Optimal order execution

Transcription

1 Optimal order execution Jim Gatheral (including joint work with Alexander Schied and Alla Slynko) Thalesian Seminar, New York, June 14, 211

2 References [Almgren] Robert Almgren, Equity market impact, Risk July 25, [Almgren and Chriss] Robert Almgren and Neil Chriss, Optimal execution of portfolio transactions, Journal of Risk (21). [Alfonsi, Fruth and Schied] Aurélien Alfonsi, Antje Fruth and Alexander Schied, Optimal execution strategies in limit order books with general shape functions, Quantitative Finance 1(2) (21). [Alfonsi, Schied and Slynko] Aurélien Alfonsi, Alexander Schied and Alla Slynko, Order book resilience, price manipulation, and the positive portfolio problem, id= (29). [Forsyth et al.] P.A. Forsyth, J.S. Kennedy, S. T. Tse, and H. Windcliff, Optimal Trade Execution: A Mean - Quadratic-Variation Approach, University of Waterloo (211). [Gatheral] Jim Gatheral, No-dynamic-arbitrage and market impact, Quantitative Finance 1(7) (21).

3 References [Gatheral and Schied] Jim Gatheral and Alexander Schied, Optimal Trade Execution under Geometric Brownian Motion in the Almgren and Chriss Framework, International Journal of Theoretical and Applied Finance 14(3) (211). [Gatheral, Schied and Slynko] Jim Gatheral, Alexander Schied and Alla Slynko, Transient linear price impact and Fredholm integral equations, Mathematical Finance forthcoming (211). [Obizhaeva and Wang] Anna Obizhaeva and Jiang Wang, Optimal trading strategy and supply/demand dynamics MIT working paper (25). [Predoiu, Shaikhet and Shreve] Silviu Predoiu, Gennady Shaikhet and Steven Shreve, Optimal execution in a general one-sided limit-order book, SIAM Journal on Finance Mathematics (211). [Weiss] Alexander Weiss, Executing large orders in a microscopic market model, (21).

4 Overview of this talk Statement of the optimal execution problem Stochastic optimal control and the HJB equation The Almgren-Chriss framework and 21 model Statically and dynamically optimal strategies Model-dependence of optimality of the static solution The Obizhaeva and Wang model The Alfonsi and Schied model Price manipulation and existence of optimal strategies Transient linear price impact

5 Overview of execution algorithm design Typically, an execution algorithm has three layers: The macrotrader This highest level layer decides how to slice the order: when the algorithm should trade, in what size and for roughly how long. The microtrader Given a slice of the order to trade (a child order), this level decides whether to place market or limit orders and at what price level(s). The smart order router Given a limit or market order, which venue should this order be sent to? In this lecture, we are concerned with the highest level of the algorithm: How to slice the order.

6 Statement of the problem Given a model for the evolution of the stock price, we would like to find an optimal strategy for trading stock, the strategy that minimizes some cost function over all permissible strategies. We will specialize to the case of stock liquidation where the initial position x = X and the final position x T =. A static strategy is one determined in advance of trading. A dynamic strategy is one that depends on the state of the market during execution of the order, i.e. on the stock price. Delta-hedging is an example of a dynamic strategy. VWAP is an example of a static strategy. It will turn out, surprisingly, that in many models, a statically optimal strategy is also dynamically optimal.

7 Bellman s principle of optimality An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. (See Bellman, 1957, Chap. III.3.)

8 Stochastic optimal control Consider a cost functional of the form [ T ] J t = E h(t, y s, v s ) ds t where y s is a state vector, v s is a vector-valued control and the evolution of the system is determined by a stochastic differential equation (SDE): dy t = f (t, y t, v t ) dt + σ(t, y t, v t ) dz t The mathematical statement of Bellman s principle of optimality is that when the control v is optimal, [ t+ ] J t = E h(t, y s, v s ) ds + J t+ t (1)

9 Now, by Itô s Lemma, t+ J t+ = J t + = J t + t t+ t dj { } J t + Lv s J ds + (local) martingale where L v t is the infinitesimal generator of the Itô diffusion: L v t = 1 2 σ(t, y, v)2 y,y + f (t, y, v) y Substituting back into (1) gives [ t+ t+ { } ] J J t = J t + E h(t, y s, v s ) ds + t t t + Lv s J ds

10 The HJB equation Dividing by and taking the limit, we obtain the HJB equation: J t + min v G {Lv t J + h(t, y t, v)} =

11 Deterministic and stochastic optimal control In deterministic optimal control, the evolution of the state vector is deterministic. In stochastic optimal control, the evolution of the state vector is stochastic.

12 Almgren and Chriss [Almgren and Chriss] model market impact and slippage as follows. The stock price S t evolves as ds t = σ dz t and the price S t at which we transact is given by S t = S t + η v t where v t := ẋ t is the rate of trading. The state vector is y t = {S t, x t }. The components of the state vector evolve as ds t = σ dz t ; dx t = v t dt;

13 Cost of trading The risk-unadjusted cost of trading (with no penalty for risk) is given by C t = E t [ T The HJB equation becomes t ] [ T ] S s v s ds = E t (S s + η v s ) v s ds t C t σ2 C S,S + min v G { C x v t + (S t + η v t ) v t } = (2) and the optimal choice of v t (the first order condition) is v t = 1 2 η (C x S t )

14 Substituting back into (2) and defining C := C x S gives the equation for the cost function: ( C x ) 2 = 4 η C t (3) with boundary conditions C(T, y T ) = C(T, {S T, }) =. The solution of this equation is The optimal control is then C = η x 2 T t (4) v t = x C 2 η = x t T t It is optimal to liquidate stock at a constant rate vt independent of the stock price S t ; the static VWAP strategy is dynamically optimal.

15 The statically optimal strategy The statically optimal strategy v s is the one that minimizes the cost function [ T ] [ T ] T C = E S s v s ds = E (S s + η v s ) v s ds = η vs 2 ds again with v s = ẋ s. The Euler-Lagrange equation is then s v s = s,s x s = with boundary conditions x = X and x T = and the solution is obviously v t = X ( T ; x t = X 1 t ) T

16 Adding a risk term [Almgren and Chriss] add a risk term that penalizes the variance of the trading cost. [ T ] T Var[C] = Var x t ds t = σ 2 xt 2 dt The expected risk-adjusted cost of trading is then given by C = η for some price of risk λ. T ẋ 2 t dt + λ σ 2 T x 2 t dt Note the analogies to physics and portfolio theory. The first term looks like kinetic energy and the second term like potential energy. The expression looks like the objective in mean-variance portfolio optimization.

17 The Euler-Lagrange equation becomes with ẍ κ 2 x = κ 2 = λ σ2 η The solution is a linear combination of terms of the form e ±κt that satisfies the boundary conditions x = X, x T =. The solution is then sinh κ(t t) x(t) = X sinh κt Once again, it turns out that the statically optimal solution is dynamically optimal.

18 What happens if we change the risk term? Suppose we penalize average VaR instead of variance. This choice of risk term has the particular benefit of being linear in the position size. The expected risk-adjusted cost of trading is then given by C = η T ẋ 2 t dt + λ σ T x t dt for some price of risk λ. The Euler-Lagrange equation becomes ẍ A = with A = λ σ 2 η

19 The solution is a quadratic of the form A t 2 /2 + B t + C that satisfies the boundary conditions x = X, x T =. The solution is then ( x(t) = X A T ) ( 2 t 1 t ) (5) T In contrast to the previous case where the cost function is monotonic decreasing in the trading rate and the optimal choice of liquidation time is, in this case, we can compute an optimal liquidation time. When T is optimal, we have C T ẋ T + A x T = from which we deduce that ẋ T =.

20 Substituting into (5) and solving for the optimal time T gives 2 X T = A With this optimal choice T = T, the optimal strategy becomes x(t) = X ( 1 t ) 2 T u(t) = ẋ(t) = 2 X ( 1 t ) T One can verify that the static strategy is dynamically optimal, independent of the stock price.

21 An observation from Predoiu, Shaikhet and Shreve Suppose the cost associated with a strategy depends on the stock price only through the term T S t dx t. with S t a martingale. Integration by parts gives [ T ] [ T ] E S t dx t = E S T x T S x x t ds t = S X which is independent of the trading strategy and we may proceed as if S t =. Quote from [Predoiu, Shaikhet and Shreve]...there is no longer a source of randomness in the problem. Consequently, without loss of generality we may restrict the search for an optimal strategy to nonrandom functions of time.

22 Corollary This observation enables us to easily determine whether or not a statically optimal strategy will be dynamically optimal. In particular, if the price process is of the form S t = S + impact of prior trading + noise, and if there is no risk term, a statically optimal strategy will be dynamically optimal. If there is a risk term independent of the current stock price, a statically optimal strategy will again be dynamically optimal. It follows that the statically optimal strategy is dynamically optimal in the following models: Almgren and Chriss Almgren (25) Obizhaeva and Wang

23 ABM vs GBM One of the reasons that the statically optimal strategy is dynamically optimal is that the stock price process is assumed to be arithmetic Brownian motion (ABM). If for example geometric Brownian motion (GBM) is assumed, the optimal strategy depends on the stock price. How dependent is the optimal strategy on dynamical assumptions for the underlying stock price process?

24 Forsyth et al. [Forsyth et al.] solve the HJB equation numerically under geometric Brownian motion with variance as the risk term so that the (random) cost is given by C = η T ẋ 2 t dt + λ σ 2 T S 2 t x 2 t dt The efficient frontier is found to be virtually identical to the frontier computed in the arithmetic Brownian motion case. The problem of finding the optimal strategy is ill-posed; many strategies lead to almost the same value of the cost function. It is optimal to trade faster when the stock price is high so as to reduce variance. The optimal strategy is aggressive-in-the-money when selling stock and passive-in-the-money when buying stock.

25 Gatheral and Schied [Gatheral and Schied] take time-averaged VaR as the risk term so that [ T T ] C(T, X, S ) = inf E vt 2 dt + λ S t x t dt, (6) v G where G is the set of admissible strategies. C(T, X, S) should then satisfy the following Hamilton-Jacobi-Bellman PDE: with initial condition C T = 1 2 σ2 S 2 C SS + λ S X + inf v R (v 2 v C X ). (7) lim C(T, X, S) = T { if X =, + if X. (8)

26 The optimal strategy under GBM Solving the HJB equation explicitly gives Theorem The unique optimal trade execution strategy attaining the infimum in (6) is ( T t ) [ xt = X λt t ] S s ds (9) T 4 Moreover, the value of the minimization problem in (6) is given by [ T { C = E (ẋ t ) 2 + λx } ] t S t dt = X 2 T λ T X S λ2 8 σ 6 S 2 ( e σ2 T 1 σ 2 T 1 2 σ4 T 2).

27 The optimal strategy under ABM If we assume ABM, S t = S (1 + σw t ), instead of GBM, the risk term becomes T λ S x t dt. (1) As we already showed, the optimal strategy under ABM is just the static version of the dynamic strategy (9) obtained by replacing S t with its expectation E[S t ] = S, a strategy qualitatively similar to the Almgren-Chriss optimal strategy.

28 Comparing optimal strategies under ABM and GBM As before, define the characteristic timescale 4 X T = λ S and choose the liquidation time T to be T. With T = T, the optimal trading rate under ABM becomes v A (t) = x t T t + X T 2 (T t) = 2 X T and the optimal trading rate under GBM becomes ( t ) 1 T (11) v G (t) = x t T t + X T 2 S t S (T t). (12)

29 Comparing optimal strategies under ABM and GBM In the following slide: The upper plots show rising and falling stock price scenarios respectively; the trading period is 2 days and daily volatility is 4%. The lower plots show the corresponding optimal trading rates from (11) and (12); the optimal trading rate under ABM is in orange and the optimal trading rate under GBM is in blue. Even with such extreme parameters and correspondingly extreme changes in stock price, the differences in optimal trading rates are minimal.

30 Stock price Stock price Trading rate Trading rate Time (days) Time (days)

31 Remarks For reasonable values of σ 2 T 1, there is almost no difference in expected costs and risks between the optimal strategies under ABM and GBM assumptions. Intuitively, although the optimal strategy is stock price-dependent under GBM assumptions but not under ABM assumptions, when σ 2 T 1, the difference in optimal frontiers is tiny because the stock-price S t cannot diffuse very far away from S in the short time available. Equivalently, as in the plots, there can only be a small difference in optimal trading rates under the two assumptions.

32 Practical comments It s not clear what the price of risk should be. More often that not, a trader wishes to complete an execution before some final time and otherwise just wants to minimize expected execution cost. In Almgren-Chriss style models, the optimal strategy is just VWAP (trading at constant rate). From now on, we will drop the risk term and the dynamics we will consider will ensure that the statically optimal solution is dynamically optimal.

33 Obizhaeva Wang order book process f(d t ) f(d t+ ) Order density f(x) E t E t+ E t Price level D t D t+ When a trade of size ξ is placed at time t, E t E t+ = E t + ξ D t = η E t D t+ = η E t+ = η (E t + ξ)

34 When the trading policy is statically optimal, the Euler-Lagrange equation applies: δc = t δu t where u t = ẋ t. Functionally differentiating C with respect to u t gives δc t = δu t T u s e ρ (t s) ds + t u s e ρ (s t) ds = A (13) for some constant A. Equation (13) may be rewritten as T u s e ρ t s ds = A which is a Fredholm integral equation of the first kind (see [Gatheral, Schied and Slynko]).

35 Now substitute into (13) to obtain u s = δ(s) + ρ + δ(s T ) δc δu t = e ρ t + ( 1 e ρ t) = 1 The optimal strategy consists of a block trade at time t =, continuous trading at the rate ρ over the interval (, T ) and another block trade at time t = T.

36 Consider the volume impact process E t. The initial block-trade causes = E E + = 1 According to the assumptions of the model, the volume impact process reverts exponentially so E t = E + e ρ t + ρ t e ρ (t s) ds = 1 i.e. the volume impact process is constant when the trading strategy is optimal.

37 The model of Alfonsi, Fruth and Schied Alfonsi, Fruth and Schied [Alfonsi, Fruth and Schied] consider the following (AS) model of the order book: There is a continuous (in general nonlinear) density of orders f (x) above some martingale ask price A t. The cumulative density of orders up to price level x is given by F (x) := x f (y) dy Executions eat into the order book (i.e. executions are with market orders). A purchase of ξ shares at time t causes the ask price to increase from A t + D t to A t + D t+ with ξ = Dt+ D t f (x) dx = F (D t+ ) F (D t )

38 Schematic of the model f(d t ) f(d t+ ) Order density f(x) E t E t+ E t Price level D t D t+ When a trade of size ξ is placed at time t, E t E t+ = E t + ξ D t = F 1 (E t ) D t+ = F 1 (E t+ ) = F 1 (E t + ξ)

39 Optimal liquidation strategy in the AS model The cost of trade execution in the AS model is given by: C = T v t F 1 (E t ) dt + t T [H(E t+ ) H(E t )] (14) where E t = t is the volume impact process and H(x) = u s e ρ (t s) ds x F 1 (x) dx gives the cost of executing an instantaneous block trade of size x.

40 Consider the ansatz u t = ξ δ(t) + ξ ρ + ξ T δ(t t). For t (, T ), we have E t = E = ξ, a constant. With this choice of u t, we would have C(X ) = F 1 (ξ ) T v t dt + [H(E +) H(E )] + [H(E T ) H(E T )] = F 1 (ξ ) ξ ρ T + H(ξ ) + [H(ξ + ξ T ) H(ξ )] = F 1 (ξ ) ξ ρ T + H(X ρ ξ T ) Differentiating this last expression gives us the condition satisfied by the optimal choice of ξ : or equivalently F 1 (X ρ ξ T ) = F 1 (ξ ) + F 1 (ξ ) ξ F 1 (ξ + ξ T ) = F 1 (ξ ) + F 1 (ξ ) ξ

41 Functionally differentiating C with respect to u t gives δc T = F 1 (E t ) + δu t = F 1 (E t ) + t T t u s F 1 (E s ) δe s δu t ds u s F 1 (E s ) e ρ (s t) ds (15) The first term in (15) represents the marginal cost of new quantity at time t and the second term represents the marginal extra cost of future trading. With our ansatz, and a careful limiting argument, we obtain δc δu t = F 1 (ξ ) + ξ F 1 (ξ ) [1 e ρ (T t)] +e ρ (T t) [ F 1 (ξ T + ξ ) F 1 (ξ ) ]

42 Imposing our earlier condition on ξ T gives δc = F 1 (ξ ) + ξ F 1 ρ (T (ξ ) [1 e t)] δu t +e ρ (T t) ξ F 1 (ξ ) = F 1 (ξ ) + ξ F 1 (ξ ) which is constant, demonstrating (static) optimality. Example With F 1 (x) = x, ξ + ξ T = F 1 (ξ +ξ T ) = F 1 (ξ )+F 1 (ξ ) ξ = ξ ξ which has the solution ξ T = 5 4 ξ.

43 Generalization Alexander Weiss [Weiss] and then Predoiu, Shaikhet and Shreve [Predoiu, Shaikhet and Shreve] have shown that the bucket-shaped strategy is optimal under more general conditions than exponential resiliency. Specifically, if resiliency is a function of E t (or equivalently D t ) only, the optimal strategy has a block trades at inception and completion and continuous trading at a constant rate in-between.

44 Optimality and price manipulation For all of the models considered so far, there was an optimal strategy. The optimal strategy always involved trades of the same sign. So no sells in a buy program, no buys in a sell program. It turns out (see [Gatheral]) that we can write down models for which price manipulation is possible. In such cases, a round-trip trade can generate cash on average. You would want to repeat such a trade over and over. There would be no optimal strategy.

45 Linear transient market impact The price process assumed in [Gatheral] is S t = S + t f (v s ) G(t s) ds + noise In [Gatheral, Schied and Slynko], this model is on the one hand extended to explicitly include discrete optimal strategies and on the other hand restricted to the case of linear market impact. When the admissible strategy X is used, the price S t is given by S t = St + G(t s) dx s, (16) {s<t} and the expected cost of liquidation is given by C(X ) := 1 G( t s ) dx s dx t. (17) 2

46 Condition for no price manipulation Definition (Huberman and Stanzl) A round trip is an admissible strategy with X =. A price manipulation strategy is a round trip with strictly negative expected costs. Proposition (Bochner) C(X ) for all admissible strategies X if and only if G( ) can be represented as the Fourier transform of a positive finite Borel measure µ on R, i.e., G( x ) = e ixz µ(dz).

47 First order condition Theorem Suppose that G is positive definite. Then X minimizes C( ) if and only if there is a constant λ such that X solves the generalized Fredholm integral equation G( t s ) dx s = λ for all t T. (18) In this case, C(X ) = 1 2 λ x. In particular, λ must be nonzero as soon as G is strictly positive definite and x.

48 Transaction-triggered price manipulation Definition (Alfonsi, Schied, Slynko (29)) A market impact model admits transaction-triggered price manipulation if the expected costs of a sell (buy) program can be decreased by intermediate buy (sell) trades. As discussed in [Alfonsi, Schied and Slynko], transaction-triggered price manipulation can be regarded as an additional model irregularity that should be excluded. Transaction-triggered price manipulation can exist in models that do not admit standard price manipulation in the sense of Huberman and Stanzl definition.

49 Condition for no transaction-triggered price manipulation Theorem Suppose that the decay kernel G( ) is convex, satisfies 1 G(t) dt < and that the set of admissible strategies is nonempty. Then there exists a unique admissible optimal strategy X. Moreover, Xt is a monotone function of t, and so there is no transaction-triggered price manipulation. Remark If G is not convex in a neighborhood of zero, there is transaction-triggered price manipulation.

50 An instructive example We solve a discretized version of the Fredholm equation (with 512 time points) for two similar decay kernels: G 1 (τ) = 1 (1 + t) 2 ; G 2(τ) = t 2 G 1 ( ) is convex, but G 2 ( ) is concave near τ = so there should be a unique optimal strategy with G 1 ( ) as a choice of kernel but there should be transaction-triggered price manipulation with G 2 ( ) as the choice of decay kernel.

51 G 1 ( ) is convex, but G 2 ( ) is concave near τ = so there should be a unique optimal strategy with G 1 ( ) as a choice of kernel but there should be transaction-triggered price manipulation with G 2 ( ) as the choice of decay kernel.

52 Schematic of numerical solutions of Fredholm equation 1 G 1 (τ) = 1 (1+t) 2 G 2 (τ) = 1 1+t In the left hand figure, we observe block trades at t = and t = 1 with continuous (nonconstant) trading in (, 1). In the right hand figure, we see numerical evidence that the optimal strategy does not exist.

53 Numerical solution of Fredholm equation Consider the Fredholm equation G( t s ) dxs = const. for all t [, T ]. (19) Define t i := i T N One way to discretize this integral equation is as follows: where v j = X tj X tj 1 N G ij v j = const.. (2) j=1 and G ij = ti t i 1 tj t j 1 G( t s ) ds dt

54 Numerical solution of Fredholm equation This discretized equation (2) is a matrix equation of the form G.v = c which can be solved numerically as v = G 1.c

55 Nonlinear Fredholm equation If the optimal strategy is differentiable, the following nonlinear version of the Fredholm equation holds: where T f t (v s ) G( t s ) ds = c t [, T ] (21) { f (vs ) if s t f t (v s ) = v s f (v t ) if s > t In a recent thesis, Ngoc-Minh Dang shows how to solve the discretized version of this equation using matrix inversion and Newton iteration with the approximation f i (v (m) j ) f i (v (m 1) j ) + ( v (m) j v (m 1) j ) f i (v (m 1) j )

56 Some remarks on the nonlinear case Dang considers the case where f (v) = v 1.1. In this case, the optimal strategy is differentiable. In the more practical case where f ( ) is concave, the optimal strategy appears to be not differentiable. In this case, not surprisingly, the numerical solution fails.

57 Now we give some examples of the optimal strategy under linear transient market impact with choices of kernel that preclude transaction-triggered price manipulation.

58 Example I: Linear market impact with exponential decay G(τ) = e ρ τ and the optimal strategy u(s) solves T u(s)e ρ t s ds = const. We already derived the solution which is u(s) = A {δ(t) + ρ + δ(t t)} The normalizing factor A is given by T u(t) dt = X = A (2 + ρ T ) The optimal strategy consists of block trades at t = and t = T and continuous trading at the constant rate ρ between these two times.

59 Schematic of optimal strategy The optimal strategy with ρ =.1 and T = 1 u(s) Time s

60 Example II: Linear market impact with power-law decay G(τ) = τ γ and the optimal strategy u(s) solves T u(s) ds = const. t s γ The solution is A u(s) = [s (T s)] (1 γ)/2 The normalizing factor A is given by T u(t) dt = X = A π ( T 2 ( ) ) γ Γ 1+γ 2 ) Γ ( 1 + γ 2 The optimal strategy is absolutely continuous with no block trades. However, it is singular at t = and t = T.

61 Schematic of optimal strategy The red line is a plot of the optimal strategy with T = 1 and γ = 1/2. u(s) s

62 Example III: Linear market impact with linear decay G(τ) = (1 ρ τ) + and the optimal strategy u(s) solves T u(s) (1 ρ t s ) + ds = const. Let N := ρ T, the largest integer less than or equal to ρ T. Then u(s) = A N i= ( 1 i ) { ( δ s i ) ( + δ T s i )} N + 1 ρ ρ The normalizing factor A is given by T u(t) dt = X = A N 2 i= ( 1 i ) = A (2 + N) N + 1 The optimal strategy consists only of block trades with no trading between blocks.

63 Schematic of optimal strategy Positions and relative sizes of the block trades in the optimal strategy with ρ = 1 and T = 5.2 (so N = ρ T = 5). u(s) Time s

64 Summary I The optimal trading strategy depends on the model. For Almgren-Chriss style models, if the price of risk is zero, the minimal cost strategy is VWAP. In Alfonsi-Schied style models with resiliency that depends only on the current spread, the minimal cost strategy is to trade a block at inception, a block at completion and at a constant rate in between. We exhibited other models for which the optimal strategy is more interesting. In most conventional models, the optimal liquidation strategy is independent of the stock price. However, for each such model, it is straightforward to specify a similar model in which the optimal strategy does depend on the stock price. With reasonable parameters and timescales, the optimal strategy is close to the static one.

65 Summary II In some models, price manipulation is possible and there is no optimal strategy. It turns out that we also need to exclude transaction-triggered price manipulation. We presented example of models for which price manipulation is possible. In the case of linear transient impact, we provided conditions under which transaction-triggered price manipulation is precluded.