Optimal Order Placement

Transcription

1 Optimal Order Placement Peter Bank joint work with Antje Fruth OMI Colloquium Oxford-Man-Institute, October 16, 2012

2 Optimal order execution Broker is asked to do a transaction of a significant fraction of the average daily trading volume over a certain period. Key challenge: Law requires optimal execution! How to guarantee this? Micro level: What order type (e.g., limit, market, cancellation... ) of what size to send to which kind of venue (lit, dark) where (New York, Chicago) and when? high-dimensional optimization problem which has to be solved fast (milliseconds) and frequently, taking into account market microstructure effects: not topic of this talk Macro level: How to optimally split a large transaction into smaller pieces to be sent to the micro trader for execution over the alotted trading period? dynamic optimal control problem to be discussed here

3 Market features to be addressed market depth: price reaction as a function of order size market resilience: permanent vs. temporary impact market tightness: spread market risk: fluctuations in portfolio value Very small selection of papers closest to this talk: Almgren & Chriss, Schied et al.,... : infinitely tight (no spread, unidirectional trades), finite depth ( quadratic costs of rates), infinitely resilient (local impact), mean-variance or utility Obizhaeva & Wang, Alfonsi et al., Predoiu et al.: finite depth determined by general, but constant shape of limit order book, finite resilience (half-life of market order impact, resilience function), market risk addressed at best partially

4 Our framework: deterministic case Obizhaeva & Wang s block shaped order book model with deterministic, but time-varying depth and resilience: Fruth, Schöneborn, Urusuov (2011), Alfonsi, Acevedo (2012) cumulative orders issued so far: X = (X t ) t 0 right-cont., increasing with X 0 0, X = x market impact of orders so far: η(x ) = (η t (X )) t 0 solution to dη t = dx t δ t r t η t dt market depth: δ = (δ t ) t 0 0 u.s.c., 0 < sup δ < resilience: r = (r t ) t 0 > 0 locally Lebesgue-integrable Optimization problem: Find an order schedule with minimal execution costs: ( C(X ) = η t (X ) + ) tx dx t min. 2δ t [0, )

5 Main result Theorem ( ) t Let ρ t exp 0 r s ds and λ t δ t /ρ t, and consider L λ u λ t t = inf, t 0. u>t sup v u λ v /ρ u sup v t λ v /ρ t Then the optimal schedule is to send orders while the resulting market impact η is no larger than yl /ρ, i.e., Xt δ s = d sup {(yl ρ u) η 0 }, t 0 s 0 u s [0,t] where y > 0 is chosen such that X = x, provided such a y can be found. Otherwise, inf X X C(X ) = 0, i.e., there is no solution.

6 Comments approach via dynamic programming possible, at least for differentiable data: HJB-equation variational characterization of optimality possible if data differentiable and good guess of general structure of optimal policy available; Euler-Lagrange equations; see thesis by Antje Fruth Pontryagin s maximum principle; FBDE our approach: convex optimization explicit result conveying trade-off between future and present market depth and resilience alternative description emerges in sketch of proof minimal assumptions illustrations later

7 Step 1: Change of control variable Proposition Y t η 0 + [0,t] dx s λ s and X t ( t where λ t δ t /ρ t, ρ t exp 0 r s ds [0,t] λ s dy s, t 0, ), defines mappings from X {X Dom(C) : X 0 = 0, X = x} into Y {Y Dom(K) : Y 0 = η 0, λ s dy s = x} [0, ) and vice versa such that for κ λ/ρ = δ/ρ 2 : C(X ) = K(Y ) 1 κ t d(yt 2 ). 2 [0, ) Hence: It suffices to minimize K(Y ) subject to Y Y.

8 Step 2: Convexification of the problem Proposition The functional K(Y ) 1 2 [0, ) κ t d(y 2 t ). is (strictly) convex in Y if and only if κ is (strictly) decreasing.

9 Step 2: Convexification of the problem Proposition The functional K(Y ) 1 2 [0, ) κ t d(y 2 t ). is (strictly) convex in Y if and only if κ is (strictly) decreasing. Lemma For any Y Y we can find a Ỹ Y such that Ỹ Y, {dỹ > 0} {d λ < 0}, K(Y ) K(Ỹ ) = K(Ỹ ) where K(Y ) 1 2 [0, ) κ t d(y 2 t ), λt sup λ u, κ t λ t /ρ t. u t

10 Step 2: Convexification of the problem (ctd.) Proposition The functional K(Y ) 1 2 [0, ) κ t d(y 2 t ) where κ t λ t /ρ t and λ t sup u t λ u is convex in Y and a minimizer Y in { } Ỹ Y Dom( K) : Y 0 = η 0, λ t dy t = x [0, ) is also a minimizer over the set Y, over which it minimizes the original functional K(Y ) as well provided {dy > 0} {λ = λ}. Hence: Need to solve first order conditions of a convex problem.

11 Step 3: First order conditions Proposition Y minimizes K over its class Ỹ where x λ [0, ) t dyt > 0 if and only if there is a constant y > 0 such that Yu d κ u y λ t for t 0 with = whenever dyt > 0. [t, )

12 is right-cont., incr., and satisfies the first-order conditions. Step 3: First order conditions Proposition Y minimizes K over its class Ỹ where x λ [0, ) t dyt > 0 if and only if there is a constant y > 0 such that Yu d κ u y λ t for t 0 with = whenever dyt > 0. [t, ) Time-change For 0 k κ 0 let τ k inf{t 0 : κ t k} Λk kρ τk with Λ 0 0 Then the concave envelope Λ of Λ over [0, κ 0 ] has a left-continuous decreasing density Λ and Y t (y Λ κt ) η 0, t 0,

13 Putting it all together... Theorem In case Λ L 2 ( κ 0 0 ( Λ k ) 2 dk) 1 2 < we can choose y > 0 uniquely such that } Xt λ 0 (y Λ κ0 η 0 ) + + λ s d {(y Λ κs ) η 0, t 0, (0,t] increases from X 0 0 to X = x; this X X is an optimal order schedule. In the special case where η 0 = 0, y = x/ Λ 2 L 2 and the minimal costs are given by C(X ) = x 2 /(2 Λ 2 L 2 ). If, by contrast, Λ L 2 = then inf X X C(X ) = 0 and there is no optimal order schedule. Remark: The quantity L t of our first theorem coincides, after a time-change, with the initial slope of the concave envelope for the restriction of Λ k = kρ k to the interval [0, κ t ], t 0.

14 Illustration I Obizhaeva & Wang model has finite horizon T > 0, constant depth δ and resilience r: λ t = λ t = δe rt 1 [0,T ] (t), κ t = κ t = δe 2rt 1 [0,T ] (t), Λ k = δk1 [δe 2rT,δ](k), Λk = δk (ke rt ), Λ k = 1 δ 2 k 1 (δe 2rT,δ](k) + e rt 1 [0,δe 2rT ](k) Λ κt = 1 2 ert = L t for t < T, Λ κt = e rt = L T Yt = η 0 for t < τ log + (2η 0 /y)/r, Yt = y 2 ert for τ t < T, YT = yert Xt = ( y 2 η 0) + δ + y 2 δ(t τ)+ for t < T, XT = y 2 δ(t τ + 1)

15 Illustration I An optimal order placement strategy in the Obizhaeva & Wang-setting: Κ Κ Λ Λ X 0 T Figure: Optimal order schedule X (black) for constant market depth δ (blue), its resilience adjustment λ = λ (red), κ = κ (green) over a finite horizon T.

16 Illustration II Fluctuating market depth, constant resilience: Λ t 0 Λ t 0 0 t 1 T Figure: The market depth δ (blue)

17 Illustration II Fluctuating market depth, constant resilience: Λ t 0 Λ t 0 Λ 0 t 1 T Figure: The market depth δ (blue), its resilience adjustment λ (purple)

18 Illustration II Fluctuating market depth, constant resilience: Λ Λ t 0 Λ t 0 Λ 0 t 1 T Figure: The market depth δ (blue), its resilience adjustment λ (purple), its decreasing envelope λ (red)

19 Illustration II Fluctuating market depth, constant resilience: Λ Λ t 0 Κ Λ t 0 Λ 0 t 1 T Figure: The market depth δ (blue), its resilience adjustment λ (purple), its decreasing envelope λ (red), κ (green)

20 Illustration II Fluctuating market depth, constant resilience: Λ Λ t 0 Κ Λ t 0 Λ X 0 t 1 T Figure: The market depth δ (blue), its resilience adjustment λ (purple), its decreasing envelope λ (red), κ (green), and the optimal schedule X (black)

21 Illustration II (ctd) Fluctuating market depth, constant resilience: Λ t 0 Λ t 0 0 Figure: The time-changed decreasing envelope of risk-adjusted market depth Λ (red)

22 Illustration II (ctd) Fluctuating market depth, constant resilience: Λ t 0 Λ t 0 0 Figure: The time-changed decreasing envelope of risk-adjusted market depth Λ (red) and the corresponding concave envelope Λ (orange)

23 Illustration II (ctd) Fluctuating market depth, constant resilience: Λ t 0 Λ t 0 0 Figure: The time-changed decreasing envelope of risk-adjusted market depth Λ (red) and the corresponding concave envelope Λ (orange) with its density Λ (black)

24 Illustration III Fluctuating market depth, very strong resilience: Λ t 0 Λ t 0 X 0 t 1 T Figure: The market depth (blue) and the optimal schedule (black) with strong resilience

25 Illustration III (ctd) Fluctuating market depth, with ever lower resilience: Λ t 0 Λ t 0 X 0 t 1 T Figure: The market depth (blue) and the optimal schedule (black) with ever lower resilience

26 Illustration III (ctd) Fluctuating market depth, with ever lower resilience: Λ t t 00 X 0 t 1 T Figure: The market depth (blue) and the optimal schedule (black) with ever lower resilience

27 Illustration III (ctd) Fluctuating market depth, with ever lower resilience: Λ t t 00 X 0 t 1 T Figure: The market depth (blue) and the optimal schedule (black) with ever lower resilience

28 Illustration III (ctd) Fluctuating market depth, with ever lower resilience: Λ t t 00 X 0 t 1 T Figure: The market depth (blue) and the optimal schedule (black) with ever lower resilience

29 Illustration III (ctd) Fluctuating market depth, with ever lower resilience: Λ t t 00 X 0 t 1 T Figure: The market depth (blue) and the optimal schedule (black) with ever lower resilience

30 Illustration IV Fluctuating market depth, no resilience: Λ X 0 t 0 t 1 T Figure: The market depth (blue) and the optimal schedule (black) Proposition If r 0 then any order schedule X X with {dx > 0} arg max δ is optimal.

31 Our framework: stochastic case Obizhaeva & Wang s block shaped order book model with stochastically varying depth and resilience: Fruth (2011) orders: X = (X t ) 0 t T X (x) right-cont, increasing, adapted with X 0 0, X T = x market impact: η(x ) = (η t (X )) 0 t T solution to dη t = dx t δ t r t η t dt market depth: δ = (δ t ) 0 t T cont., adapted, bounded, > 0 resilience: r = (r t ) 0 t T Lebesgue-integrable, adapted, > 0 Optimization problem: Find an order schedule with minimal expected execution costs: ( C(X ) = E η t (X ) + ) tx dx t min s.t. X X (x). 2δ t [0,T ]

32 Step 1: Change of control variable Proposition With λ δ/ρ, we get the one-to-one correspondence dx s Y t η 0 + and, resp., X t λ s dy s, 0 t T, λ s [0,t] [0,t] between X X and Y Y where { Y (Y t ) t [0,T ] η 0 : RC., incr., adapted such that for κ λ/ρ = δ/ρ 2 : C(X ) = K(Y ) 1 2 E [0,T ] [0,T ] κ t d(y 2 t ). λ t dy t = x } Hence: It suffices to minimize K(Y ) subject to Y Y (x).

33 Step 2: Convexification of the problem Proposition The functional K(Y ) 1 2 E [0,T ] κ t d(y 2 t ). is (strictly) convex in Y if and only if κ > 0 is a (strict) supermartingale.

34 Step 2: Convexification of the problem Proposition The functional K(Y ) 1 2 E [0,T ] κ t d(y 2 t ). is (strictly) convex in Y if and only if κ > 0 is a (strict) supermartingale. Unfortunately: We cannot assume this without loss of generality because of a puzzling counter-example; see Fruth (2011).

35 Step 2: Convexification of the problem Proposition The functional K(Y ) 1 2 E [0,T ] κ t d(y 2 t ). is (strictly) convex in Y if and only if κ > 0 is a (strict) supermartingale. Unfortunately: We cannot assume this without loss of generality because of a puzzling counter-example; see Fruth (2011). Hence: We assume that resilience is strong enough to ensure the supermartingale property of κ = δ/ρ 2 = δ/ exp ( 2. 0 r s ds ).

36 Step 3: First order conditions Proposition Y minimizes K over its class Y (x), x [0,T ] λ t dyt > 0, if and only if {dy > 0} { K(Y ) = λm(y )} where t K(Y ) E [ ] Yu dκ u F t [t,t ] (0 t T ) is the gradient of K at Y and where M(Y ) denotes the martingale part of the lower Snell-envelope S t (Y ) = M t (Y )+A t (Y ) ess inf S t E [ S K(Y )/λ S F t ] (0 t T ).

37 Step 3: First order conditions Proposition Y minimizes K over its class Y (x), x [0,T ] λ t dyt > 0, if and only if {dy > 0} { K(Y ) = λm(y )} where t K(Y ) E [ ] Yu dκ u F t [t,t ] (0 t T ) is the gradient of K at Y and where M(Y ) denotes the martingale part of the lower Snell-envelope S t (Y ) = M t (Y )+A t (Y ) ess inf S t E [ S K(Y )/λ S F t ] (0 t T ). Problem: How to construct such a Y?

38 Step 4: A family of representation problems Proposition Let M > 0 be a martingale and suppose L M is an optional process with u.r.c. paths satisfying [ ] E sup L M t d( κ u ) F S = λ S M S [S,T ] t [S,u] for any stopping time S T. Then any Y M of the form Yt M = η 0 sup s [0,t] L M s, t [0, T ], satisfies the first order condition {dy M > 0} { K(Y M ) = λm(y M )} and M conincides with the martingale part M(Y M ) of the Snell envelope associated with K(Y M )/λ.

39 Step 4: A family of representation problems Proposition Let M > 0 be a martingale and suppose L M is an optional process with u.r.c. paths satisfying [ ] E sup L M t d( κ u ) F S = λ S M S [S,T ] t [S,u] for any stopping time S T. Then any Y M of the form Yt M = η 0 sup s [0,t] L M s, t [0, T ], satisfies the first order condition {dy M > 0} { K(Y M ) = λm(y M )} and M conincides with the martingale part M(Y M ) of the Snell envelope associated with K(Y M )/λ. Current work: How to construct M such that [0,T ] λ dy M = x?

40 Conclusion optimal order schedule with time-varying depth and resilience execution costs not convex in general, but in the deterministic case we can find a convex majorant whose solutions minimize original costs in determinstic case first order conditions solved via time-change and concave envelopes explicit construction of solution for general specifications of depth and resilience unde minimal assumptions Obizhaeva & Wang s scheduling shape not generic stochastic case: convexity has to be assumed: example in Fruth (2011) with counterintuitive structure of optimal schedule granted convexity: first order conditions involving Snell envelope of cost gradient, solved by representation problem

41 Conclusion optimal order schedule with time-varying depth and resilience execution costs not convex in general, but in the deterministic case we can find a convex majorant whose solutions minimize original costs in determinstic case first order conditions solved via time-change and concave envelopes explicit construction of solution for general specifications of depth and resilience unde minimal assumptions Obizhaeva & Wang s scheduling shape not generic stochastic case: convexity has to be assumed: example in Fruth (2011) with counterintuitive structure of optimal schedule granted convexity: first order conditions involving Snell envelope of cost gradient, solved by representation problem Thank you very much!