Y y t)f Y y t))ς rγ ρ ) + ς rγ ρ )vt) ] dt + Y y t)ςdw t), 7) Y y 0) = y, 0 <y<ŷ, and the value function [ ] θy v )) uy) = inf E e [ δ rt Y y t) + rk + v ) v t) dt 0 ] + e rθyv )) K Yyθ y v )))=0, 8) where θ y v )) = inf{t : Y y t) is outside 0, ŷ)} and it is finite a.s.). Obviously, uy) > 0ifŷ< Kr δ, and we also have uy) K. For vt), we can take the same class as in problem 7) 8).

9 Real Options with Competition and Incomplete Markets 37 Theorem There exists a unique value ŷ such that u ŷ) = 0, ŷ Kr δ. 9) The value function uy) cf. 8)) extended by zero beyond ŷ is the unique solution of V.I. 5). It is C and piecewise C. Referring back to 4), from Theorem, we have obtained that there exists a unique solution of 4) such that gy) C and piecewise C. There exists a unique ŷ such that y ς g g yα λςρ) + y ς rγ ρ )g + rg = 0, y <ŷ, gy) = fy) K, y ŷ, g ŷ) = f 30) ŷ), g0) = 0. Note that gy) 0 since uy) fy)+ K. We generate the main result that the value function given by 0) is indeed of the form 3). Theorem 3 The function Fx,y) defined by 3) coincides with the value function given by 0)..3 Follower s Optimal Stopping Rule We next define the optimal stopping rule as: ˆτy)= inf{t : Y y t) ŷ}, 3) where Y y t) is the process defined in ) and ŷ is the unique value defined by the V.I. 9) the smooth matching point). We must note that the follower s stopping time ˆτy) is the follower s optimal entry if he can enter in the market at time zero. Since the follower can enter only after the leader who starts at time θ), for finite θ, the follower will enter at time: 3 ˆτ θ = θ +ˆτ Y y θ) ). 33) 3 For any test function Ψx,s),wehavetheformula: E [ ΨY y ˆτ θ ), ˆτ θ ) F θ ] = ΨYy θ), θ) Yy θ) ŷ + Yy θ)<ŷe [ Ψŷ,t +ˆτy)) ] y=yy θ),t=θ. 3)

10 38 A. Bensoussan and S.C.) Hoe 3 Leader s Problem and Solution After solving the follower s optimal policy, we are now ready to solve the leader s problem, which is complicated by the fact that he must share the market project value) upon the follower s optimal entry at ˆτ θ. Thus, by investing K, the leader expects to receive a continuous cash flow δ Yt)per unit time prior to the follower s entry, and δ Yt)per unit time afterwards. The leader s wealth evolves according to the following system of stochastic equations: dxt) = πt)xt)σ λdt + dwt) ) + rxt)dt Ct)dt, t < θ, Xθ) = Xθ 0) K, dxt) = πt)xt)σ λdt + dwt) ) + rxt)dt + δ Yt)dt Ct)dt, θ<t<ˆτ θ, dxt) = πt)xt)σ λdt + dwt) ) + rxt)dt + δ Yt)dt Ct)dt, t > ˆτ θ, dyt) = Yt) αdt + ς ρdwt) + ρ dw 0 t) )), X0) = x, Y0) = y, with x R, y 0, 34) where θ and ˆτ θ are stopping times chosen optimally by the leader and the follower, respectively. The leader s problem is to maximize his expected discounted utility from consumption by choosing stopping time θ, consumption rate C, and investment strategy π. As in the follower s case, we have to solve leader s complete utility maximization problem in two steps. 3. Postinvestment Utility Maximization Suppose that θ = 0, the leader s wealth is x, and the cash flow y>0; then the leader s wealth becomes immediately x K since he must pay the fixed cost of entry, K. The leader must share the market upon follower s entry at ˆτy). Thus, for a generic initial wealth x, the leader s wealth evolves as follows: dx L t) = πt)x L t)σ λdt + dwt) ) + rx L t) + δ Yt) Ct) ) dt, t<ˆτy), X L 0) = x, dx t) = πt)x t)σ λdt + dwt) ) + rx t)dt + δ Yt)dt Ct)dt, t>ˆτy), X ˆτy) ) = X L ˆτy) ). 35)

11 Real Options with Competition and Incomplete Markets 39 If θ = 0 and y ŷ, the follower enters immediately, and the leader s problem is identical to the follower s, i.e., 0). So, we consider the function L x, y) = ) rγ e rγ x+fy) + μ+ λ r, 36) where f is the solution of 6), 5) on the interval [0,εy+ M ε ]. 4 If θ = 0 and y<ŷ, the leader s problem is described as follows. The wealth process is described by X L in 35) and the cash flow process follows ). Recall that θ 0 = inf{t : Y y t) = 0} and ˆτy) = inf{t : Y y t) ŷ}. 5 If θ 0 < ˆτy), the follower never invests, and the leader s value function at time θ 0 corresponds to FX L θ 0 )) cf. )). If ˆτy) < θ 0, the leader s value function corresponds to L X L ˆτy)), Y ˆτ y))), at the follower s entry time, ˆτy). Thus, to a pair of C ), π )), we associate the objective function J C ), π ) ) [ ˆτy) θ 0 = E e μt UCt))dt + F X L θ 0 ) ) e μθ 0 θ 0 ˆτy) 0 + L X L ˆτy) ),Y ˆτy) )) ] e μ ˆτy) ˆτy)<θ 0, 37) and we consider the value function: L x, y) = sup {π ),C )} U x,y J C ), π ) ), 38) where Ux,y ={π, C) : I ; τ = lim τ ˆτy) θ 0 N a.s.} with I and τ defined in 7) and 8) by replacing N X with X L respectively. We associate the value function with the Bellman equation: μl + L rx + δ y) + L + sup π [ πxσ λ L + ςρy L y L x, ŷ) = L x, ŷ). y αy + L ς y + sup y C UC) C L ) ) + π x σ L ] = 0, We study the Bellman equation 39) fory ]0, ŷ[ and we define: L x, y) = L x, y), if y>ŷ, where L x, y) is defined in 36). The extension is continuous but not C.Also, we note that for y = 0, then Yt) = 0 for all t, the problem then reduces to the 39) 4 See footnote for the definition of M ε. 5 Here Y y t) is the process defined in ).

12 40 A. Bensoussan and S.C.) Hoe classical investment-consumption portfolio optimization problem; thus we have the boundary condition: L x, 0) = Fx). 40) We look for a solution of the form L x, y) = ) rγ e rγ x+qy) + μ+ λ r 4) with q solving the problem y ς q + α λςρ)yq r γ y ς ρ )q rq + δ y = 0, 0 <y<ŷ, q0) = 0, qŷ) = fŷ). 4) where fy)is the solution of 6), 5) on the interval [0,εy+M ε ]. We extend qy) by fy)for y>ŷ. The function L x, y) is continuous but not C. The study of 4) is similar to 6), but it is simpler because it is defined on a bounded interval. Similar to the study of 6), we can show that qy) may be interpreted as a function of a control problem, and there exists a unique solution which is C 0, ŷ).forδ >δ,wehave: qy) fy). 43) Theorem 4 The function L x, y) defined by 4) coincides with the value function given in 38). 3.. The Leader s Pre-investment Utility Maximization We now turn to the leader s optimal stopping problem i.e., choice of θ) with obstacle defined by L x, y), the solution to the postinvestment utility maximization. Before the stopping time θ, the leader s wealth and the cashflow process evolve as 5) and ) respectively. At time θ θ 0, the leader stops. If θ 0 θ, the leader never takes the investment and receives FX 0 θ 0 )) cf. )). If θ<θ 0, the leader receives L X 0 θ) K,Yθ)) cf. 4)) at θ. Therefore, the objective function is [ ) θ θ 0 J x,y C ), π ), θ = E U Ct) ) e μt dt + L X 0 θ) K,Yθ) ) e μθ θ<θ F X 0 θ 0 ) ) ] e μθ 0 θ 0 θ. 44)

13 Real Options with Competition and Incomplete Markets 4 We define the value function where Lx,y) = sup {π ),C ),θ} U 0 x,y J x,y C ), π ), θ ), 45) U 0 x,y ={C,π,θ): I 0 ; θ θ 0 < a.s.; τ = lim τ 0 N θ θ 0 a.s.} with I 0 and τ 0 N defined in 7) and 8) respectively. As a consequence of Dynamic Programming, assuming sufficient smoothness of L, we can associate the strong formulation of V.I. to the value function Lx,y) as: μl + rx L L + αy y + ς y ) L + sup y c UC) C L [ ) + sup π πσx λ L + ςρy L y + π x σ ] L 0, Lx,y) L x K,y), Lx,y) L x K,y) )[ μl + rx L L + αy y + ς y L + sup C UC) C L ) + supπ [ πσx λ L + ςρy L y We have the boundary condition: We look for a solution of the form y ) + π x σ L ]] = 0. 46) Lx, 0) = Fx). 47) Lx,y) = ) rγ e rγ x+hy) + μ+ λ r, 48) with hy) satisfying the following V.I.: y ς h + h yα λςρ) y ς rγ ρ )h rh 0, hy) qy) K, )[ hy) qy)+ K y ς h + h yα λςρ) y ς rγ ρ )h rh ] = 0, h0) = 0. 49) We encounter a new difficulty that does not occur in the follower s problem. We observe that the leader s obstacle qy) K is C 0 but not C. We cannot as in 5) consider uy) = hy) qy) + K since qy) is not sufficiently smooth. We will consider nonetheless the function uy) = hy) fy)+ K

14 4 A. Bensoussan and S.C.) Hoe which satisfies the following problem: y ς u y α λςρ yς rγ ρ )f ) u + ς rγy ρ )u + ru δ y + rk, u m, u m) [ y ς u y α λςρ yς rγ ρ )f ) u + ς rγy ρ )u + ru+ δ y rk ] = 0, u0) = K. 50) In 50), the function m = qy) fy)is the solution of the problem y ς m y α λςρ yς rγ ρ )f ) m + ς rγy ρ )m + ry = δ δ )y, 0 <y<ŷ, m0) = mŷ) = 0, 5) and my) is extended by 0 for y>ŷ. The function m is continuous but its derivative is discontinuous at ŷ. The difficulty is that one cannot interpret uy) as the value function of a control problem. Instead, it is, more appropriately, the value function of a stochastic differential game. Theorem 5 We assume r α+λςρ γς ρ ) >δ ŷ. There exists a unique uy) C 0, ), piecewise C, solving 50). This function vanishes for y sufficiently large. Moreover, it is the value function given by ) uy) = inf J y v ), θ 5) sup v ) θ with the controlled diffusion and objective function given by dy y t) = Y y t) α λςρ ς rγ ρ )Y y t)f Y y t)) + vt)ς rγ ρ ) ) dt + ςy y t)dw t), Y y 0) = y, ) [ θ J y v ), θ = E 0 θ 0 δ Y y t) + rk + v t) ) e rt dt + Ke rθ0 θ 0 <θ + m Y y θ) ) e rθ θ<θ 0], where θ 0 = inf{t : Y y t) = 0}, and m is the solution of 5) extended by zero for y>ŷ. 6 We next state that the solution of 50) is characterized by two intervals. 53) 6 For vt), we can take the same class as in problem 7) 8).

15 Real Options with Competition and Incomplete Markets 43 Theorem 6 The solution uy) of 50) is of the form y ς u y α λςρ yς rγ ρ )f ) u + ς rγy ρ )u + ru = δ y + rk, 0 <y<y and y <y<y 3, 54) with the value matching and smooth pasting conditions: uy ) = my ), u y ) = m y ), uy ) = my ), u y ) = m y ), uy 3 ) = 0, u y 3 ) = 0, 55) where my) = gy) fy), the solution of 5) and extended by 0 for y>ŷ. There exists a unique triple y,y,y 3 with 0 <y <y < ŷ<y 3 such that 54), 55) hold. Proof We know that u, the solution to 50), vanishes for ȳ>ŷ, ȳ sufficiently large. Since u0)>m0) and uȳ) = mȳ) = 0, there exists a first point y < ȳ such that uy ) = my ).Wemusthavey < ŷ. Otherwise, y =ȳ and u coincides with the solution of 5), i.e., the same system 50) with m = 0. But then ȳ =ŷ, hence y =ŷ. In this case, ũ = u m satisfies the equation y ς ũ y α λςρ yf + m )ς rγ ρ ) ) ũ with the boundary conditions + y ς rγ ρ )ũ ) + rũ = δ y + rk 56) ũ0) = K, ũŷ) = 0, and since u ŷ) = 0, ũ ŷ 0) = m ŷ 0) which implies ũ ŷ 0)>0. It follows that ũy) < 0fory close to ŷ, which is impossible since it must be positive. Therefore, y < ŷ. We claim also that δ y rk. Indeed, set ũy) = uy) my), then it satisfies 56) with the boundary conditions ũ0) = K, ũy ) = 0, ũ y ) = 0. 57) The matching of the derivatives comes from the fact that ũy) is C and ũy) > 0, ũy ) = 0. So y is a local minimum, hence ũ y ) = 0. Suppose δ y <rk, then using 56), we see that ũ y 0)<0; hence, ũy) < 0 for y<y, close to y. This is impossible. Since uŷ) > mŷ) = 0, there exists an interval in which ŷ is contained and such that the equation holds on this interval. One of the extremities of this interval is y 3 =ȳ. Cally the other extremity, such that uy ) = my ). Therefore, y y < ŷ. Necessarily, y >y. Otherwise, u will be the solution of the equation on 0,y 3 ),

16 44 A. Bensoussan and S.C.) Hoe which is the case studied at the beginning of the proof, which is impossible. But then we have uy ) = my ), u y ) = m y ). On the other hand, on the interval y,y ), m satisfies 5) and the right-hand side δ δ )y > δ y + rk, since δ y>rk, by virtue of y>y and δ y >rk. Thus, m satisfies all conditions on y,y ). Therefore, u = m on y,y ).By the uniqueness of u Theorem 5), the triple y,y,y 3 is necessarily unique. We note the property that uy) fy)+ K, which implies hy) 0. It remains to show that Lx,y) defined by 48) is the value function 45). Theorem 7 The function Lx,y) defined by 48) coincides with the value function 44). 3. Leader s Optimal Stopping Rule The optimal stopping rule for the leader is defined as: inf{t : Y y t) y }, if 0 <y<y, 0, if y ˆθy)= y y, inf{t : Y y t) y or Y y t) y 3 }, if y <y<y 3, 0, if y y 3, 58) where Y y t) is the process defined in ). 4Conclusion We study a problem similar to the one presented in Bensoussan et al. []. Although we consider the investment payoffs governed by a geometric Brownian motion dynamics like the lump-sum payoff case in Bensoussan et al. [], we do not encounter additional regularity issues encountered in the lump-sum payoff case, which results from indifference consideration for overcoming the comparison of gains and losses at different times in the incomplete markets. On the contrary, we are able to characterize a two-interval solution for the leader s optimal investment rule as the arithmetic Brownian motion cashflow payoff case presented in Bensoussan et al. []. The choice of a geometric Brownian motion cashflow process is motivated by the specification of an uncertain payoff arising from a stochastic demand process for the project s output, common in the financial economics literature see, for example, Dixit and Pindyck [3] and Grenadier [4]). We note that to study cashflow process in terms of a geometric Brownian motion process rather than an arithmetic Brownian motion process invokes additional nontrivial mathematical consideration. Comparing with the arithmetic Brownian motion cashflow payoff case, the current study requires additional absorbing barrier consideration as well as an additional

17 Real Options with Competition and Incomplete Markets 45 intermediate study of non-linear nd order differential equation, which turns out to be a solution to a minimization problem. The economic interpretation of the leader s two-interval solution for the Stackelberg game is interesting. Below the lower threshold, neither player will invest because the output value is too low. Above the upper threshold, both players invest as soon as possible because output value is very high. Around the middle threshold, output value is attractive to the follower, who invests as soon as possible. As a result the leader will have little or no time to exploit their monopoly position in the output market. Since output value is below the upper threshold, the leader prefers to invest at a lower threshold value, thus decreasing the follower s interest. This allows the leader to maintain a monopoly position in the output market for a longer time. This result, understandable but not necessarily intuitive, can be revealed only through the mathematics of the V.I. Acknowledgement The first author acknowledges support of National Science Foundation DMS and of the Research Grants Council of HKSAR CityU 5003). References. Bensoussan, A.: Applications of Variational Inequalities in Stochastic Control. Elsevier/North- Holland, Amsterdam 978). Bensoussan, A., Diltz, J.D., Hoe, S.: Real options games in complete and incomplete markets with several decision makers. SIAM J. Financ. Math. ), ) 3. Dixit, A., Pindyck, R.S.: Investment Under Uncertainty. Princeton University Press, Princeton 994) 4. Grenadier, S.: The strategic exercise of options: development cascades and overbuilding in real estate markets. J. Finance 55), ) 5. Kinderlenher, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, San Diego 980)

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