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Optimal execution
The problem
Optimal execution of a large block of shares has long been a major problem for cash traders and brokers. Indeed, a trader wishing to buy or sell a large amount of shares necessarily faces adverse price movements – generally called “market impact” in the literature – as a direct consequence of her trades. To minimize these costs, she has to find a way to “optimally” split her orders into smaller ones. Indeed, trading slowly will reduce her market impact, but it exposes her to the risk that the price moves over the course of the execution process. She therefore needs to find a trade-off between price risk and execution costs.
To our knowledge, the first attempt to solve this problem is due to Bertsimas and Lo [30] in 1998. However in this paper, the authors neglected the price risk by only minimizing the expected cost of execution. As a consequence, the optimal strategy in their model often boils down to a constant execution speed, which is sub-optimal when the price risk is taken into account. Moreover, their model does not produce different strategies for assets with different liquidity, all else equal, whereas it seems obvious, intuitively, that a trader should execute more rapidly when the asset is liquid than when it is not. In 1999, Almgren and Chriss [11] proposed a new framework which became the basis of a lot of optimal execution algorithms. Instead of minimizing the cost of trading, they max-imized a mean-variance objective function. This allows to take into account not only the expected execution cost but also the variance of the strategy, which is of major importance in the case of a volatile asset. Moreover, they obtain closed-form formula for the optimal execution strategy.
Since then, the so-called Almgren-Chriss model has been largely generalized, to incorporate random execution costs as in Almgren [7], to study stochastic liquidity and volatility [8, 9], or to better take into account the presence of the LOB (Guéant et al. [64], Obizhaeva and Wang [89], Predoiu et al. [92]). Various types of execution strategies have been built: maximizing a risk-adjusted function of the PnL as in Almgren and Chriss [11] corresponds to strategy of the Implementation Shortfall (IS) type, whereas Guéant and Royer [67] or Konishi [76] focus on Volume-Weighted Average Price (VWAP) execution, and Guéant [61] also studied Percentage of Volume (POV) and Target Close (TC) orders.
A first model: Almgren-Chriss
We present briefly here a continuous-time version of the Almgren-Chriss model. For more details, one can refer to Guéant [61] or Cartea et al. [39].
We consider the case of a trader with a single-stock portfolio. Her initial inventory (either long or short) is given by q0 2 R, and we denote by (qt)t2[0;T ] its inventory process over the time period [0; T ], with T > 0: The trader controls the speed of execution, modeled by a process (vt)t2[0;T ] that is progressively measurable, with dqt = vtdt, bounded and such that R T vtdt = q0, i.e. the trader wishes to have an empty inventory at time T: 0
Almgren and Chriss separate two types of market impact. The first one is the permanent impact, which impacts durably the price of the asset (with no resilience). It is shown in Gatheral [55] that the absence of dynamic arbitrage implies a linear permanent market impact. Hence the price process (St)t2[0;T ] verifies dSt = dWt + kvtdt; with the initial price S0 > 0 given, > 0 the volatility of the asset, (Wt)t2[0;T ] a standard Brownian motion and k 0 a parameter modeling the magnitude of the permanent impact.
The second type of market impact is the temporary impact. In the Almgren-Chriss framework, it represents the costs associated with the bid-ask spread or other execution costs, and it is instan-taneous in the sense that the price process does not change. This impact is only modeled in the cash process (Xt)t2[0;T ] of the trader, and therefore everything works as if the price was impacted by the transaction but returned instantaneously to its original state. The dynamics of (Xt)t2[0;T ] is given by dXt = vtStdt L(vt)dt; where L is the function modeling the temporary impact, which satisfies some natural assumptions (L(0) = 0, L is strictly convex with a minimum in 0, and L is asymptotically superlinear). In their paper, Almgren and Chriss choose a quadratic function for L.
Several optimization criteria have been studied in the literature. Let us consider here the case of a CARA utility function. For a given > 0 modeling her risk aversion, the trader wishes to maximize E e XT over the set of admissible controls (vt)t2[0;T ]:
Schied et al. [97] proved that we can, without loss of generality, reduce our study to the case of deterministic execution strategies. It is then easy to prove that, in that case, the problem boils down to minimizing the function J given by J(q) = Z 0 T L(q0(t)) + 2 2q(t)2 dt; over the set of absolutely continuous functions q 2 W 1;1(0; T ) satisfying q(0) = q0 and q(T ) = 0. In particular, the permanent impact modeled by the parameter k plays no role in the resolution of the problem.
If the Legendre-Fenchel transform H of the function L is differentiable, the Hamiltonian character-ization of the optimal strategy q is given by 8p0(t) = 2q (t);
In the particular case of quadratic execution costs as proposed by Almgren and Chriss, i.e. L(v) = v2 for a given > 0, q is then the unique solution to the ODE q 00(t) = 2 q (t); with boundary conditions q (0) = q0 and q (T ) = 0. From this, we get the formula of Almgren and Chriss sinh 2 (T t)
with the associated optimal control v (t) = q 0(t) = q0 2 2 : We observe in q 2sinh q 2 T particular that q is convex if q0 > 0, and concave if q0 < 0, which differs from the constant execution speed obtained by Bertsimas and Lo [30].
Our contribution
The closed-form deterministic formula obtained by Almgren and Chriss relies mostly on the Brown-ian dynamics of the price. For more complex dynamics, in general, the optimal strategy depends on the price and therefore becomes stochastic. Moreover, although multi-asset optimal execution was addressed in Almgren and Chriss [11], the literature on this topic remains quite limited. Liquidating (or buying) large orders of different assets simultaneously is however a major problem, for instance for funds facing large withdrawals (or subscriptions), and neglecting the cross-asset relations can lead to very sub-optimal strategies in terms of price and liquidity risk. Some extensions of the multi-asset Almgren-Chriss model have been proposed: for instance, Bismuth et al. [31] studied the problem of optimal liquidation of a portfolio with Bayesian learning techniques, and Lehalle [82] studied a similar problem with a constraint of balance between the portfolio lines over the course of the execution.
The multi-asset Almgren-Chriss model considers that the asset prices follow standard Brownian dy-namics with a given correlation matrix. It ignores, however, the presence of co-integration between the prices, which can have an important impact on the optimal execution strategy. Optimal trading of a co-integrated pair of assets is presented in Cartea et al. [39]. More recently, Cartea et al. [37] considered a portfolio of assets following a multidimensional Ornstein-Uhlenbeck dynamics. This is a particularly interesting problem as, in addition to the simple optimal execution problem, it allows to build statistical arbitrage strategy to take advantage of the mean reversion. In their approach, they maximize the expected PnL of the trader minus a running risk penalty. Their problem boils down to a system of Riccati ODEs, for which they prove global existence and uniqueness, and they can therefore characterize the optimal strategy using a standard verification argument.
In Chapter 6, we consider a model similar to the one of Cartea et. al [37], but we maximize an expected utility function of the PnL. As in their paper, we characterize the value function by a system of Riccati ODEs. We prove existence and uniqueness of a solution to this system by a comparison argument.
We consider a trader in charge of d 2 N assets over a period of time [0; T ], with T > 0. The trader controls the execution speed for each of her assets, represented by a process (vt)t2[0;T ] = 1 d | in R d . Her inventory process (qt)t [0;T ] = q 1 ; : : : ; q d | verifies therefore the (vt ; : : : ; vt ) t2[0;T ] 2 t t t2[0;T ] dynamics dqt = vtdt; with q0 2 Rd given.
The price process (St) = S1 ; : : : ; Sd | [0;T ] of the assets verify a multidimensional Ornstein- t2[0;T ] t t t Uhlenbeck dynamics given by 2 dSt = R(S St)dt + V dWt; ; : : : ; Wtk | with S0 2 R+ d given, R 2 Md(R), V 2 Md;k(R) and (Wt)t2[0;T ] = Wt1 is a t2[0;T ] k-dimensional standard Brownian motion (with independent coordinates), for some k 2 N . We denote by = V V | the quadratic covariation matrix of the price process.
Finally, the cash process (Xt)t2[0;T ] of the trader verifies the dynamics: dXt = vt|Stdt L(vt)dt; with X0 2 R given, where L : Rd ! R+ is a strictly convex function, decreasing on R and in-creasing on R+ with L(0) = 0, asymptotically superlinear representing the temporary impact of (XT +qT| ST the trader.
We then aim at maximizing over the set of admissible controls, where > 0 represents the risk aversion of the trader, and ‘ : Rd ! R+ is a continuous penalty function for the terminal inventory.
The HJB equation associated with the above problem is given by:
0 =@tw(t; x; q; S) + sup f (v|S + L(v))@xw(t; x; q; S) + v|@qw(t; x; q; S)g + (S S)|R|@Sw(t; x; q; S) + 1 T r @SS2w(t; x; q; S) ;
for all (t; x; q; S) 2 [0; T ) R Rd Rd with the terminal condition:
w(T; x; q; S) = e (x+q|S ‘(q)) 8(x; q; S) 2 R Rd Rd:
We first make the following ansatz
w(t; x; q; S) = e (x+q|S+ (t;q;S)) 8(t; x; q; S) 2 [0; T ] R Rd Rd; and get a new PDE for : [0; T ] Rd Rd ! R: 0 =@t (t; q; S) + S R q + S R @S (t; q; S) S R q S R @S (t; q; S) + T r
2 T r ( (q + rS (t; q; S))(q + rS (t; q; S))|) + H (@q (t; q; S)) ; for all (t; q; S) 2 [0; T ) Rd Rd with terminal condition:
(T; q; S) = ‘(q) 8(q; S) 2 Rd Rd;
where H : Rd ! R is the Legendre-Fenchel transform of the function L, i.e. H(p) = sup v|p L(v)
If we then assume that the functions L and ‘ are of the form L(v) = v | v and ‘(q) = q| q , for some 2 Sd+(R); 2 Sd++(R); we see that the supremum in the above definition is reached at a unique v = 12 1p and we get 8p 2 Rd H(p) = 14 p| 1p:
We can then use a second ansatz: (t; q; S) = q|A(t)q + q|B(t)S + S|C(t)S + D(t)|q + E(t)|S + F (t) 8(t; q; S) 2 [0; T ] Rd Rd; (8)
which allows us to characterize the value function of the problem by the following system of ODEs
8 A0(t) = (B(t) + Id) (B(t)| + Id) A(t) 1A(t)
B0(t) = (B(t) + Id)R + 2 (B(t) + Id) C(t) A(t) 1B(t)
> C0 (t) = R C(t) + C(t)R + 2 C(t) C(t) B(t) B(t)
> D0(t) = (B(t) + Id)RS + (B(t) + Id) E(t) A(t) D(t)
> E0(t) = 2C(t)RS + R E(t) + 2 C(t) E(t) B(t1 2 1 D(t);
> F (t) = S|R|E(t) T r( C(t)) + 2 E(t)| E(t) 4 D(t)|
> 0 | 1D(t)
where Id denotes the identity matrix in Md(R), with terminal conditions:
A(T) = ; B(T) = C(T) = D(T) = E(T) = F(T) = 0:
Using comparison techniques, we prove existence and uniqueness of a solution to this system, and by a standard verification argument we are then able to characterize the optimal control (vt )t2[0;T ] by vt = 12 1 (2A(t)qt + B(t)St + D(t)) :
We then illustrate this strategy with several numerical examples to show the importance of taking the non-brownian dynamics into account.
Table of contents :
Introduction
1 Market making on OTC markets
1.1 The problem of the market maker
1.2 A first model: Avellaneda-Stoikov
1.3 Main existing results
1.4 Our contribution
2 Optimal tick sizes
2.1 The problem
2.2 The model with uncertainty zones
2.3 Our contribution
3 Optimal execution
3.1 The problem
3.2 A first model: Almgren-Chriss
3.3 Our contribution
I Optimal market making on OTC markets
1 Size matters for OTC market makers: general results and dimensionality reduction techniques
1.1 Introduction
1.2 Market making with marked point processes
1.2.1 Modeling framework and notations
1.2.2 Existence and uniqueness of a solution to (1.2)
1.2.3 Verification theorem
1.3 Solving the multi-asset market making problem with factors
1.3.1 A low-dimensional approximation
1.3.2 A Monte-Carlo method to take the residual risk into account
1.4 Numerical results
1.4.1 The case of two assets: one factor vs. two factors
1.4.2 Dealing with 30 assets
1.5 Appendix
1.5.1 A particular case: hedging the market factors
1.5.2 On the construction of the processes Ji;b and Ji;a
2 From passive to active: when market makers are also market takers
2.1 Introduction
2.2 A model for active market makers
2.2.1 Modeling framework and notations
2.2.2 Preliminary results
2.3 Viscosity solution to (HJ)
2.3.1 Existence
2.3.2 Uniqueness
2.4 Numerical results
2.5 Appendix
2.5.1 Proof of Lemma 3
2.5.2 Proof of Proposition 4
3 Algorithmic market making for options
3.1 Introduction
3.2 Description of the problem
3.2.1 The market
3.2.2 The optimization problem of the market maker
3.2.3 Assumptions and approximations
3.3 An approximate solution to the problem
3.3.1 Change of variables: beating the curse of dimensionality
3.3.2 Hamilton-Jacobi-Bellman equation and optimal controls
3.4 Numerical results
3.4.1 Model parameters
3.4.2 Optimal quotes
3.4.3 Conclusion
3.5 Appendix
3.5.1 An alternative to the -hedging assumption
3.5.2 Beyond the constant-vega assumption
4 Closed-form approximations in multi-asset market making
4.1 Introduction
4.2 The multi-asset market making model
4.2.1 Model setup
4.2.2 The optimization problems
4.2.3 The Hamilton-Jacobi-Bellman and Hamilton-Jacobi equations
4.2.4 Existing theoretical results
4.3 A quadratic approximation of the value function and its applications
4.3.1 Introduction
4.3.2 An approximation of the value function in closed form
4.3.3 From value functions to heuristics and quotes
4.4 Beyond the quadratic approximation: a perturbative approach
4.5 A multi-asset market making model with additional features
4.5.1 A more general model
4.5.2 The Hamilton-Jacobi equation
4.5.3 Quadratic approximation
4.5.4 From value functions to heuristics and quotes
4.6 Numerical results
4.6.1 Value function and optimal quotes
4.6.2 Closed-form approximations
II Other problems in mathematical finance
5 On bid and ask side-specific tick sizes
5.1 Introduction
5.2 The model with uncertainty zones
5.3 High frequency market making under side-specific tick values and interaction with the exchange
5.3.1 The market maker’s problem
5.3.2 The platform’s problem
5.4 Numerical results
5.4.1 Similar tick values on both sides
5.4.2 Side-specific tick values: additional opportunities for the market maker
5.4.3 Side-specific tick values: effect of
5.5 Conclusion
5.6 Appendix
5.6.1 Proof of Proposition 7
5.6.2 Proof of Theorem 5
5.6.3 Effects of the uncertainty zones on h
6 Optimal execution and statistical arbitrage under Ornstein-Uhlenbeck dynamics
6.1 Introduction
6.2 The optimal liquidation problem
6.2.1 Modelling framework and notations
6.2.2 Hamilton-Jacobi-Bellman equation
6.2.3 Main mathematical results
6.3 Numerical results
6.3.1 Single-asset case
6.3.2 Multi-asset case
6.4 Appendix – Multi-asset optimal execution with correlated Brownian motions and execution costs
6.5 Appendix – Merton portfolio optimization problem under Ornstein-Uhlenbeck dynamics and exponential utility
6.5.1 Modelling framework
6.5.2 HJB equation
Bibliography
