Optimal exit strategies for investment projects. 7th AMaMeF and Swissquote Conference

Optimal exit strategies for investment projects Simone Scotti Université Paris Diderot Laboratoire de Probabilité et Modèles Aléatories Joint work with : Etienne Chevalier, Université d Evry Vathana Ly Vath, Université d Evry and ENSIIE Alexandre Roch, UQAM 7th AMaMeF and Swissquote Conference Ecole Polytechnique Fédérale de Lausanne

Introduction Motivations : Optimal Exit Strategy A firm has an investment project which is unprofitable and for which the liquidation costs evolve stochastically.

Introduction Motivations : Optimal Exit Strategy A firm has an investment project which is unprofitable and for which the liquidation costs evolve stochastically. A recent example : Shale oil extraction (via hydraulic fracturing) in North Dakota with high production costs ( 60 USD) in front of an oil price having high oscillations (now 50 USD). 1 Patient firms - Continuously estimate the theoretical/fair price - Wait the time when the project becomes profitable again or equivalently wait for a buyer at that price. 2 Impatient firms - Find a buyer for the project - Immediately sell even at a discounted price w.r.t. the theoretical/fair price 3 Mixed strategies - Continuously estimate the theoretical/fair price - Wait for a buyer at that price - If a buyer proposes an acceptable discounted price, immediately sell

Introduction Parallelism with finance : Selling an illiquid asset Impatient trading : Optimal portfolio selection with transaction costs, Optimal portfolio liquidation,.. the trader pays liquidity costs. Patient trading : Optimal portfolio liquidation with limit orders (Bayraktar, Ludkowski 2012) the asset is sold at a random time τ (execution and inventory risks). Mixed trading : Market making (Guilbaud, Pham 2011) tradeoff between costs and risks

Introduction Model and problem formulation Dynamic programming system and properties of v i Logarithmic utility Power utility

Model and problem formulation Model Basic problem definition We consider a probability space (Ω, F, P) with a filtration F = (F t) t 0 satisfying the usual conditions. Let W and B be two correlated (F t)-brownian motions, with correlation ρ. Theoretical/fair value of the firm project evolving according a positive process S, which may be written as S t := exp(x t), where the process X is governed by the following s.d.e. dx t = µ(x t)dt + σ(x t)db t X 0 = x, where µ and σ are two Lipschitz functions with sub-linear growth conditions.

Model and problem formulation Model Termination costs liquidation process Y, is a mean-reverting process and governed by the following s.d.e. dy t = α(y t)dt + γ(y t)dw t, Y 0 = y, where α and γ are locally Lipschitz functions on R +. Stochastic discount factor : (f (Y t)) t 0, where f is a positive, continuous and decreasing function defined on R + [0, 1], and satisfies the following conditions : f (0) = 1 lim f (y) = 0 y Discount price : Should the firm manager decide to sell immediately the asset at the discount price, she would obtain a cash-flow of S tf (Y t).

Model and problem formulation Model The recovery time We introduce an external time τ representing the time when the project becomes profitable again or the arrival of a buyer accepting to pay the fair price. Intensity regimes : Let L be a continuous time, time homogenous, irreductible Markov chain, independent of W and B, with m + 1 states. The generator of the chain L under P is denoted by A = (ϑ i,j ) i,j=0,...n. Here ϑ i,j is the constant intensity of transition of the chain L from state i to state j. Break-even time : The arrival time, denoted by τ, is defined as the first jump time of a Cox process with an intensity (λ Lt ) t 0. τ is independent of W and B and, without lost of generality we assume λ 0 λ 1... λ m>0

Model and problem formulation Model Utility function Classical assumptions : - U : R + R, is non-decreasing, concave and belongs to C 2 (R + ). - U has the following behavior lim s 0 s U (s) < +

Model and problem formulation Model Utility function Classical assumptions : - U : R + R, is non-decreasing, concave and belongs to C 2 (R + ). - U has the following behavior lim s 0 s U (s) < + Supermeanvalued utility : (Dinkyn, Oksendal) The firm manager is coherent and rational : we suppose that U is supermeanvalued w.r.t. S, i.e. U(s) E s [U(S θ )] for s 0 and any stopping time θ T where T is the collection of all F stopping times.

Model and problem formulation An optimal stopping problem with regime switching An optimal stopping problem with regime switching Maximizing the expected utility of the wealth received from the sales of the asset. Objective function : For x R, y R +, i {0,..., m}, we set v(i, x, y) := sup E i,x,y [ ] h(x θ, Y θ ) I θ τ + U(e Xτ ) I θ>τ, θ T where E i,x,y denotes the flow with initial condition X 0 = x, Y 0 = y and L 0 = i and h(x, y) = U(exp(x)f (y)).

Dynamic programming system and properties of the value functions Dynamic programming system and properties of v i Model and problem formulation Dynamic programming system and properties of v i Logarithmic utility Power utility

Dynamic programming system and properties of the value functions DPP and associated VI system Objective functions bounds and monotonicity Proposition The value functions v i are non-decreasing in x, non-increasing in y and satisfy Proof : Max ( h(x, y), E x [U(e Xτ )] ) v(i, x, y) U(e x ) on R R +. Monotonicity of U and f. Uniqueness of SDEs solution, continuity of processes X and Y (up to ξ y := inf{t > 0, Y y t = 0}) Supermeanvalued assumption Remarks - We have v(i, x, 0) = U(e x ) : it is optimal to immediately exit from the project when discount factor reaches 1. - If E x [U(e Xτ )] = U(e x ) : the optimal policy is to wait until the arrival of the break-even time

Dynamic programming system and properties of the value functions DPP and associated VI system Continuity of the value functions Continuity of the value functions The value functions v i are continuous on R R + and satisfy : Proof Lemma lim v i (u, y) = v i (x, 0) = U(e x ). (u,y) (x,0 + ) Continuity of stochastic flows (up to inf{t > 0, Y y t = 0}). There exists an optimal stopping time θi,x,y such that v(i, x, y) := E i,x,y [ h(x θ i,x,y, Y θ i,x,y ) I θ i,x,y τ + U(e Xτ ) I θ i,x,y >τ Moreover, θ i,x,y ξy τ where ξy := inf{t 0; Yy t = 0}. ] Supermeanvalued utility assumption lim s 0 s U (s) < +

Dynamic programming system and properties of the value functions DPP and associated VI system Viscosity Characterization of value function Theorem The value functions v i, i {0,..., m}, are the unique continuous viscosity solutions on R R + with growth condition v i (x, y) U(e x ), and boundary data lim y 0 v i (x, y) = U(e x ), to the system of variational inequalities : [ ] min Lv(i, x, y) G i v(., x, y) J i v(., x, y), v(i, x, y) h(x, y) = 0, where, for functions ϕ : {0,..., m} R R + R with ϕ(i,, ) C 2 (R R + ) for all i {0,..., m}, we have set Lφ(x, y) = µ(x) φ φ + α(y) x y + 1 2 σ2 (x) 2 φ x 2 + ργ(y)σ(x) 2 φ x y + 1 2 γ2 (y) 2 φ y 2. and G i and J i act on functions ϕ : G i ϕ(., x, y) = j i ϑ i,j (ϕ(j, x, y) ϕ(i, x, y)) J i ϕ(., x, y) = λ i (U(e x ) ϕ(i, x, y)).

Dynamic programming system and properties of the value functions DPP and associated VI system Proof of uniqueness Comparaison principle Let (φ i ) 0 i m (resp. (ψ i ) 0 i m ) a family of continuous subsolution (resp. super solution) of the VI system on R R + satisfying the following growth conditions on {0,.., m} R R + Max ( h(x, y), E i,x [U(e Xτ )] ) φ(i, x, y) (resp. ψ(i, x, y)) U(e x ) on R R + and lim y 0 + φ i (x, y) lim y 0 + ψ i (x, y). We have φ ψ on {0,.., m} R R +. Proof. Step 1. Construction of strict super-solutions to the system with suitable perturbations of ψ i : - ψ γ i = (1 γ)ψ i + γη i - (ψ γ i ) (i=1,...,n) is a strict super-solution to the VI system. Step 2. It suffices to show (by contradiction) that for all γ (0, 1) : max sup (φ i ψ γ i ) 0, i {0,..,m} R R +

Dynamic programming system and properties of the value functions DPP and associated VI system Proof of uniqueness Technical point : Construct a strict super solution which dominates φ i and ψ i. The following function is a strict super solution which dominates U(e x ) { s w(s) = 4 + k + A 0 + A 1 s + 1 2 A 2s 2 s 0 s 4 + k + U(e s ) ln (4 + s) s > 0 where A 0, A 1, A 2 and k are strictly positive constants. Existence of such strict super solution follows from characterization of E i,x [U(e Xτ )] as solution of a system of PDE.

Dynamic programming system and properties of the value functions DPP and associated VI system Stopping and continuation regions Stopping and continuation regions E = { (i, x, y) {0,..., m} R R + v(i, x, y) = h(x, y) } C = {0,..., m} R R + \ E. We also define the (i, x) sections for every (i, x) {0,..., m} R by E (i,x) = {y (0, + ) v(i, x, y) = h(x, y)} and C (i,x) = R + \ E (i,x). Optimal exit time : θixy = inf { u 0 ( Lu i, Xx u, } u) Yy E. Properties of the stopping region Let (i, x) {0,..., m} R. If E i,x [U(e Xτ )] = U(e x ), then, for all y R +, v(i, x, y) = U(e x ) and E (i,x) = {0}. If E i,x [U(e Xτ )] < U(e x ), then x 0 such that E (i,x0 ) \ {0} = and ȳ (i, x) := sup E (i,x) < +.

Logarithmic utility Model and problem formulation Dynamic programming system and properties of v i Logarithmic utility Power utility

Logarithmic utility Application to logarithmic utility Logarithmic utility Throughout this section we assume that U(s) = ln(s) on R +. X and Y solutions of the following SDEs : dx t = µdt + σ(x t)db t; X 0 = x dy t = κ (β Y t) dt + γ Y tdw t; Y 0 = y Remarks The supermean value assumption implies that µ 0. If µ = 0, we have seen that v(i, x, y) = U(e x ) and E (i,x) = {0}

Logarithmic utility Application to logarithmic utility Dimension reduction Proposition For (i, y) {1,..., m} R + we define the function : w(i, y) = sup E i,y [µ(θ τ) + ln (f (Y θ )) I {θ τ} ], θ T L,W where T L,W is the set of stopping times with respect to the filtration generated by (L, W). We have v(i, x, y) = x + w(i, y) on {1,..., m} R R +. Proof. On {0,..., m} R R +, we have v(i, x, y) = sup E i,x,y [X θ τ + ln (f (Y θ )) I {θ τ} ]. θ T We prove that v (i, x, y) = 1 then v(i, x, y) = x + φ(i, y) x An optimal stopping time is θixy = inf{t 0 : φ(li t, Yy t ) = ln(f (Yt y ))}, which belongs to T L,W. We obtain φ = w.

Logarithmic utility The case with no regime switch No regime switch Let i {1,..., m}. Throughout this section, we shall assume that ϑ i,j = 0 i j and that there exists 0 < y i such that E (i,x) = [0, y i ]. Proposition y i is the solution of the following equation g(y i ) µ λ i g (y i ) the function w(i, ) is given by = γ2 2λ i g(y) g(y w(i, y) = i ) µ λ ( i Ψ λi α, 2αβ γ 2, 2α γ 2 y i ( Ψ λi α, 2αβ γ 2 ), 2α γ 2 y i ( Ψ λi 2αβ + 1, α γ 2 + 1, 2α γ 2 y i ( λi ) Ψ α, 2αβ γ 2, 2α γ 2 y where Ψ denotes the confluent hypergeometric function of second kind ). y y i ) + µ y > y i λ i

Logarithmic utility The case with two regimes Two regimes We assume that m = 1, ϑ 0,1 ϑ 1,0 0 and that, for i {0, 1}, there exists y i E (i,s) = [0, y i ]. > 0 such that Proposition We can show that y 0 y 1 Let Λ be the matrix ( ) λ0 + ϑ Λ = 0,1 ϑ 0,1 ϑ 1,0 λ 1 + ϑ 1,0 As ϑ 0,1 ϑ 1,0 > 0 it is easy to check that Λ has two eigenvalues λ 0 and λ 1 < λ 0. Let Λ = P 1 ΛP be the diagonal matrix with diagonal ( λ 0, λ 1 ). The transition matrix P is denoted by ( p 0 P = 0 p 0 ) p 1 0 p 1. 1

Logarithmic utility The case with two regimes Two regimes Proposition The function w(1, ) is given by w(1, y) = g(y) y [0, y 1 ] [ ( λ0 p 1 0 êψ α, 2αβ γ 2 +p 1 1 [ f Ψ ( λ1 ) ], 2α γ 2 x + µ λ0 α, 2αβ γ 2 ) ], 2α γ 2 y + µ λ1 y (y 1, ) where Ψ denotes the confluent hypergeometric function of second kind, I is a particular solution to the non-homogeneous confluent differential equation.

Logarithmic utility The case with two regimes Two regimes Proposition The function w(0, ) is given by w(0, y) = g(y) y [0, y 0 ] ( λ0 + ϑ 0,1 ĉφ α +I [ ( λ0 p 0 0 êψ α, 2αβ γ 2 +p 0 1, 2α ) ( γ 2 y + dψ λ0 + ϑ 0,1 α, 2αβ ( γ 2 2α γ 2, β, 2 λ 0 + ϑ 0,1 γ 2, 2 ϑ 0,1g( ) + µ γ 2 [ f Ψ ( λ1 ) ], 2α γ 2 y + µ λ0 α, 2αβ γ 2 ) ], 2α γ 2 y + µ λ1, 2αβ γ 2 ) (y), 2α ) γ 2 y y (y 0, y 1 ] where Φ and Ψ denote respectively the confluent hypergeometric function of first and second kind, I is a particular solution to the non-homogeneous confluent differential equation. y (y 1, )

Power utility Model and problem formulation Dynamic programming system and properties of v i Logarithmic utility Power utility

Power utility Power utility Throughout this section we assume that U(s) = s a on R + with 0 < a 1. There exists µ and σ in R s.t. X is solution of the following sde : dx t = µdt + σdb t Remarks The supermean value assumption implies that µa + σ2 2 a2 0. If µa + σ2 2 a2 = 0, we have seen that v(i, x, y) = U(e x ) and E (i,x) = {0}

Power utility Dimension reduction Proposition For (i, y) {1,..., m} R + we define the function : We have u(i, y) = sup E i,y [e (µ+(1 ρ 2 ) σ 2 2 )(θ τ)+ρσw ( ) θ τ I {θ>τ} + g(y θ )I {θ τ} ]. θ T L,W v(i, x, y) = e ax u(i, y) on {1,..., m} R R +. (u(i, )) 0 i m are the unique continuously differentiable viscosity solutions of the system of equation : [ min Lu(i, y) λ i (1 u(i, y)) ] ϑ i,j (u(j, y) u(i, y)), u(i, y) g(y) = 0. j i where we have set g(y) = (f (y)) a and, for φ C 1 (R + ), Lφ(y) = 1 2 γ2 y 2 φ [ y 2 + α(β y) + ρσγa ] φ [ σ 2 y y + a 2 ] + µa φ(y). 2

Power utility Conclusions Study of a optimal exit strategy from a firm project Mathematical characterization of the objective function Explicit solutions for specific utility functions (power and logarithm)

Power utility Thank you for your attention Paper available on Journal of Mathematical Analysis and Applications (or send me an email)