Optimal Execution: II. Trade Optimal Execution

Transcription

1 Optimal Execution: II. Trade Optimal Execution René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Purdue June 21, 212

2 Optimal Execution Set-Up Goal: sell x > shares by time T > X = (X t ) t T execution strategy X t position (nb of shares held) at time t. X = x, X T = Assume X t absolutely continuous (differentiable) Pt mid-price (unaffected price), P t transaction price, I t price impact P t = P t + I t e.g. Linear Impact A-C model: I t = γ[x t X ] + λẋt Objective: Maximize form of revenue at time T Revenue R(X) from the execution strategy X R(X) = ( Ẋt)P t dt

3 Specific Challenges First generation: Price impact models (e.g. Almgren - Chriss) Risk Neutral framework (maximize ER(X)) versus utility criteria More complex portfolios (including options) Robustness and performance constraints (e.g. slippage or tracking market VWAP) Second generation: Simplified LOB models Simple liquidation problem performance constraints (e.g. slippage or tracking market VWAP) and using both market and limit orders

4 Optimal Execution Problem in A-C Model with C(X) = R(X) = = ( Ẋt)P t dt = x P + Ẋ t I t dt. Interpretation Ẋ t Pt dt Ẋ t I t dt X t d P t C(X) x P (initial) face value of the portfolio to liquidate X td P t volatility risk for selling according to X instead of immediately! C(X) execution costs due to market impact

5 Special Case: the Linear A-C Model R(X) = x P + X t d P t λ Ẋt 2 dt γ 2 x 2 Easy Case: Maximizing E[R(X)] E[R(X)] = x P γ 2 x 2 λe Ẋt 2 dt Jensen s inequality & constraints X = x and X T = imply Ẋ t = x T trade at a constant rate indpdt of volatility! Bertsimas - Lo (1998)

6 More Realistic Problem Almgren - Chriss propose to maximize E[R(X)] αvar[r(x)] (α risk aversion parameter late trades carry volatility risk) For DETERMINISTIC trading strategies X E[R(X)] αvar[r(x)] = x P γ ( ) ασ 2 x X t 2 + λẋ t 2 dt maximized by (standard variational calculus with constraints) Ẋt sinh κ(t t) ασ 2 = x for κ = sinh κt 2λ For RANDOM (adapted) trading strategies X, more difficult as Mean-Variance not amenable to dynamic programming

7 Maximizing Expected Utility Choose U : R R increasing concave and maximize E[U(R(X T )] Stochastic control formulation over a state process (X t, R t ) t T. v(t, x, r) = sup E[u(R T ) X t = x, R T = r] ξ Ξ(t,x) value function, where Ξ(t, x) is the set of admissible controls { } ξ = (ξ s ) t s T ; progressively measurable, ξs 2 ds <, ξ s ds = x t t X s = X ξ s = x and (choosing P t = σw t ) R s = R ξ s = R+σ s s X u dw u λ t t t ξ u du, s ξ 2 udu, Ẋ s = ξ s, X t = x dr s = σx s dw s λξ 2 s ds, R t = r

8 Finite Fuel Problem Non Standard Stochastic Control problem because of the constraints Still, one expects ξ s ds = x. For any admissible ξ, [v(t, X ξ t, Rξ t )] t T is a super-martingale For some admissible ξ, [v(t, X ξ t If v is smooth, and we set V t = v(t, X ξ t, Rξ t, R ξ t )] t T is a true martingale ), Itô s formula gives dv t = ( tv(t, X t, R t) + σ2 2 2 rr v(t, X t, R t) ) λξt 2 r v(t, X t, R t) ξ t xv(t, X t, R t) dt + σ xv(t, X t, R t)dw t

9 Hamilton-Jabobi-Bellman Equation One expects that v solves the HJB equation (nonlinear PDE) t v + σ2 2 2 xxv inf ξ R [ξ2 λ r v + ξ x v] = in some sense, with the (non-standard) terminal condition { U(r) if x = X v(t, x, r) = otherwise

10 Solution for CARA Exponential Utility For u(x) = e αx and κ as before v(t, x, r) = e αr+x 2 αλκ coth κ(t t) solves the HJB equation and the unique maximizer is given by the DETERMINISTIC ξt cosh κ(t t) = x κ sinh κt Schied-Schöneborn-Tehranchi (21) Optimal solution same as in Mean - Variance case Schied-Schöneborn-Tehranchi s trick shows that optimal trading strategy is generically deterministic for exponential utility Open problem for general utility function Partial results in infinite horizon versions

11 Shortcomings Optimal strategies are DETERMINISTIC do not react to price changes are time inconsistent are counter-intuitive in some cases Computations require solving nonlinear PDEs with singular terminal conditions

12 Recent Developments Gatheral - Schied (211), Schied (212) In the spirit of Almgren-Chriss mean-variance criterion, maximize [ ] E R(X) λ X t P t dt The solution happens to be ROBUST Pt can be a semi-martingale, optimal solution does not change

13 Recent Developments Almgren - Li (212), Hedging a large option position g(t, P t) price at time t of the option (from Black-Scholes theory) Revenue R(X) = g(t, P T T ) + X T PT P t Ẋ tdt λ Ẋt 2 dt Using Itô s formula and the fact that g solves a PDE, R(X) = R + [X t + xg(t, P T t)]dt λ Ẋt 2 dt R = x P +g(, P ) Introduce Y t = X t + xg(t, P t) for hedging correction { d P t = γẋtdt + σdwt dy t = [1 + γ 2 xxg(t, P t)]dt + σ 2 xxg(t, P t)dw t Minimize [ ( ) ] σ 2 E G( P T, Y T ) + 2 Y t 2 2 γẋtyt + λẋt dt Explicit solution in some cases (e.g. 2 xxg(t, x) = c, G quadratic)

14 Transient Price Impact Flexible price impact model Resilience function G : (, ) (, ) measurable bounded Admissible X = (X t ) t T cadlag, adapted, bounded variation Transaction price t P t = P t + G(t s) dx s Expected cost of strategy X given by x P + E[C(X)] where C(X) = G( t s )dx s dx t

15 Transient Price Impact: Some Results No Price Manipulation in the sense of Huberman - Stanzl (24) if G( ) positive definite Optimal strategies (if any) are deterministic Existence of an optimal X solvability of a Fredholm equation Exponential Resilience G(t) = e ρt dxt = x ( ) δ (dt) + ρdt + δ T (dt) ρt + 2 X purely discrete measure on [, T ] when G(t) = (1 ρt) + with ρ > dx t = x 2 [δ (dt) + δ T (dt)] if ρ < 1/T dx t = x n n+1 i= δ it /n(dt) if ρ < n/t for some integer n 1 Obizhaeva - Wang (25), Gatheral - Schied (211)

16 Optimal Execution in a LOB Model Unaffected price P t (e.g. P t = P + σw t ) Trader places only market sell orders Placing buy orders is not optimal Bid side of LOB given by a function f : R (, ) s.t. f (x)dx =. At any time t b a f (x)dx = bids available in the price range [ P t + a, P t + b] The shape function f does not depend upon t or P t Obizhaeva - Wang (26), Alfonsi - Schied - Schulz (211),Predoiu - Shaikhet - Shreve (211)

17 Optimal Execution in a LOB Model (cont.) Price Impact process D = (D t ) t T adapted, cadlag At time t a market order of size A moves the price from P t + D t to P t + D t where Dt D t f (x)dx = A Volume Impact Q t = F(D t ) where F (x) = x f (x )dx. LOB Resilience: Q t and D t decrease between trades, e.g. dq t = ρq t dt, for some ρ > At time t, a sell of size A will bring Dt Dt ( P t + x)f (x)dx = A P t + xdf (x) D t D t Qt = A P t + ψ(x)dx = A P t + Ψ(Q t) Ψ(Q t ) Q t if ψ = F 1 and Ψ(x) = x ψ(x )dx.

18 Stochastic Control Formulation Holding trajectories / Trading strategies { } Ξ(t, x) = (Ξ s ) t s T : càdlàg, adapted, bounded variation, Ξ t = x Ξ ac (t, x) = { (Ξ s ) t s T : Ξ s = x+ dx t dq t dr t s = dξ t = dξ t ρq tdt t } ξ r dr for (ξ s ) t s T bounded adapted = ρq tψ(q t)dt σξ tdw t

19 Value Function Approach State space process Z t = (X t, Q t, R t ), value function v(t, x, q, r) = v(t, z) = sup ξ Ξ(t,x) E[U(R T Ψ(Q T )] First properties U(r Ψ(q + r)) v(t, x, q, r) U(r Ψ(q)) v(t, x, q, r) = U(r Ψ(q + r)) for x = and t = T Functional approximation arguments imply v(t, x, q, r) = sup E[U(R T Ψ(Q T )] ξ Ξ(t,x) = sup E[U(R T Ψ(Q T )] ξ Ξ ac(t,x) = sup E[U(R T Ψ(Q T )] ξ Ξ d (t,x)

20 QVI Formulation As before Assume v smooth and apply Itô s formula to v(t, X t, Q t, R t ) v(t, X t, Q t, R t ) is a super-martingale for a typical ξ implies x v q v t v + σ2 2 x 2 2 rr v ρqψ(q) r v ρq q v QVI (Quasi Variational Inequality) instead of HJB nonlinear PDE min[ t v + σ2 2 x 2 2 rr v ρqψ(q) r v ρq q v, x v q v] = with terminal condition v(t, x, q, r) = U(r Ψ(x + q)) Existence and Uniqueness of a viscosity solution R.C. - H. Luo (212)

21 Special Cases Assuming a flat LOB f (x) = c and U(c) = x v(t, x, q, r) = r q2 (1 e 2ρs ) 2c (x + qe ρs) 2 c(2 + ρ(t t s) with s = (T t) inf{u [, T ]; (1 + ρ(t t u))qe ρu x} Still with f (x) = c but for a CARA utility U(x) = e αx v(t, x, q, r) = exp [ αr α2c ] (αcσ2 xx 2 +q 2 (1 e 2ρs )+ϕ(t+s)(x+qe ρs ) 2 where ϕ is the solution of the Ricatti s equation ϕ(t) = ρ 2 2ρ + αcσ 2 ϕ(t)2 + 2ραcσ2 2ρ + αcσ 2 ϕ(t) 2ραcσ2 2ρ + αcσ 2, ϕ(t ) = 1 and s = (T t) inf{u [, T ]; (αcσ 2 + ρϕ(t + u))x ρ(2 ϕ(t + u))qe ρu }