Optimal high-frequency trading with limit and market orders

We propose a framework for studying optimal market-making policies in a limit order book (LOB). The bid–ask spread of the LOB is modeled by a tick-valued continuous-time Markov chain. We consider a small agent who continuously submits limit buy/sell orders at best bid/ask quotes, and may also set limit orders at best bid (resp. ask) plus (resp. minus) a tick for obtaining execution order priority, which is a crucial issue in high-frequency trading. The agent faces an execution risk since her limit orders are executed only when they meet counterpart market orders. She is also subject to inventory risk due to price volatility when holding the risky asset. The agent can then also choose to trade with market orders, and therefore obtain immediate execution, but at a less favorable price. The objective of the market maker is to maximize her expected utility from revenue over a short-term horizon by a trade-off between limit and market orders, while controlling her inventory position. This is formulated as a mixed regime switching regular/impulse control problem that we characterize in terms of a quasi-variational system by dynamic programming methods. Calibration procedures are derived for estimating the transition matrix and intensity parameters for the spread and for Cox processes modelling the execution of limit orders. We provide an explicit backward splitting scheme for solving the problem and show how it can be reduced to a system of simple equations involving only the inventory and spread variables. Several computational tests are performed both on simulated and real data, and illustrate the impact and profit when considering execution priority in limit orders and market orders.     

Cross-impact and no-dynamic-arbitrage

We extend the ‘No-dynamic-arbitrage and market impact’-framework of Gatheral [Quant. Finance, 2010, 10(7), 749–759] to the multi-dimensional case where trading in one asset has a cross-impact on the price of other assets. From the condition of absence of dynamical arbitrage we derive theoretical limits for the size and form of cross-impact that can be directly verified on data. For bounded decay kernels we find that cross-impact must be an odd and linear function of trading intensity and cross-impact from asset i to asset j must be equal to the one from j to i. To test these constraints we estimate cross-impact among sovereign bonds traded on the electronic platform MOT. While we find significant violations of the above symmetry condition of cross-impact, we show that these are not arbitrageable with simple strategies because of the presence of the bid-ask spread.     

Modelling the shape of the limit order book

This article develops a parsimonious way to use the shape of the limit order book to produce an estimate of the asset price. The posited model captures and describes the evolution of the distribution of limit orders on the bid and ask sides of the LOB during the trading session and provides estimates of the execution asset price over time. The performance of the model is evaluated against some existing standards from the market microstructure literature during the trading session. Empirical evidence on listed companies confirm a strong contribution of our methodology to the innovation in asset prices, according to the information share coefficients. We also document a significant improvement relative to the Hasbrouck [J. Finance, 1991, 46, 179–207] model when our model estimates are included as regressors.     

Prospect theory and a manager's decision to trade a blind principal bid basket

We analyze asset managers' decisions to execute substantial baskets of stocks. Such transactions can be executed in two ways. The first method is through blind auctions with unique features known as blind principal bids (BPBs). The manager learns the trading cost once the auction process determines the commission. The second method is to use an agency trade. In this case, the asset manager faces an actual trading cost that is unknown before execution due to market impact uncertainty. Using proprietary data, we investigate the manager's choice between BPBs and agency trade in a natural experiment with large monetary stakes. The volume traded in each sample basket is significant, which can amount to over 0.5% of the NYSE daily share volume. Our findings show that managers' behavior is more consistent with prospect theory than the expected utility framework. When asset managers trade BPB baskets, they are reluctant to realize the potential higher cost of agency trade and opt for the fixed cost of BPBs instead. Thus, our results complement the literature on the disposition effect by demonstrating that managers' trading cost decisions exhibit “play it safe” behavior.     

A simulation analysis of the microstructure of an order driven financial market with multiple securities and portfolio choices

In this paper we propose an artificial market where multiple risky assets are exchanged. Agents are constrained by the availability of resources and trade to adjust their portfolio according to an exogenously given target portfolio. We model the trading mechanism as a continuous auction order-driven market. Agents are heterogeneous in terms of desired target portfolio allocations, but they are homogeneous in terms of trading strategies. We investigate the role played by the trading mechanism in affecting the dynamics of prices, trading volume and volatility. We show that the institutional setting of a double auction market is sufficient to generate a non-normal distribution of price changes and temporal patterns that resemble those observed in real markets. Moreover, we highlight the role played by the interaction between individual wealth constraints and the market frictions associated with a double auction system to determine the negative asymmetry of the stock returns distribution.     

Intraday Price Discovery in the DJIA Index Markets

Abstract:  This paper explores the dynamics of price discovery between the Dow Jones Industrial Average (DJIA) index and its three derivative products: the DIAMOND exchange-traded fund (ETF), the floor-traded regular futures, and the electronically traded mini futures. Even though the American Stock Exchange is the primary listing exchange for the ETF, the analysis indicates that the electronically traded ETF on the Archipelago (ArcaEx) electronic communications network dominates the price discovery process for DIAMOND shares. The E-mini futures contribute the most to price discovery, followed by the ArcaEx DIAMOND. The DJIA index and regular futures contribute least to price discovery. The analysis is repeated using the derivatives of the S&P 500 index as a robustness check. The results indicate that multi-market trading ensures greater pricing efficiency. Informed traders favor electronic trading because of immediate and anonymous trade execution.     

Trading costs and price discovery

The price discovery roles of a set of related markets or securities have been investigated in many different settings where trading costs effect is often commingled with other trading arrangement factors. In Hong Kong, regular futures and mini futures contracts as well as their underlying spot asset are all traded on a same electronic trading platform. The trading arrangements thus provide us with a unique setting where we can isolate the impacts of transaction costs on price discovery. Using Hasbrouck’s (J Finance 50:1175–1199, 1995) information share approach, it is found that in Hong Kong, the regular futures contracts market plays a dominant role in price discovery while the mini futures and cash index markets play minor roles. The results in this paper provide an unequivocal support to the trading costs hypothesis.     

Does information vault Niagara Falls? Cross-listed trading in New York and Toronto

We document differential private information in cross-border asset pricing using the probability of informed trading (PIN) for Canadian shares traded on both sides of Niagara Falls. Relative to the New York Stock Exchange (NYSE), the Toronto Stock Exchange (TSX) has more informed trades and a larger information share. This cross-border information imbalance is associated with small but positive price premiums in New York as predicted by a model. The dynamics of these premiums depends on trade informativeness. Lastly, the PIN for TSX trading typically rises upon cross-listing on the NYSE, which is consistent with the negative event-study response.     

Automating the price discovery process: Some international comparisons and regulatory implications

Automated trade execution systems are examined with respect to the degree to which they automate the price discovery process. Seven levels of automation of price discovery are identified, and 47 systems are classified according to these criteria. Systems operating at various levels of automation are compared with respect to age, geographical location, and type of securities traded. Information provided to market participants and asymmetries of information between traders with direct access to the automated market and outside investors also are examined. It is found, for example, that the degree of asymmetric information increases with the level of automation of price discovery. The potential for trading abuses related to prearranged trading, noncompetitive execution, and trading ahead of customers is analyzed for each level of automation. Certain levels of automation widen the opportunities for trading abuses in some respects, but may narrow them in others.     

Nasdaq and The Chicago Stock Exchange: An Analysis of Multiple Market Trading

We analyze a set of 97 NASD-listed securities that trade on both the Nasdaq and Chicago Stock Exchange (CHX) to determine if trading costs and price improvement differ between the two markets. We find that order execution costs, which we define by the traded spread and the signed effective half-spread, are significantly lower on the CHX. This difference is consistent over trade types and for trades of at least 1,000 shares. Also, we find that trades occurring on the CHX receive more price improvement than do those occurring on Nasdaq.     

Arbitrage Chains

A privately informed trader will engage in costly arbitrage, that is, trade on his knowledge that the price of an asset is different from the fundamental value if: (1) his order does not move the price immediately to reflect the information; and (2) he can hold the asset until the date when the information is reflected in the price. We study a general equilibrium model in which all agents optimize. In each period, there may be a trader with a limited horizon who has private information about a distant event. Whether he acts on his information, and whether subsequent informed traders act, is shown to depend on the possibility of a sequence or chain of future informed traders spanning the event date. An arbitrageur who receives good news will buy only if it is likely that, at the end of his trading horizon, a subsequent arbitrageur's buying will have pushed up the expected price. We show that limited trading horizons result in inefficient prices, because informed traders do not act on their information until the event date is sufficiently close. We also show that limited horizons can arise because of the cost-carry associated with holding an arbitrage portfolio over an extended period of time.     

Insights from bitcoin trading

We examine commonality in returns and volume for Bitcoin–fiat currency pairs, each trading in a country with a single time zone. Bitcoin has substantial volume and obeys the theory related to commonality, liquidity, and price discovery. We find evidence that one common factor explains 68% of the variance in hourly volume. Though trading is higher on weekdays, there is substantial weekend trading, reflecting high retail participation. Volume is higher on exchanges during local working hours, as seen in forex markets, supporting the view that trading patterns depend on the location of trade rather than the location of the asset traded.     

Adaptive Learning in an Expectational Difference Equation with Several Lags: Selecting among Learnable REE

It is demonstrated that adaptive learning in the least squares sense may be incapable of satisfactorily reducing the number of attainable equilibria in a rational expectations model when focusing on the forward‐solutions to the model. The model examined, as an illustration, is a basic asset pricing model for exchange rate determination that is augmented with technical trading in the currency market in the form of moving averages since it is the most commonly used technique according to questionnaire surveys. The forward‐solutions to such a model are preferable to the backward‐solutions that are normally utilized since announcement effects is an important feature in currency trade. Because of technical trading in foreign exchange, the current exchange rate depends on j _max lags of the exchange rate, meaning that the model has j _max+1 rational expectations equilibria, where several of them are adaptively learnable in the least squares sense. However, since past exchange rates should not affect the current exchange rate when technical trading is absent, it is possible to single out a unique equilibrium among the adaptively learnable equilibria that is economically meaningful. It is worth noting that the model examined can also be viewed as a model for stock price determination in which the forward‐solutions to the model are preferable to the backward‐solutions since the importance of announcement effects is a common characteristic for currency and stock markets.     

Optimal Hedging and Valuation of Nontraded Assets

This paper examines a number of valuation problems faced by an expected-utility-maximizing investor who, over a given time horizon, is constrained to hold an asset which cannot be replicated by dynamic trading and which therefore does not have a unique no-arbitrage price. We first derive the private valuation which the investor assigns to the nontradedasset in order to determine his optimal investment in the traded assets. We thereby show that, as part of this portfolio, the investor hedges the private valuation process of the nontraded asset, rather than its market price process. We also study the price at which the investor would be willing to sell the nontraded asset if he were subsequently prohibited from trading in it, as well as the amount the investor would be willing to pay to removethe trading restriction. All three values are shown to depend in an intuitive manner on the investor's risk aversion, the residual risk of the nontraded asset unhedged by the traded assets, the difference between the constrained holding and optimal unconstrained holding of the asset and the length of the time horizon over which the asset cannot be traded.     

Dynamic Asset Allocation and the Informational Efficiency of Markets

Markets have an allocational role; even in the absence of news about payoffs, prices change to facilitate trade and allocate resources to their best use. Allocational price changes create noise in the signal extraction process, and markets where such trading is important are markets in which we may expect to find a failure of informational efficiency. An important source of allocational trading is the use of dynamic trading strategies caused by the incomplete equitization of risks. Incomplete equitization causes trade. Trade implies the inefficiency of passive strategies, thus requiring investors to determine whether price changes are informational or allocational.     

Optimal Hedging and Valuation of Nontraded Assets

This paper examines a number of valuation problems faced byan expected-utility maximizing investor who, over a given timehorizon, is constrained to hold an asset which cannot be replicatedby dynamic trading and which therefore does not have a uniqueno-arbitrage price. We first derive the private valuation whichthe investor assigns to the nontraded asset in order to determinehis optimal investment in the traded assets. We thereby showthat, as part of this portfolio, the investor hedges the privatevaluation process of the nontraded asset, rather than its marketprice process. We also study the price at which the investorwould be willing to sell the nontraded asset if he were subsequentlyprohibited from trading in it, as well as the amount the investorwould be willing to pay to remove the trading restriction. Allthree values are shown to depend in an intuitive manner on theinvestor’s risk aversion, the residual risk of the nontradedasset unhedged by the traded assets, the difference betweenthe constrained holding and optimal unconstrained holding ofthe asset and the length of the time horizon over which theasset cannot be traded. JEL Classification: G11     

Performance of utility-based strategies for hedging basis risk

The performance of optimal strategies for hedging a claim on a non-traded asset is analysed. The claim is valued and hedged in a utility maximization framework, using exponential utility. A traded asset, correlated with that underlying the claim, is used for hedging, with the correlation ρ typically close to 1. Using a distortion method (Zariphopoulou 2001 Finance Stochastics 5 61–82) we derive a nonlinear expectation representation for the claim’s ask price and a formula for the optimal hedging strategy. We generate a perturbation expansion for the price and hedging strategy in powers of ε²?=1?ρ². The terms in the price expansion are proportional to the central moments of the claim payoff under the minimal martingale measure. The resulting fast computation capability is used to carry out a simulation-based test of the optimal hedging program, computing the terminal hedging error over many asset price paths. These errors are compared with those from a naive strategy which uses the traded asset as a proxy for the non-traded one. The distribution of the hedging error acts as a suitable metric to analyse hedging performance. We find that the optimal policy improves hedging performance, in that the hedging error distribution is more sharply peaked around a non-negative profit. The frequency of profits over losses is increased, and this is measured by the median of the distribution, which is always increased by the optimal strategies. An empirical example illustrates the application of the method to the hedging of a stock basket using index futures.     

The Market Evaluation of Information in Directors' Trades

The purpose of this paper is to examine the propensity, characteristics and performance of directors' trades. Consistent with prior research we show that on average, directors outperform the market. However, we also find that there exist a large number of trades which do not share these abnormal share price returns and consequently have little information content. This has important consequences for market participants who use director trading activity as a signal for their own trading strategies. Using different measures of directors' trades based on trade characteristics, we report that purchases by directors are more informative than sales. In addition, the number of directors trading within a twenty day window and the percentage of the directors' holding that is being traded are both important factors in the abnormal share price performance following the trade.