# Static vs adapted optimal execution strategies in two benchmark trading models

## Abstract

We consider the optimal solutions to the trade execution problem in the two different classes of i) fully adapted or adaptive and ii) deterministic or static strategies, comparing them. We do this in two different benchmark models. The first model is a discrete time framework with an information flow process, dealing with both permanent and temporary impact, minimizing the expected cost of the trade. The second model is a continuous time framework where the objective function is the sum of the expected cost and a value at risk (or expected shortfall) type risk criterion. Optimal adapted solutions are known in both frameworks from the original works of Bertsimas and Lo (1998) and Gatheral and Schied (2011). In this paper we derive the optimal static strategies for both benchmark models and we study quantitatively the improvement in optimality when moving from static strategies to fully adapted ones. We conclude that, in the benchmark models we study, the difference is not relevant, except for extreme unrealistic cases for the model or impact parameters. This indirectly confirms that in the similar framework of Almgren and Chriss (2000) one is fine deriving a static optimal solution, as done by those authors, as opposed to a fully adapted one, since the static solution happens to be tractable and known in closed form.

AMS Classiï¬cation Codes: 60H10, 60J60, 91B70;

JEL Classiï¬cation Codes: C51, G12, G13

Keywords: Optimal trade execution, Optimal Scheduling, Algorithmic Trading, Calculus of Variations, Risk Measures, Value at Risk, Market Impact, Permanent Impact, Temporary Impact, Static Solutions, Adapted Solutions, Dynamic Programming.

## 1 Introduction

A basic stylized fact of trade execution is that when a trader buys or sells a large amount of stock in a restricted amount of time, the market naturally tends to move in the opposite direction. If one assumes an unaffected price dynamics for the traded asset, trading activity will impact this price and lead to an affected price. Supply and demand based analysis says that if a trader begins to buy large amounts, other traders will notice and the affected price will tend to increase. Similarly, if one begins to sell large amounts, the affected price will tend to decrease. This is particularly important when the market is highly illiquid, since in that case no trade goes unnoticed. The goal of optimal execution, or more properly optimal scheduling, is to find how to execute the order in a way such that the expected profit or cost is the best possible, taking into account the impact of the trade on the affected price.

As far as we are concerned in this paper, there are two main categories of trading strategies: deterministic, also called static in the execution literature jargon, and adapted, or adaptive. We will use static / deterministic and adapted / adaptive interchangeably. Deterministic strategies are set before the execution, so that they are independent of the actual path taken by the price. They only rely on information known initially. Adapted strategies are not known before the execution. The amount executed at each time depends on all information known up to this time. Clearly market operators, in reality, will monitor market prices and trade based on their evolution, so that the adapted strategy is the more natural one. However, in some models it is much harder to find an optimal trading strategy in the class of adapted strategies than in the class of static ones.

In 1998, Bertsimas and Lo [6] have defined the best execution as the strategy that minimizes the expected cost of trading over a fixed period of time. They derive the optimal strategy by using dynamic programming, which means that they go backwards in time. The optimal solution is therefore sought in the class of adapted strategies, as is natural from backward induction, but is found to be deterministic anyway. However, once an information process is added, influencing the affected price, the optimal solutions are adapted and no longer static. This approach minimizes the expected trade cost only, without including any risk in the criterion to be optimized. In particular, the criterion does not take into account the variance of the cost function.

Two years later, Almgren and Chriss [2] consider the minimization of an objective function that is the sum of the expected execution cost and of a cost-variance risk criterion. Unlike the previous model, this setting includes in the criterion the possibility to penalize large variability in the trading cost. To solve the resulting mean-variance optimization, Almgren and Chriss assume the solution to be deterministic from the start. This allows them to obtain a closed-form solution. This solution, however, is only the best solution in the class of static strategies, and not in the broader and more natural class of adapted ones.

Gatheral and Shied [12] later solve a similar problem, the main difference being that they assume a more realistic model for the unaffected price. Gatheral and Schied derive an adapted solution by using an alternative risk criterion, the time-averaged value-at-risk function. They obtain closed-form expressions for the strategy and the optimal cost. The solution is not static. However, this does not seem to lead to a solution that is very different, qualitatively, from the static one. Indeed, Brigo and Di Graziano (2014), adding a displaced diffusion dynamics, find that in many situations only the rough statistics of the signal matter in the class of simple regular diffusion models [7]. In this paper we will compare the static and fully adapted solutions in detail.

Since the solutions obtained in the setting of Almgren and Chriss [2] are deterministic, they may be sub-optimal in the set of fully adapted solutions under a cost-variance risk criterion, so several papers have attempted to find adapted solutions by changing the framework slightly. This allows one to take the new price information into account during the execution, and to have more precise models. For example, in 2012 Almgren [5] assumes that the volatility and liquidity are random. He numerically obtains adapted results under these assumptions. Almgren and Lorenz [4] obtain adapted solutions by using an appropriate dynamic programming technique.

Similarly, in this paper we will focus on what one gains from adopting a more general adapted strategy over a simple deterministic strategy in the classic discrete time setting of Bertsimas and Lo [6] with information flow and in the continous time setting of Gatheral and Schied with time-averaged value-at-risk criterion [12].

The paper is structured as follows. In Section 2 we will introduce the discrete time model by Bertsimas and Lo, looking at the cases of temporary and permanent market impact on the unaffected price, and including the solution for the case where the price is also affected by an information flow process. We will derive and study the optimal static and fully adapted solutions and compare them, quantifying in a few numerical examples how much one gains from going fully adapted.

In Section 3 we will introduce the continuous time model as in Gatheral and Schied, allowing for both temporary and permanent impact and for a risk criterion based on value at risk. We will report the optimal fully adapted solution as derived in [12] and we will derive the optimal static solution using a calculus of variation technique, similar for example to the calculations in [10]. We will compare the two solutions and optimal criteria in a few numerical examples, to see again how much one gains from going fully adapted.

Section 4 concludes the paper, summarizing its findings, and points to possible future research directions.

## 2 Discrete time trading with information flow

### 2.1 Model formulation with cost based criterion

Let be the number of units left to execute at time , such that is the initial amount and at the final time . In this section we consider a buy order, so that the purpose of the strategy is to buy an amount of asset by time , minimizing the expected cost of the trade. The amount to be executed during the time interval is . We expect to be non-negative, since we would like to implement a pure buy program. However we do not impose a constraint of positivity on , so that the optimal solution, in principle, might consider a mixed buy/sell optimal strategy.

Since the problem is in discrete time, it is only updated every period so we will assume that the price does not change between two update times.

With that in mind, we assume that the unaffected mid-price process is given by

(2.1) | ||||

(2.2) |

where the information coefficient , and the volatilities and are positive constants, and are independent standard Brownian motions and the parameter is in . We define , .

would be the price if there were no impact from our executions. It follows an arithmetic Brownian motion (ABM) to which an information component has been added. The information process is an AR(1) process. It could be for example the return of the S&P500 index, or some information specific to the security being traded. represents the relevance of that information, that is how much it impacts the price.

There are two dynamics that we will consider for the real price , depending on whether the market impact is assumed to be permanent or temporary. We will explain what those terms mean when defining the price dynamics below. We assume that the market impact is linear in both settings, which means that the market reacts proportionally to the amount executed.

In the case of permanent market impact the mid-price dynamics are changed by each execution. This means that when we compute the trade cost, the unaffected price is replaced, during the execution, by the impacted or affected price :

(2.3) |

where the permanent impact parameter is a positive constant.

In the case of temporary market impact each execution only changes the price for the current time period. The mid-price is still given by (2.1), and the effective price is derived from each period. has the following dynamics:

(2.4) |

where the temporary impact parameter is a positive constant.

###### Remark 2.1.

Since one case assumes that the impact lasts for the whole trade, and the other assumes that the impact is instantaneous and affects only an order at the time it is done, both are limit cases of a more general impact pattern that is more progressive, see for example Obizhaeva and Wang [17].

We will keep the two more stylized impact cases and analyze them separately. The problem in both cases is to minimize the expected cost of execution. Since we are considering a buy order, is the number of units left to buy. Hence the optimal expected execution cost at time is

(2.5) |

subject to , .

###### Remark 2.2.

As we mentioned earlier, we do not enforce any constraint on the sign of , which means that we are allowed to sell in our buy order.

We now present some calculations deriving the optimal solution of problem (2.5) in the cases of permanent and temporary impact. Our calculations in the general setting follow essentially Bertsimas and Lo [6] but with a slightly different notation, as done initially in Bonart, Brigo and Di Graziano [8] and Kulak [15]. We further derive the optimal solution in the static class, using a more straightforward method.

### 2.2 Permanent market impact: optimal adapted solution

In this section, we solve problem (2.5) reproducing the solution of Bertsimas and Lo [6], assuming that the market impact is permanent, which means that the affected price follows (2.3). In the adapted setting, the problem is solved recursively. At any time , we consider the problem as if was the initial time, and the execution was optimal from time . We only have to make a decision for the period , ignoring the past and having already solved the future.

For any , the execution cost from time onward is the sum of the cost at time and the cost from time onward.
Taking the minimum of the expectation, this can be written as the Bellman equation:

(2.6) |

Since the execution should be finished by time T (), all the remaining shares must be executed during the last period :

Substituting this value into the Bellman equation (2.6) taken at gives us the optimal expected cost at time :

where we used the fact that , and are known at time , as well as the null expectation of standard Brownian motion increments.

We now move one step backward to obtain the optimal strategy at time , plugging the expression above in (2.6) taken at and noting that .

In order to find the minimum of this expression, we set to zero its derivative with respect to :

The solution of this equation is the optimal amount to execute at time :

The optimal expected cost at time is

We resume the recursion using the expression above, and obtain the optimal strategy at time .

In order to find the minimum of this expression, we set to zero its derivative with respect to :

The solution of this equation is the optimal amount to execute at time :

We then compute the optimal expected cost at time :

More generally, we can see a pattern emerging from the three previous optimal strategies and expected costs results, which can be proven formally by induction.

###### Proposition 2.3 (Optimal execution strategy).

For any the optimal execution strategy at time is

###### Remark 2.4.

can be simplified to

###### Proof.

Let .

and . ∎

###### Proposition 2.5 (Optimal expected cost).

For any , the minimum expected cost at time is

###### Remark 2.6.

can be simplified to

###### Proof.

Let .

∎

###### Corollary 2.1.

In particular, the optimal expected cost at time is

(2.7) |

###### Remark 2.7.

Although this strategy is adapted, it does not take into account the price, but only the information process. This makes sense because if there was no information, the optimal strategy would be deterministic as shown in [6].

### 2.3 Permanent market impact: optimal deterministic solution

We will now constrain the solutions of (2.5) to be deterministic, so that the strategy is known at time and can be executed with no further calculations, independently of the path taken by the price.

###### Theorem 2.8 (Optimal deterministic execution strategy).

When we restrict the solutions to the subset of deterministic strategies, the optimal strategy is

(2.8) |

###### Proof.

To solve (2.5), we will simply assume that every is known at time and compute the expected cost at time :

Problem (2.5) can be rewritten as

To find the minimum, we set to zero the partial derivatives of the expected cost with respect to , …, . For it gives us

We obtain the difference equation

(2.9) |

with boundary conditions and .

The solution of (2.9) is of the form for some constants , and . Plugging this expression back in the equation yields

From the boundary conditions we have

and

Combining those, we obtain the closed-form formula of the optimal deterministic strategy. ∎

###### Remark 2.9.

If (no initial information), (information is just noise) or (information is irrelevant), the strategy consists in splitting the execution in orders of equal amounts over the period . This is a particular case of the strategy more generally known as VWAP (volume-weighted average price), and is the strategy obtained when there is no information.

###### Theorem 2.10 (Optimal expected cost associated with the deterministic strategy).

The expected cost at time associated with the optimal deterministic strategy is

(2.10) |

###### Proof.

For lighter calculations, we set

The optimal expected cost at time is

(2.11) |

We compute the two sums in (2.11) separately for clarity:

The second sum is

The first sum is