A discretionary stopping problem with applications to the optimal timing of investment decisions.

Transcription

1 A discretionary stopping problem with applications to the optimal timing of investment decisions. Timothy Johnson Department of Mathematics King s College London The Strand London WC2R 2LS, UK Tuesday, 24th November Joint work with Mihail Zervos and Karen Watson. timothy.johnson@kcl.ac.uk Web : johnson

2 The concept of real options was introduced by McDonald and Siegel in Consider a project the value of which depends on the price of a single asset, such as the price of oil. Classical finance tells you to invest in the project if it has a positive NPV, if it doesn t the project has no value. McDonald and Siegel showed that if you have the option to delay project start, the value of the project can be positive even if the NPV is not, but you should initiate at a higher asset price than under the NPV rule.

3 Classical Finance

4 Project Value Project's value Project's payoff Investment region Asset's Price

5 Real Options

6 Project Value Project's value Project's payoff Investment region Asset's Price

7 Other applications include When should you sell an asset that you have been endowed with? X t models the asset s price at time t The payoff, g, is simply g(x) = x, or could be adjusted for tax / utility. Exercise of perpetual American options The payoff, g, is g(x) = (x K) +, where K > 0 is the strike

8 This problem has been widely studied, notably by Dixit & Pindyck, Trigeorgis and Cortazar & Schwartz Mainly for assets driven by a geometric Brownian motion, but some mean reverting diffusions. Affine payoffs. Constant discount rates. We consider General one-dimensional Itô diffusions. General payoffs, that are increasing above a specific value of the asset s price

9 State dependent discounting.

10 Suppose the asset s price is driven by GBM dx t = bx t dt + σx t dw t, X 0 = x > 0 and the payoff function, g, is given by g(x) = x K and r is constant In this case and it follows that e rt X t = xe (b r)t e 1 2 σ2 t+σw t if b > r then v(x) = and there may be no optimal strategy if b < r and K = 0 then v(x) = x and τ 0 is optimal if b < r and K > 0 then then the solution is not totally trivial

11 Strong evidence of mean-reversion in real asset prices, exchange and interest rates. We should consider diffusions other than GBM. State process dx t = b(x t ) dt + σ(x t ) dw t, X 0 = x > 0, W is a standard, one-dimensional Brownian motion and b, σ are given deterministic functions such that X t 0, for all t > 0 A stopping strategy is a collection S x = ((Ω, F,F t, P x, X, W), τ) Given an initial condition x > 0, the objective is to maximise [ t J(S x ) := E x e Λτ g(x τ )1 {τ< } ], where Λ t = r(x s)ds 0

12 over all stopping strategies S x g is increasing for large x and r is positive and bounded The value function is v(x) := sup Sx, J(S x) for x > 0.

13 Observe that [ E x e Λ t g(x t ) ] [ t ] = g(x) + E x 0 e Λs f(x s )ds. where f(x) := 1 2 σ2 (x)g (x) + b(x)g (x) r(x)g(x) if f(x) < 0 for all x then v(x) = g(x) and τ 0 is optimal. if f(x) > 0 for all x then v and the optimal strategy may not exist. We therefore assume that there exists x 1 > 0 such that f(x) = 1 2 σ2 (x)g (x) + b(x)g (x) r(x)g(x) > 0, for x < x 1, < 0, for x > x 1.

14 At time 0 the project s management has two options Capitalise, and receive g(x), so v(x) g(x). (1) Wait, for a short time and then continue optimally, which yields Using Itô s formula, v(x) v(x) + E x [ t v(x) E x [ e Λ t v(x t ) ]. 0 e Λ s ( ) ] 1 2 σ2 v + bv rv (X s )ds. Dividing by t and letting t 0 we obtain 1 2 σ2 (X s )v (X s ) + b(x s )v (X s ) r(x s )v(x s ) 0 (2)

15 Since capitalising or waiting are the only two options, one must be optimal., and either (1) or (2) must hold with equality, i.e. g(x) v(x) σ2 (X s )v (X s ) + b(x s )v (X s ) r(x s )v(x s ) 0 These heuristic arguments suggest that the value function, v, should identify with a solution, w, to the HJB equation { } 1 max 2 σ2 (x)w (x) + b(x)w (x) r(x)w(x), g(x) w(x) = 0 We verify that w = v.

16 Every solution to the ODE 1 2 σ2 (x)w (x) + b(x)w (x) r(x)w(x) = 0 is a linear combination of the functions ψ and φ defined by ψ(x) := φ(x) := for a given choice of z > 0 { Ex [e Λ τz] for x < z, 1/E z [e Λ τx] for x z, { 1/Ez [e Λ τx] for x < z, E x [e Λ τz] for x z, (increasing) (decreasing)

17 When X is a geometric Brownian motion, dx t = bx t dt + σx t dw t, then ψ(x) = x n and φ(x) = x m where m < 0 < n are appropriate constants. When X is a square root mean reverting proces (CIR), dx t = κ(θ X t )dt + σ X t dw t, then ψ(x) = 1 F 1 ( r κ, 2κθ σ 2 ; 2κ σ 2x ) and φ(x) = U ( r κ, 2κθ σ 2 ; 2κ ) σ 2x.

18 When X is a geometric Ornstein-Uhlenbeck process, dx t = κ(θ X t )X t dt + σx t dw t, then and ( ψ(x) = x n 1F 1 n,2n + 2κθ σ 2 ; 2κx ) σ 2 φ(x) = x n U ( n,2n + 2κθ σ 2 ; 2κx ) σ 2.

19 When X is an exponential Ornstein-Uhlenbeck process, dx t = (κ(θ ln X t ) + 12 ) σ2 X t dt + σx t dw t, or X t = e Y t with dy t = κ(θ Y t )dt + σdw t, then and ψ(x) = φ(x) = Γ( r+κ 2κ ) π U ( r 2κ, 1 2 ; κ σ 2(θ ln x)2), for x e θ 1F 1 ( r 2κ, 1 2 ; κ σ 2(θ ln x)2), for x > e θ, 1F 1 ( r Γ( r+κ 2κ ) π 2κ, 1 2 ; κ σ 2(θ ln x)2), for x e θ U ( r 2κ, 1 2 ; κ σ 2(θ ln x)2), for x > e θ.

20 We consider solutions to the HJB equation is given by w(x) = { Aψ(x) + Bφ(x), if x < x, g(x), if x x, Since the pay-off function is increasing, we expect that the value function is increasing, implying B = 0. To specify A and x, we use smooth-pasting that requires the value function to be C 1 at the free boundary point, x, so Aψ(x ) = g(x ) and Aψ (x ) = g (x ), which is equivalent to q(x ) = 0 where q(x) := g(x)ψ (x) g (x)ψ(x), x > 0 (3)

21 Solving the HJB Equation Project Value Psi Project's payoff Wait x* Capitalise Asset's Price

22 We can deduce from (3) that lim x 0 q(x) 0 and so there is a solution to q(x) = 0 if limsup x q(x) > 0. Observe that q(x) = g 2 (x) d dx ( ψ(x) g(x) ), for x > 0. Now, if limsup x q(x) 0 then this calculation implies ( ) d ψ(x) lim 0. x dx g(x) However, this inequality and the continuity of ψ and g imply that lim x ψ(x) g(x) <.

23 We can show that lim x ψ(x) g(x) =. under the assumption that there exists a function l such that lim x l(x) =, and 1 2 σ2 (x)(lg) (x) + b(x)(lg) (x) r(x)(lg)(x) 0, for x > x 2 > x 1 Fairly easy to find candidates for l (because x 2 can be arbitrarily large). Consider g(x) = g(x) = ξx η K, for x > 0 and b(x) < 0 for x > x 3, where ξ, η, K > 0 are constants and x 3 > (K/ξ) 1/n The diffusions already given

24 Then l(x) = ( ln x ln x 3 ) + is suitable.

25 This result is useful in solving other investment problems Consider the decision to invest in a project that provides a running payoff and can be abandoned v(x) = sup S x E x [ e Λ τ i g i (X τi )1 {τi < } + τo + e Λ τog o (X τo )1 {τo < } τ i ]. e Λ t h(x t )dt This can be decomposed into two problems [ ( τo [ g o (x) + h σ2 g ṽ(x) = sup S x E x v(x) = sup S x E x 0 e Λ t [e Λ τ i ( g i (X τi ) + ṽ(x τi ) ) ] 1 {τi < } o + bg o rg o ](X t )dt. ) ] 1 {τo < }