Optimal Execution Tracking a Benchmark

Transcription

1 Optimal Execution Tracking a Benchmark René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Princeton, June 20, 2013

2 Optimal Execution Market Set-Up R.C. - M. Li Goal: sell v > 0 shares by time T > 0 (finite horizon) P t mid-price (unaffected price), t P t = P 0 + σ(u)dw u, 0 t T, 0 V (t) volume traded in the market up to (and including) time t Market VWAP= 1 T V 0 P t dv (t) Fraction of shares still to be executed in the market X(t) = V V (t) V = T t T (deterministic V (t) used to change clock). Convenient simplification!

3 Broker Problem v t volume executed by the broker up to time t x t = v v t v fraction of shares left to be executed by the broker at time t Where x t = 1 l t m t l t cumulative volume executed through limit orders m t cumulative volume executed through market orders Broker average liquidation price vwap= 1 ( T v 0 P t S 2 ) dm t + Objective: Minimize discrepancy between vwap and VWAP ( ) P t + S dl 2 t

4 Naive Model for the Dynamics of the Order Book Controls of the broker: (m t ) 0 t T non-decreasing adapted process (L t ) 0 t T predictable process t N t l t = y L u µ(du, dy) = Y i L τi 0 [0,1] i=1 where µ(du, dy) point measure (Poisson) compensator ν t (du)ν(t)dt. t N t x t = 1 y L u µ(du, dy) m t = 1 Y i L τi m t 0 [0,1] i=1 So the dynamics of x t are given by dx t = y L t µ(dt, dy) dm t, [0,1] with initial condition x 0 = 1.

5 Optimization Problem Goal of the broker [ ] sup (L,m) A E U(vwap VWAP), For the CARA exponential utility, approximately [ ( ( inf E S exp γ (L,m) A 2 + T 0 [x L,m u X(u)]dP u S dm u ) ], We will work with a Mean - Variance criterion [ T inf E γ σ(u)2 [xu L,m X(u)] 2 du + S m T ], (L,m) A 0 2 S spread X(u) = (T u)/t fraction of shares left to be executed in the market.

6 Stochastic Control Problem Singular control problem of a pure jump process Value function where [ T J(t, x, L, m) = E t J(t, x) = inf J(t, x, L, m) (L,m) A(t,x) γ σ(u)2 [xu L,m X(u)] 2 du + Sm T ]. 2 J(t, x) is non-decreasing in t for x [0, 1] fixed. (A(t 2, x) A(t 1, x) whenever t 1 t 2 )

7 Tough Luck: Problem is NOT Convex The set A of admissible controls is not convex. For any number l (0, 1), the two controls (L 1, m 1 ) and (L 2, m 2 ) by: L 1 t = 1 {t τ1 } + x τk 1 1 {τk 1 <t τ k }, and mt 1 = x T 1 {T t}, k=2 and: L 2 t = l 2 1 {t τ 1 } + x τk 1 1 {τk 1 <t τ k }, and mt 2 = x T 1 {T t}, k=2 are admissible, but the pair (L, m) defined by IS NOT L t = 1 2 (L1 t + L 2 t ), and m t = 1 2 (m1 t + m 2 t ),

8 Closest Related Works Poisson random measure µ(dt, dy) for claim sizes Y t insurer pays Y t α t up to a retention level α t re-insurer covers the excess (Y t α t ) + Wealth process of the Insurance Company t t t X t = x + p(α s)ds y α s µ(ds, dy) dd s p(α) insurer net premium (after paying the reinsurance company) D t cumulative dividends paid up to (and including) time t [ τ ] sup E e ru dd u (α t ) t,(d t ) t 0 time of bankruptcy τ = inf{t 0; X t 0} Jeanblanc-Shyryaev (1995) optimal dividend distribution for Wiener process, Asmunssen- Hjgaard-Taksar (1998) optimal dividend distribution for diffusion, Mnif-Sulem (2005) prove existence and uniqueness of a viscosity solution, Goreac (2008) multiple contracts

9 Similarities & Differences Similarities α t standing limit orders L t D t cumulative market orders m t Differences We work in a finite horizon (PDEs instead of ODEs) We use a Mean - Variance criterion We exhibit a classical solution (as opposed to a viscosity solution) We derive a system of ODEs identifying the value function the optimal stratagy

10 Technical Assumptions ν t (dy)ν(t)dt intensity of Poisson measure µ(dt, dy) with ν t ([0, 1]) = 1. T 0 σ(t)2 dt < sup 0 t T ν(t) < t σ(t)2 (X(t) x) is increasing for each x [0, 1] ν(t) t 1 ν(t) ν t ( ) is decreasing (in the sense of stochastic dominance)

11 Hamilton-Jabobi-Bellman Equation (QVI) min [ [Aφ](t, x), t φ(t, x) + [Bφ](t, x) ] = 0. where [Aφ](t, x) = S x φ(t, x) and [Bφ](t, x) = γ σ(t)2 2 [X(t) x]2 + ν(t) inf [φ(t, x y L) φ(t, x)]ν t (dy) 0 L x [0,1] with terminal condition and boundary condition: φ(t, x) = Sx, (notice that φ(t, x) = 0) T φ(t, 0) = t γσ(u) 2 X(u)du. 2

12 Classical Solution Theorem The value function is the unique solution of [ J(t, x) = min inf J(t, x), 0 y x γ σ(t)2 2 [X(t) x]2 + ν(t) [0,1] ] [J(t, (x y) L(t, y)) J(t, x)]ν t (dy) with where t J(t, 0) = γ 0 σ(u) 2 X(u) 2 du, and J(T, x) = Sx 2 L(t, x) = arg min J(t, y) 0 y x J is C 1,1 x J(t, x) convex for t fixed t J(t, x) non-decreasing for x fixed x J(t, x) 0

13 Free Boundary (No-Trade Region) [0, T ] [0, 1] = A B C with A = {(t, x); x J(t, x) < 0} = {(t, x); 0 t < τ l (x)} B = {(t, x); 0 x J(t, x) S} = {(t, x); τ l (x) t τ m(x)} C = {(t, x); x J(t, x) = S} = {(t, x); τ m(x) t} where τ l (x) = inf{t > 0; x J(t, x) 0} τ m(x) = inf{t > 0; x J(t, x) S} τ l (x) T (1 x) τ m(x)

14 Optimal Trading Strategy If t > τ m(x t ) i.e. (t, x t ) C (never happens) place market orders m t > 0 (just enough to get into B) If t = τ m(x t ) i.e. (t, x t ) C place market orders at a rate dm t = τ m(x t )dt (just enough so not to exit B) If τ l (x t ) t < τ m(x t ) i.e. (t, x t ) B A place L t = x t L(t) limit orders (as much as possible without getting ahead too much) If t < τ l (x t ) i.e. (t, x t ) A (never happens) no trade

15 Special Case I: Large Fill Distribution ν t (dy) = δ 1 (dy): the crossings, when they occur, fill all the requested limit orders. Theorem The value function solves [ ] J(t, x) = min inf J(t, x), γ σ(t)2 0 y x 2 [X(t) x]2 + ν(t)[j(t, L(t, x)) J(t, x)] with t J(t, 0) = γ 0 σ(u) 2 X(u) 2 du, and J(T, x) = Sx 2

16 Special Case II: Arrival Price Benchmark This specific model corresponds to the case X(τ) = 0 for all τ [0, T ]. Theorem The value function is the unique solution of [ J(t, x) = min inf J(t, x), γ σ(t)2 0 y x 2 x 2 + ν(t) [0,1] ] [J(t, (x y) + ) J(t, x)]ν t (dy) with t J(t, 0) = γ 0 σ(u) 2 X(u) 2 du, and J(T, x) = Sx 2

17 Special Case III: Stationary Approximation When (t, x) is far enough from the corners (0, 1) and (T, 0), J looks like a function of x X(t) (deviation from the benchmark). Stationarity assumption ν 1 (dt) = λdt for some constant λ > 0 ν t (dy) = ν(dy) for all t [0, T ]. σ(t) = σ for all t [0, T ] Look for an approximation of the form for some function w to be determined. J(t, x) α + βx + w(x X(t)) True in the Large Fill case (use the Lambert function)

18 The Discrete Case and Approximation Results The integer v denotes the quantity of shares (expressed as a number of lots) the broker has to sell by time T, Trades can only be in multiples of one lot. t x t looks like a staircase starting from x 0 = 1 and ending at x T = 0. In units of v lots, the measures ν t (dy) are supported by the grid {1/v, 2/v,, (v 1)/v, 1} The process x = (x t ) 0 t T. and the controls L = (L t ) 0 t T and m = (m t ) 0 t T take values in the grid I v := {0, 1/v,, (v 1)/v, 1}. The sets of admissible controls are defined accordingly. Identify functions ϕ on the grid I v with finite sequence (ϕ i ) 0 i v where ϕ i = ϕ(i/v). Denote by Iϕ the piecewise linear continuous function [0, 1] x [Iϕ](x) which coincides with ϕ on the grid I v and which is linear on each interval [i/v, (i + 1)/v]. (ϕ i ) 0 i v is said to be convex if Iϕ is convex For any integers v and v, and functions ϕ and ϕ on the grids I v and I v, we have: Iϕ Iϕ = sup [Iϕ](x) [Iϕ ](x) = x [0,1] sup x I v I v [Iϕ](x) [Iϕ ](x).

19 Characterization of the Solution The operators A and B become and [Aϕ] i (t) = S ϕ i (t) + ϕ i 1 (t), i = 1,, v, [Bϕ] i (t) = γ σ(t)2 2 [X(t) i/v]2 + ν(t) min 0 l i so the HJB QVI remains the same: v [ϕ i j l (t) ϕ i (t)]ν t (j/v) j=1 min [ [Aϕ] i (t), ϕ i (t) + [Bϕ] i (t) ] = 0, i = 1,, v. As before we have existence and uniqueness of a C 1 functions of t [0, T ] satisfying T ϕ i (t) = Si/v + min [Bϕ] i (u)du, i = 0, 1,, v. t 0 j i Interpreting the solution ϕ as a function on [0, T ] I v defined by ϕ(t, i/v) = ϕ i (t), since ϕ i (T ) = Si/v and: ϕ i (t) = min [Bϕ] j (t) 0 j i we get ϕ i (t) + [Bϕ] i (t) 0, i = 0, 1,, v and ϕ i (t) = max t ϕ j (t) 0 j i so that i ϕ i (t) is non-decreasing and [ ] ϕ i (t) = min min ϕ j (t), [Bϕ] i (t). 0 j i

20 Characterization of the Value Function in the Discrete Case Theorem The value function J of the problem can be identified to the sequence (J i ) 0 1 v of C 1 functions of t [0, T ] satisfying: J 0 (t) = T t γσ(u) 2 2 X(u) 2, J i (T ) = Si/v, i = 0, 1,, v [ t J i (t) = min t J i 1 (t), ν(t) v j=1 [ϕ (i j) l i (t) (t) ϕ i (t)]ν t (j) + γσ(t)2 2 [X(t) i/v] 2 ] where l i (t) = min{l; ϕ l (t) = min 0 j i ϕ j (t)}

21 Optimal Solution in the Discrete Case τi m = inf{t [0, T ]; J i (t) J i 1 (t) < S/v} τi l = inf{t [0, T ]; J i (t) J i 1 (t) < 0} τ m i Tx(i 1 2 ) < τ l i