*Ms. Shimizu, P. is now a housewife, and does not have any other occupation.
In recent years, as the increase of trading opportunities as the stock trading “venue,” it has also been diversified the ways of trading for the institutional investors who execute a large amount of their order. An institutional investor, referred to as the large trader, has to take into account of market impacts because of the liquidity of supply and demand at the stock exchange. That is to say, if the large trader wants to purchase a large amount of her order of a specific stock at the stock exchange, it is hard for her to find counter-sellers who satisfy her orders from unspecified number of trading participants. Thus, she must consider the risk of price shift up. On the contrary, if someone who wants to sell a large amount, it is necessary to take into account the risk of price down. This type of problem has been tackled by many researchers and practitioners so far. For examples, seminal papers   and  derived optimal execution strategies only in the stock exchange, which considered the trade-off between the market impact risk and the volatility (price change) risk. On the other hand, alternative trading platforms at the off-exchange referred to dark pools and internal crossing have been attached the attention. For more details refer to  . Since the invisibility of liquidity at the off-exchange trading does not impact on the price, that trading venues are popular platforms for the large trade. When the large trader adopts her executions at an off-exchange trading venue, especially, internal crossing with a broker, it is necessary to obtain some consents with the counter-party about the execution date (time), price, and volume. For example, the large trader executes with the broker (counter party) α shares at noon on specific day with the opening price of that day. Although it is also possible to make a contract using the closing price, which we call “closing price guaranteed execution”, as mentioned in  , the closing price guaranteed execution is misalignment because the broker is able to accumulate his holdings at the stock exchange with changing the price to be desirable as possible. That is, the broker could make a profit by round trip trade as in  . Thus, in theoretical and practical points of view, VWAP (Volume Weighted Average Price) is often used for an execution price at the off-exchange trading venue as in  . But it is difficult to treat the VWAP in a dynamic optimization problem. Thus, the VWAP guaranteed execution problem is often treated in a static framework.
In this paper, we derive an optimal execution strategy at the traditional trading and off-exchange trading venues where we adopt a closing price guaranteed execution (more accurately called after-hour-trading like ToSTNeT at Tokyo Stock Exchange). First of all, we establish the framework in which a single large trader makes a contract with a broker to execute her holdings at the specific intraday closing price before executing at the traditional stock exchange. That is, after the large trader executes at the traditional stock exchange within the day, she makes a trade of remaining volume with the broker at the closing price of that traditional stock exchange based on the contract. In this situation, the contract about the trading volume does not reach an agreement. However, using the approach of  that shows an optimal execution strategy making decisions dynamically is in the class of static ones, we derive the optimal allocations of both traditional stock exchange trading volume and off-exchange trading volume at the initial time. Therefore, we show that we can contain not only time and price but also the execution volume in this contract (agreement) at the off-exchange trading. However, since we cannot deny the possibility of the price manipulation, we will put the strong assumption about the broker.
This paper is organized as follows. A general stock price model is introduced on the specific stock exchange and we give an assumption on off-exchange trading in Section 2. If we do not impose such an assumption then the large trader is always defeated in the broker, accordingly no agreement in over-the-counter trading would be reached. Section 3 presents the wealth process of a single large trader and derives the optimal execution strategy in both trading venues. This main idea enables the large trader to make a contract (agreement) with a broker to trade predetermined volume at the closing price before the intraday trading in the stock exchange. We also show that this problem has another explanation. In Section 4, numerical examples are represented, and Section 5 concludes this paper.
In this section, we define the stock price model in the exchange which is partially based on  and assume some conditions. We fix a probability space and assume that all stochastic processes are defined on this space with a filtration. In particular, the i.i.d. random sequence represents the public news effect and follows as .
2.1. Price Model in the Exchange Trading
Let denote the stock price at time and the execution price at time . We assume that if the large trader submits the large volume, the stock price is immediately sifted up and her submitted orders are always executed entirely. That is, the order book with block shape is fully liquid for both the bid and ask sides but we define the depth of the book of this stock to be always for all price at time , where represents the price change per unit execution. Then, if we define as the submitted order volume of the large trader then
Here, we define the price process as
In this paper we derived the optimal execution strategy for a large trader who affects the stock price considering both the traditional stock exchange and off-exchange trading venues. We set the large trader makes a contract with a broker to be guaranteed her execution at the closing price on the intraday trading before she submits her order to the exchange. Then we showed that the large trader could control her order volumes before executing at the traditional stock exchange. However, without the requirement for the broker not to manipulate the stock price, since the large trader can perceive to suffer a loss, she does not make a contract with the broker at the off-exchange trading venue. On the other hand, when the large trader can estimate that the cost of off-exchange trading is lower than the market impact cost, she also manipulates the market. Moreover, we indicated another point of view whether to carry over the execution of the large trader to the next day or not. Similarly, in this case, we showed that the carryover of the execution depends on the market impact cost level at the trading day. Concerning more realistic but more complicated cases about the price manipulation of the large trader, game theoretical analysis of the VWAP guaranteed execution and the effects of the broker’s strategic behavior is left for our future research.
We thank an anonymous referee and the editorial staffs for their constructive comments.