Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs

Transcription

1 Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs Mihai Sîrbu, The University of Texas at Austin based on joint work with Jin Hyuk Choi and Gordan Žitković UT Austin The Mathematics of the New Financial System Minneapolis, May 19th, 2012

2 Outline Objective Statement of the problem Literature Shadow prices Our shadow price approach Solution of the HJB Main results Conclusions

3 Objective revisit the classic problem of Davis and Norman using the so called shadow price approach treat all possible values of parameters (and therefore, study the well posedness of the problem)

4 The (Davis and Norman) model One risky asset (stock) ds t = S t (µ dt + σ db t ), t [0, ) with S 0 > 0. Transaction costs: λ (0, 1) and λ > 0: one gets only S t = (1 λ)s t for one share of the stock pays S t = (1 + λ)s t for it Bond pays zero interest rate: r = 0.

5 Investment/consumption strategies (Kallsen-Muhle Karbe notation) ϕ 0 t : number of bonds held at time t (after the transaction, if any, took place) ϕ t : number of stocks held at time t (same as above) c t consumption rate The processes ϕ and ϕ 0 are right-continuous and of finite variation and c is nonnegative and locally integrable. Distinguish between the initial values (ϕ 0 0, ϕ 0 ) and the values (ϕ 0 0, ϕ 0) (after which the processes are right-continuous). Initial position (ϕ 0 0, ϕ 0 ) = (η B, η S ). Remark: the initial jump appears even if there is no transaction cost.

6 Investment/consumption strategies cont d Self-financing strategy (ϕ 0, ϕ, c): t t t ϕ 0 t = ϕ 0 0 S u dϕ u + S u dϕ u c u du, (1) where ϕ = ϕ 0 + ϕ ϕ is the pathwise minimal (Hahn-Jordan) decomposition of ϕ. Remark: there is a possible jumps at time zero, as we assume that ϕ 0 = ϕ 0 = 0. Admissible strategy: can always be liquidated ϕ 0 t + ϕ + t S ϕ t S t 0, t 0.

7 Optimal investment/consumption For p (, 1), we consider the utility function U : [0, ) [, ) of the power (CRRA) type { { 1 p U(c) = cp, c 0, p 0 0, p > 0, and U(0) = log(c), c 0, p = 0,, p 0 Optimization problem: u = sup E (ϕ 0,ϕ,c) [ 0 ] e δt U(c t ) dt, where δ > 0 stands for the (constant) impatience rate.

8 Literature Constantinides and Magill (76): heuristic solution Davis and Norman (90): analytic solution Shreve and Soner (94): removed some technical conditions, making use of viscosity theory. Key assumption: well posedness u < Conclusion: there exists a wedge around the Merton proportion such that jump to the boundary of the wedge do not trade inside the wedge

9 Shadow prices Introduced by Jouini and Kallal (95) Lamberton, Pham and Schweizer (98) 1. Consistent price process: S t S t S t, for all t 0. Investor with wealth η B + η S S 0 can certainly do better trading in S without transaction costs, than trading in S with transaction costs. 2. If one can find a consistent price process S such that the corresponding optimal investment strategy (without transaction costs) trades as buy stocks only when S = S sell stocks only when S = S then such a trading strategy is actually optimal in the problem with transaction costs. The price process S is called shadow price.

10 Can this idea be used for the Davis and Norman problem? Yes, in some cases Kallsen and Muhle Karbe, for p = 0. For fixed S use the explicit solution of the log investor without transaction costs to find a shadow price Herczegh and Prokaj (parallel to our work): use the solution in Shreve and Soner (the primal value function) to construct a shadow price for p 1. The shadow price is constructed in terms of the gradient of the primal value function. Both interpret the problem as a one dimensional free-boundary problem in terms of the variable π log( 1 π ). Both require restriction on parameters: NT region is in the first quadrant (no singularity in our coming approach).

11 Our shadow price approach Exploit the fact that the shadow price is a worst case scenario among consistent prices S S S. Parametrize consistent processes S by (θ, Σ) such that d S t = S t (σ + Σ t ) (db t + θ t dt), S 0 = s 0. Pass to the logarithmic scale Y t = log( S t /S t ), whose dynamics is given by dy t = α 0 (θ t, Σ t ) dt + Σ t db t, Y 0 = y = log( S 0 S 0 ). on the natural domain Y t [y, y]. Here, y = log(1 λ), y = log(1 + λ) ( ) and α 0 (Σ, θ) = θσ µ Σ 1 2 Σ + σ θ.

12 Shadow prices as a game Recall that, η B, η S are fixed. Fix y [y, y] and θ, Σ admissible. 1. Solve the optimal investment problem for an investor with initial wealth w = η B + η S S 0 e y i.e. maximize over π, c (proportion of investment in S and rate of consumption) the expected discounted utility E[ 2. Minimize first over (θ, Σ) 3. Minimize over y [y, y]. 0 e δt U(c t )]. Summarize: up to the last minimization over y, we have a problem v(w, y) := inf sup θ,σ π,c E[ 0 e δt U(c t )].

13 Game cont d We can write the Isaacs equation for the game with a two dimensional state (W, Y ). The problem actually scales as v(w, y) = w p p h(y)1 p, and the Isaacs equation reduces to a one-dimensional equation for h(y), y [y, y]. Actually, from the duality theory for complete markets, we can easily see that h(y) = inf E (Σ,θ) P(y) [ 0 ) ] e δt V (e δt E ( θ B) t dt The reduced Isaacs equation for h(y) is actually an HJB (this is what we use in the paper). Remark: the sup and inf separate in the game.

14 The one-dimensional HJB ( inf 1 Σ,θ 2 Σ2 h (y) + α q (Σ, θ)h (y) β(θ)h(y) + γ(θ)) = 0, y (y, y) w (y) = w (y) = +. ( ) q = p 1 p, α q(σ, θ) = θσ µ Σ 1 2 Σ + σ θ(1 + q), { ( 1 β(θ) = (1 + q) δ 1 2 qθ2), and γ(θ) = 2 θ2, p = 0, sgn(p), p 0. (2) The boundary conditions ensure the state constraint Y [y, y].

15 Change of variable/order reduction The HJB has the form H(h, h, h) = 0, y [y, y]. Expect h to be increasing negative Change the variable to x = h (y). The (fixed) boundaries y, y change to x = h (y), x = h (y). Rewrite the HJB by g(x) = h(y) as ( 1 2 Σ2 x g (x) α q(σ, θ)x β(θ)g(x) + γ(θ) inf Σ,θ R ) = 0, x (x, x), g (x) = g (x) = 0 and x x g (x) x dx = log( 1+λ 1 λ ) = y y.

16 Solving for g The equation for g can be rewritten as g = P(x, g) Q(x, g), where P, Q are second order polynomials. Idea of the solution: fix (α, g(α)) {P = 0} evolve the solution of the ODE until it meets again {P = 0} at x = β α move α to meet the integral constraint βα α g (x) dx = y y. x For α 0 the integral is. The integral is decreasing in α. Then, choose x = α, x = β α.

17 δ π < 1, p < (1 p)σ2 δ 2 π < 1, p > (1 p)σ2 2 π = 1 π > 1 Figure: 0 < p < 1, µ < G.

18 π < 1 π = 1 π > 1, α > x P Figure: 0 < p < 1, µ < G π > 1, α x P

19 Possible singularity A singularity appears when the NT region contains the axis η B = 0.

20 Is there always a solution g? No, as, for some values of the parameters, we have βα lim α α This value can be computed explicitly! g (x) dx = C(µ, σ, δ, p) > 0 x y y = log( 1+λ 1 λ ) C(µ, σ, δ, p) means the value function is infinite, and no solution g exists y y = log( 1+λ 1 λ ) > C(µ, σ, δ, p) means the value function is finite, and there is a solution g Full characterization of well posedness.

21 π < 1 Figure: 0 < p < 1, G µ < A π > 1, α > x P

22 How do we solve the original problem? Have g(x). Change back the variable to y = f (x) such that f (x) = g (x), x y = f (x), y = f (x). With these notation, the value function of the game is v(w, y) = w p p g(x)1 p. Also, whenever the state Y is at y x it is optimal for the investor to hold π(w, x) = π(x) proportion in the stock (driven by the factor Y ), without transaction costs.

23 Solution of the original problem cont d For fixed initial y, the investor will not buy/sell stocks invest expect when Y = y, y. The edges of the NT region have slopes π(x), π(x).

24 Choosing the intial y Recall Y 0 = y, and denote w(x) = η B + η S S 0 e f (x). i.e. the total wealth at time t = 0 if the consistent prices starts at S 0 e y. Choose y (or x) to minimize v(w(x), x) = w(x)p g(x) 1 p. p Denote again, r(x) = η S S 0 e f (x) π(x)w(x), i.e. the difference between how much the investor has in the stock at time t = 0 and what will have a time t = 0 investing in the consistent price process S y. We can compute easily d dx v(w(x), x) = A(x)r(x). Therefore if possible, choose x such that r(x) = 0 (start inside the NT region) if r(x) > 0 x, then choose x = x (jump to the upper edge of the NT region) if r(x) < 0 x, then choose x = x (jump to the lower edge of the NT region)

25 Main results Theorem (Well Posedness) Given the parameters µ, σ (0, ) and the transaction costs λ (0, 1), λ > 0, the following statements are equivalent: (1) The problem is well posed, i.e < u <. (2) The parameters of the model satisfy one of the following three conditions: - p 0, - 0 < p < 1 and µ < - 0 < p < 1, 2δ(1 p)σ 2 p, 2δ(1 p)σ 2 p µ < δ p + (1 p)σ2 2 and C(µ, σ, p, δ) < log( 1+λ 1 λ ), where C(,,, ) admits an explicit (closed-form) expression.

26 Figure: The well-posedness region. Remark: generalizes Shreve and Soner, who had this condition only for σ = 0

27 Main results, cont d Theorem If the problem is well posed, there exists a solution g : [x, x] R and a shadow price S t = S t e Yt, that leads to the optimal solution of the original problem.

28 Conclusions We have a new way to solve the problem of optimal investment/consuption with transaction costs for power utilitites: self contained and direct (no dynamic programming principle) more elementary complete: treats all parameter values, explicitly characterize well posedness Lays the foundation for expansion as power series of λ 1 3 of any order: see the paper of Jin Hyuk Choi. Closed form solution as a power series.

29 Future work Represent general singular stochastic control problems as (absolutely continuous) games (with state constraints).