Kaj Nyström, Uppsala University

2010-2015

To start this outline we recall the following summary of parts of the original purpose of the project.

`Any financial institution has to address the reality of liquidity risk and we claim that the nature of liquidity risk, though liquidity risk is fundamental to the financial system, to a large extent remains to be understood. In particular, we claim that adequate models for assessing liquidity risk, on the instrumental level, on portfolio level as well as on the level of the financial system as a whole, remain to be developed. The purpose of this research project is to address the problem of liquidity risk on these three levels and to design models and frameworks for liquidity risk which are theoretically sound and applicable to the real underlying industrial problems'.

Throughout the project we have remained faithful to its original purpose with one important additional restriction: at one stage we decided to narrow our focus and to restrict the project to the context of high-frequency trading (HFT). We decided to choose this angle for several reasons but mainly due to the availability of precise and accurate high-frequency data. In fact, through discussions with NasdaqOMX we were able to get hold of all transactions in one or several stocks during a sequence of trading days. Through this very detailed dataset we were able to build the order books in several stocks and hence to study the dynamics of the order books. Note that if you know the dynamics of the order book, then you essentially have a complete description of all aspects of the supply and demand for liquidity as well as of the price formation process. In the context of high frequency trading some of the goals of the project were formulated as follows.

o Use statistical and econometric tools to model high-frequency data.
o Develop market models for trading networks, in the context of high-frequency trading, with a particular focus on market-making/liquidity issues.
o Develop market models for trading networks which accounts for the fundamentals of market microstructure theory: inventories, information asymmetries, market making strategies,....
o Derive and analyze implications of the models on the formation of price, liquidity, volatility,...

In the project we have worked with three papers and these papers constitute the main results of the projects.

Firstly, we have developed and analyzed a model for market making and optimal portfolio liquidation under uncertainty. Market making and optimal portfolio liquidation in the context of electronic limit order books are of considerably practical importance for high frequency (HF) market makers as well as for more traditional brokerage firms supplying optimal execution services for clients. In general the two problems are based on probabilistic models defined on certain reference probability spaces. However, due to uncertainty in model parameters or in periods of extreme market turmoil, ambiguity concerning the correct underlying probability measure may appear and an assessment of model risk, as well as the uncertainty on the choice of the model itself, becomes important, as for a market maker or a trader attempting to liquidate large positions, the uncertainty may result in unexpected consequences due to severe mispricing. We focus on the market making and the optimal liquidation problems using limit orders, accounting for model risk or uncertainty. Both problems are formulated as stochastic optimal control problems, with the controls being the spreads, relative to a reference price, at which orders are placed. The models consider uncertainty in both the drift and volatility of the underlying reference price, for the study of the effect of the uncertainty on the behavior of the market maker, accounting also for inventory restrictions, as well as for the optimal liquidation using limit orders.

Secondly, we have proposed and developed a modeling framework for the dynamics of a reduced form order book in event time and based on event sizes. Compared to previous works, in our framework for the order book we allow the best bid ask spread to be larger than one tick. Based on the modeling assumption that the best bid as well as the best ask price can only move by at most one tick (up or down), when an event occurs, we show that the dynamics of this simplified order book is completely described by a non-linear transformation of two processes (X,Y). A key challenge in the modeling is the empirical fact that the high frequency order flow is strongly autocorrelated, a fact we have to deal with in the modeling of (X,Y). The core of our framework is a semi-linear regression type model for (X,Y), influenced by more classical econometric models, and one important degree of freedom is the potentially non-linear basis functions used in the regression. To understand some of the statistical properties of our models we use results from the theory of random iterative function systems. All components are worked through and explained in an application and the predictability of the model for order flows and price moves are analyzed in the context of a high frequency dataset.

Thirdly, we have considered Hawkes processes as a tool to model high-frequency data. Compared with low frequency data, high frequency data exhibits different empirical properties, for instance, essentially discontinuous evolution paths, time-varying intensities, and self-exciting features. This makes it a challenge to model appropriately the dynamics associated with high frequency data such as order arrival and price formation. To capture the microscopic structures and properties pertaining to limit order books, we have also focused on the modeling high frequency data using Hawkes processes. We have worked with two Hawkes-based models, one with exponential kernels and the other with power-law kernels. The models have been implemented and compared from different perspectives including the goodness of fit to the empirical data. Studies based on both multiple-trading-day data of one stock and multiple-stock data on one trading day indicate that Hawkes processes with slowly-decaying kernels are able to reproduce the intensity of jumps in the price processes more accurately. The results suggest that Hawkes processes with power-law kernels and their implied long memory nature of self-excitation phenomena could, on the level of microstructure, serve as a realistic model for high frequency data.

All of the three results outlined above are frameworks/models to be used to model and analyze the formation of the order book and its dynamics using stochastic control theory and tools from econometrics and statistics. In general all models have some stylized features to make them computable and applicable to the real applications. Subsequent research may try to refine the frameworks and to incorporation additional realistic features. For example, concerning the market maker problem, more sophisticated models can proposed including for example richer dynamics of market orders, impact on the limit order book, adverse selection effects and predictability, to deal with high frequency market making. Further generalizations include market orders and limit orders at best as well as next to best bid and ask together with stochastic spreads.


The problems and context of this project has currently become an internationally very active area of research as HF market makers on many exchanges today make up a substantial part of the total HFT activity. In particular, market making is a core part of the HFT business and there is an increasing interest to understand the principle on which HF market makers base their business. To be part of the international scene and to inform about the projects the members of the project have taken part in a number of international activities some of which we here list.

o December 2010, Zurich, Switzerland. Liquidity risk Europe (Kaj Nyström).
o 2011, Stockholm, Sweden. Information gathering and interviews at Riksbanken, Swedbank, NasdaqOMX and others (Kaj Nyström).
o September 2012, Yerevan, Armenia. Workshop on Stochastic and PDE Methods in Financial Mathematics (Sidi Mohamed Ould Aly).
o September 2012, at Ulm University, Germany. International Summer Academy 2012 on Advanced Stochastic Methods to Model Risk (Changyong Zhang).
o September 2012, New York, USA. High-frequency trading strategies (Kaj Nyström).
o December 2012, Paris, France. 2nd "Market Microstructure: confronting many viewpoints" conference (Sidi Mohamed Ould Aly, Kaj Nyström and Changyong Zhang).

Publications

•    Kaj Nyström, Sidi Mohamed Ould Aly and Changyong Zhang, Market Making and Portfolio Liquidation under Uncertainty, International Journal of Theoretical and Applied Finance, 2014, vol. 17, issue 05, 33 pages.
•    Kaj Nyström, Sidi Mohamed Ould Aly, A Framework for the Modeling of Order Book Dynamics based on Event Sizes, submitted.
•    Kaj Nyström, Sidi Mohamed Ould Aly, and Changyong Zhang, Hawkes Processes with Power-Law Kernels: applications to high frequency data, in preparation.

Papers be downloaded from the homepage of Kaj Nyström at the department of mathematics, Uppsala University, http://katalog.uu.se/empInfo?id=N94-1986, or accessed by sending an email to kaj.nystrom@math.uu.se.