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This article shows the execution performance of the risk-averse institutional trader with constant absolute risk aversion (CARA) type utility by using the condition of no price manipulation defined in the risk neutral sense. From two linear price impact models both satisfying that condition, we have derived the unique explicit optimal execution strategy calculated backwardly with dynamic programming equations. And our study shows that the optimal execution strategy exists in the static class. The derived solution can be decomposed into mainly two components, each giving an explanation of the property of optimal execution volume. Moreover we propose two conditions in order to compare the performance of these two price models, and illustrate that the performances of the two models are surprisingly different under certain conditions.

In the competitive market paradigm, it is assumed that security markets are perfectly elastic and all orders can be executed instantaneously. However in real markets, since institutional traders (large traders) usually submit orders of considerable sizes, such traders thus influence the price by their own dealings (called market (price) impact) and create the execution time lag for their orders. Thus the large trader often divides her holdings (orders) into small pieces considering the tradeoff between market impact risk due to her fast execution and volatility risk due to her slow execution. In [

In this paper, under no price manipulation condition, we consider mainly two types of price model depending on how the price is reverted to its previous price level for the buy trade. Let’s call one of them the permanent (impact) price model (as in e.g. [

The main goal of this paper is to derive the optimal execution strategies for these two price models. Then in the equidistance discrete trading time grid setting, we show that the optimal execution strategy of the risk-averse large trader with each price model exists in the static class by deriving backwardly the explicit solution with the dynamic programming equation. This result is similar to the one found in [

The rest of the paper is organized as follows. In Section 2, we present two price dynamics and two definitions of the price manipulation. In Section 3, we describe the optimization problem and derive explicit solutions for the two price models. Furthermore, we show the property of the optimal execution strategy and illustrate it using the comparative statics. In Section 4, we consider the relationship between two price models. The transient price model is more realistic but a little bit complicated therefore it takes much time when we simulate the execution performance, on the other hand the permanent price model is unrealistic but simple enough to be able to make high-speed trading decision in algorithmic trading system. For that reason, we suggest how to incorporate the intrinsic parameter of the transient price model into the permanent price model. More concretely, we propose two conditions that exist between those two price models under the TWAP (Time Weighted Average Price) strategy, when we attempt to compare the performance of those two price model in the same market. Section 5 contains a conclusion. Calculations and proofs are complicated but can be proved in a straightforward way.

In this section, we explain two existing price models in the discrete time setting. One is the permanent impact (price) model proposed by [

Suppose that

Moreover,

The lifted price by the large order reverts to the previous price level to a certain extent.

In the permanent price model, the execution price diminishes instantly to the permanent impact level and the expected price is maintained until the next trading time. That is,

Using Equation (3) and (4),

where

All information available to the large trader before her trading at time t are

In the permanent price model, the price impact, the temporary impact and the permanent impact are repre- sented respectively by

The transient price model, on the other hand, is the same as the permanent price model until the submitted order is executed. However the price reversion toa permanent level is not immediate but gradual. We set the time independent rate

where

Here, we define S as

where

In this transient price model, the price impact and the transient impact are

Remark 1: The economic interpretation of

The reason why we use these specific two price models is its viability, as it will explained in the next subsection. The main difference between these two models is whether the effect of the present execution is completely incorporated in the price immediately or not. In the transient price model, since the price after the present execution fall down gradually to the permanent level (in this case 0), the effect of the present execution is partially incorporated in the price at the following trading time, and is completely incorporated after a certain period.

In this subsection, we introduce the concept of price manipulation from the perspective of the feasibility of the price model. This is because the market can easily crash with the price manipulation of the large traders in the current market environment where the high-frequency trading is becoming a main stream. So the construction of the feasible price model is essential to limit such a price manipulation. In the following we introduce two concepts of price manipulation.

Definition 1 ((Pure) Price manipulation [

It is shown in [

Definition 2 (Transaction-triggered price manipulation [

Definition 2 states a stronger condition of the price manipulation than the one given by Definition 1. That is to say, even if the price model satisfies the absence of pure price manipulation, it may not satisfy the absence of the transaction-triggered price manipulation, such as buy and sell oscillation trades.

In this paper, we use an exponential resilience for the transient price model. This does not admit transaction- triggered price manipulation. As shown below in Remark 2, our control for the risk-averse large trader describes that when we apply the round trip trade. 0 trade is always optimal. So, both price models satisfy the condition of the absence of pure price manipulation.

In this section, we show that the optimal execution strategy exists in the static class by deriving the explicit solution with a dynamic programming equation. Suppose that a risk-averse large trader with CARA (Constant Absolute Risk Aversion) type utility of which the risk aversion coefficient is R submits large amount of market orders in equally time intervals over the maturity T. We consider the problem of the dynamic execution strategy that maximizes the large trader’s expected utility from her terminal wealth. Here, we show the optimal execution strategy based mainly on the transient price model. For the permanent price model, we only provide the result since it requires simpler calculation.

In this case, we define the large trader’s expected utility under the trading strategy

where

where the subscript t of the expectation represents the condition where all the information up to time t is available to the large trader.

Because of the Markov property of the dynamics and path independency of the large trader’s utility at the final period,

We derive the sequence of the optimal execution volumes which attains

Theorem (Optimal Execution Strategy with the Transient Price Model): When we use the transient price model, the optimal execution volume of a large trader at time t denoted _{t} at that time. Then at time t, the optimal execution volume and the corresponding optimal value function are respectively

and

where we set

Then a deterministic execution strategy becomes optimal.

Secondary, we provide the optimal execution strategy for the permanent price model as following corollary.

Corollary (Optimal Execution Strategy with permanent Price Model):

When we use the permanent price model, the optimal execution volume of a large trader at time t denoted

and

where

We provide a short proof of this Theorem in the appendix. For the proof of the Corollary, refer to [

Remark 2: For both price models,

and for the permanent price model

The purpose of this subsection is to give an intuitive and intelligible analysis of the optimal strategies mainly for the permanent price model as it is difficult to give an analytical proof for the optimal execution strategy using transient price model. However we can show this intuition and confirm it using some numerical examples. To this end, we set some time-homogeneity assumptions for the impact

Lemma 1 (Monotone Decrease Property): If

For the proof of Lemma1, refer to [

Proposition 1(Risk Aversion Effect): Suppose

If

Proposition 2 (Risk Neutral Trader): Suppose

Moreover, for the transient price model, the optimal execution strategy is time symmetric. Then we form the following property,

Remark 3: The optimal execution strategy for the transient price model does not have the monotone decrease property (Lemma 1). However from the numerical experiment shown in

However, there is analytical difficulty for the proof of this property because the terms of

mutually on each other over time. In fact, when we express the optimal execution volume at time t + 1 with the states at time t,

Under the time-homogeneity of

So far, we considered two price models, the permanent and the transient with intrinsic parameter

For the two price models describing a real market, if the expected costs derived from these two price models respectively with the same execution volume at the same intervals are different from each other, an arbitrage opportunity may occur between these two models. We should then unify how the information after each trade is incorporated into the price, when we compare the performance of the two price models. So, in order to standardize the market, we should find the relationship between

Suppose that the expected cost using TWAP strategy over the maturity T with the permanent and the transient price model are respectively

Definition 3 (TWAP Cost Equivalent): If

However, this condition does not satisfy the law of indifference which is a fundamental economic principle. As a stronger condition, we define TWAP equivalent condition as below.

Definition 4 (TWAP Equivalent): If

We can afterward derive following conditions using Equations (3), (5), (9), (10), and letting q = constant in order to adapt the transient price model according to the permanent price model.

Condition 1: If the market is TWAP cost equivalent, then the following condition holds:

Condition 2: If the market is TWAP equivalent, then the following condition holds:

The upper (lower) half of

The calculations of these conditions are straightforward. Within Condition 1, the mean of the accumulated transient impact at each time using the transient price model is regarded as the permanent impact, and then is assigned equally to

averse large trader corresponding to the value of

Remark 4: When

Therefore the optimal execution strategy for the transient price model is the same as the permanent price model one with

In a discrete time setting, we derived an explicit solution for the two price models by solving a dynamic programming equation backwardly from the maturity time. Under the assumptions of a large trader with CARA utility type and public news effects on price modeled as normal random variables, the optimal execution strategy exists in the static class. In particular, since the optimal execution volume for the transient price model consists of two components, that is tradeoff between impact risk and volatility risk, and the expectation of the price reversion, that solution gives consideration to the existence of transaction-triggered price manipulation. From the comparative statics, we also illustrated how the large trader’s risk aversion affects the optimal execution strategy. Furthermore, with TWAP strategy we compared the performances of the two price models where the time-ho- mogeneity of the parameters α and ρ plays a significant role in the absence of price manipulation. But it is impossible to capture completely the essence of the price process with parameters using in this study. In recent years, an order driven market becomes mainstream in various trading venues around the world. Therefore, we should specify the shape of limit order book endogenously or exogenously in order to construct the price model. Further research consists on creating more practical models that takes for instance into consideration the intraday liquidity effect among other effects and the nonlinear impact function as empirically stated in [

The authors would like to thank participants of the International Conference on Industrial Engineering Theory, Applications and Practice (IJIE2013) at Pusan National University, the RIMS Workshop on Theory and Application of Mathematical Decision Making under Uncertainty 2013 at Kyoto University, and the Symposium on Stochastic Models 2014 at Tokyo University of Science for their constructive comments.

Short proof of Theorem:

We can derive the optimal execution volume by backward induction from the maturity time T. For t = T, since the large trader must finish her purchases

Then,

where we define the maturity condition as

and the value function is

and we set

where A, B and K are the coefficients of

Next, for

where we use

where

and the value function is

where

Proceeding similarly for a general time t, we obtain the desired results (17), (19) with backward induction.

Proof of Proposition 1

From Lemma 1 and Remark 2, we show that if

Denote the terms which does not depend on R in

From Remark 2, we have

Proof of Proposition 2

When

Suppose that if

We will show for

So,

From the assumption of Equation (45) and Equation (22), we get,

Moreover, from the assumption of Equation (45)

Then,

Therefore, from Equation (22),

Then, by substituting the above

That is Equation (46).