HFT and asymptotics for small risk aversion in a Markov renewal model 

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Directional bets and the power of information in market making

We address the problem of optimal market making for assets with small tick. Even though market making is not a directional strategy per se, an agent detaining information on the future evolution of the stock price should be able to take advantage from it, and adapt her nondirectional strategy. Most of the existing literature on market making assumes that the stock price is an undrifted Brownian motion, which simplies the HJB equation and substantially implies that the optimal strategy is independent from the current mid-price. But what happens in the general case? And in particular, if the martingale dynamic is slightly deformed, how does it deform the market making optimal policy? And what about stochastic volatility? We will see that the agent information translate the agent spread centre in the market direction, creating a mix directional/market making strategy, while volatility widens the spread.
The market is made by an uncontrolled process, i.e. the stock price, which is assumed exogenous and independent from the agent strategy which does not impact them (small agent assumption), and the controlled ones, describing the agent portfolio and how her limit orders are matched.

Semi Markov model for market microstructure

For large tick assets, the stock price cannot be assumed continuous anymore, and point processes replaces It^o diusion. This approach is general, and suitable for small tick assets as well. We use a Markov renewal process (MRP) to describe the random sequence (Tn; Jn). Markov renewal theory [53] is largely studied in reliability for describing failure of systems and machines, and one purpose is to show how it can be applied to market micro-structure. By considering a suitable Markov chain modeling for (Jn), we are able to reproduce the mean-reversion of price returns, while allowing arbitrary jump size, i.e. the price can jump of more than one tick. On the other hand, the counting process (Nt), which may depend on the Markov chain, models the jump activity, and in particular the alternation of high and low activity phases. Finally, MRPs ensure a Brownian motion behavior of the price process at macroscopic scales, which is consistent with the classic diusive approach in the case of daily observations. Our MRP model is a rather simple but robust model, easy to understand, simulate and estimate, both parametrically and non-parametrically. An important feature of the MRP approach is the semi-Markov property: the price process can be embedded into a Markov system with few additional and observable state variables, ensuring the tractability of the model for applications to market making and statistical arbitrage. We underline that one of our main prerogative is to provide a model involving only observable variables, so that non-parametric estimation is easy and no optimization routine is necessary (as in [19] e.g.), which is a common issue when hidden components are present. MRPs are powerful tools to model the dependence between jump timestamps and marks, dependence which is empirically shown (and theoretically supported by [25]), and for which an explanation relying on the nature of the LOB is provided.

HFT and asymptotics for small risk aversion in a Markov renewal model

We study an optimal high frequency trading problem for large tick assets, using a market microstructure model reaching a good compromise between accuracy and tractability. The stock price is driven by a Markov Renewal Process (MRP), as previously described, while market orders arrive in the limit order book via a point process correlated with the stock price itself. Market data are taken from tick-by-tick observation of the 3-month future EUROSTOXX50, on February 2011, from 09:00:000 to 17:00:00.000 (CET). We shall rely on the previous section (see also [36]), where we show how Markov Renewal processes are an extremely exible and pertinent tool to model the stock price at high frequency, as well as easy to estimate, simulate and understand. We assume that the bid-ask spread is constantly one tick and the stock price jumps of one tick, which is consistent with liquid large tick assets.
By modeling the stock price by a pure jump process, we are able to introduce probabilistic and mechanical dependences between the price evolution and the trades arrival. We can easily introduce correlation between the next price jump and proportion of ask/bid trades before the next jump, as well as the jump risk. In the context of option pricing, jumps represent a source of market incompleteness, leading to unhedgeable claims: similarly, jumps of the stock price in the electronic market are a real source of risk for the market maker. More precisely, the agent faces two kind of risk due to the stock price jump.
i) Market risk: when the price suddenly jumps, the whole agent inventory is re-evaluated, changing the portfolio value in no time (i.e. a nite amount of risk in no time, whereas the Brownian motion has quadratic variation proportional to the interval length).
ii) Adverse selection risk: in our model, we assume that an upwards (downwards) jump at time t corresponds to a big market order clearing the liquidity on the best ask (bid) price level. If the agent posts a small limit order, say on the bid side, the latter has to be executed, since (we guess that) the goal of the big market order was to clear all the available liquidity rather than consuming a xed amount of it. In this sense, the agent does not aect the market dynamics. In this scenario, the agent is systematically penalized, since she sells liquidity at t􀀀 for less than its value at t. This risk, known as adverse selection, can and will be incorporated, measured and hedged, thanks to our market model.

Long memory patterns in high-frequency trading

We present a long memory model, based on marked point processes, which describes the stock price in the limit order book: its marks are driven by a VLMC (an ecient modeling of high-order Markov chains), while the inter-arrival times (between two consecutive jumps of the price) are represented by a self-exciting and independent univariate Hawkes process. Instead of modeling directly the mid-price, we introduce the fair price process, with the intent to reduce the microstructure noise and represent the fundamental value of the asset: empirical evidences show that micro-structural trends, a source of statistical arbitrage, emerge after noise reduction. The agent participates to the market both via impulsive (pseudo) market orders and limit orders, whose execution is modeled by Cox processes. Once the system is embedded into a Markovian one, we illustrate a trading algorithm based on optimal control techniques: we reduce the complex HJB equation associated to the problem to a simple system of variational ODE’s, for which an explicit Euler scheme gives a computationally non-expensive solution. Finally, Monte Carlo simulations show that the VLMC strategy over-performs the benchmark of a uninformed agent modeling the fair price as a simple Markov chain.
We address a high-frequency trader wanting to detect short-term patterns of the stock price, controlling the market risk associated to her portfolio. We work with liquid large tick assets (see [31] for a rigorous quantitative denition), where the price jumps and the bid-ask spread are unitary.

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The market framework and the control problem

Let us recall the setting of the market-making problem, taken from [6], where the agent places, continuously and on both sides, limit orders of unitary size at a certain distance (()t) from the opposite best policies; orders are matched by two independent Cox processes having intensities decaying with (()t). For the rest of the paper we assume that the time variable t ranges in [0; T], T < 1.

The spread width and the round-trip execution

The width of the optimal spread is constant over time, and does not depend on any state variable. It depends on the shape parameter k of the limit order book, saying that faster is the intensity decay, the smaller is the agent spread. As expected, xed costs widen the optimal spread, since the agent has to compensate the market participation cost: for ‘ ! 1, the agent eventually avoids the market. On the contrary, the prot of a round-trip execution does not depend on the transaction costs: the market maker always expects the same reward from a single round-trip execution, no matter the value of ‘, even thought the probability of the event decreases with ‘, as shown by the term c(A; k; ‘) given in (2.2.3), which is an exponential discount (in ‘) of the value function.

The spread center

The center of the optimal spread is impacted by the directional bet through the martingale deviation « (t; z). If « (t; z) > 0 (resp. < 0), the agent estimates that a holding a long (resp. short) position until T is protable. In order to increase her chances to buy (resp. sell) and decreases those to sell (resp. buy), she entirely translates her spread of « (t; z), becoming more aggressive on the bid (resp. ask) and more passive on the opposite one. Thanks to this \spread shift », the agent reaches an optimal trade-o between anticipating the market and providing liquidity, creating a mixed directional/liquidity strategy.

The martingale policies and the information advantage

By denition, strategies for no risk aversion are extremely exposed to market and inventory risk, since the directional component is extremely strong. Inventory is not necessarily bounded (notice that the optimal policy does not depend on the current inventory level) and the directional bets incorporated in the optimal policy lead to high-risk-high-reward proles. Directional bets comes from the agent awareness of the non-martingality of the stock price: under no information, the agent guesses (committing an error) that the mid-price is a martingale, i.e. that  » 0, and her trading policy is given by (0;0) = 1=k + ‘ .

Table of contents :

1 Introduction 
1.1 A introduction to the limit order book
1.2 Directional bets and the power of information in market making
1.3 Semi Markov model for market microstructure
1.4 HFT and asymptotics for small risk aversion in a Markov renewal model
1.5 Long memory patterns in high-frequency trading
2 Directional bets and the power of information in market making 
2.1 The market framework and the control problem
2.2 The explicit solution in the no risk aversion case
2.2.1 A nancial interpretation of the value function
2.2.2 A nancial interpretation of the optimal controls
2.3 The perturbation approach for positive risk aversion
2.3.1 A nancial interpretation of the value function
2.3.2 A nancial interpretation of the optimal controls
2.3.3 The special case of small martingale deviation
2.4 Examples
2.4.1 The Ornstein-Uhlenbeck process
2.4.2 The arithmetic Heston model
2.5 Numerical experiments
2.6 Appendix: the multi-asset model
2.6.1 The no risk aversion case
2.6.2 The case of small risk aversion under small martingale deviation
3 Semi Markov model for market microstructure 
3.1 Semi-Markov model
3.1.1 Price return modelling
3.1.2 Tick times modeling
3.1.3 Statistical inference
3.1.4 Price simulation
3.1.5 Semi-Markov property
3.1.6 Comparison with respect to Hawkes processes
3.2 Scaling limit
3.3 Mean Signature plot
3.4 Appendix: the mean signature plot
3.5 Appendix: a comparison to the Eurostoxx50
4 HFT and asymptotics for small risk aversion in a Markov renewal model 
4.1 Stock price in the limit order book
4.1.1 Markov renewal model
4.1.2 The stock price conditional mean and the trend indicator
4.2 Market order ow modeling and adverse selection
4.3 The market making problem
4.4 Value function and optimal controls: a perturbation approach
4.4.1 The no risk aversion case
4.4.2 The small risk aversion case
4.5 Appendix: the trade intensity function
4.6 Appendix: the estimation of the agent execution distribution #(dk;L)
4.7 Appendix: properties of the function T (t; s)
4.7.1 The PDE representation
4.7.2 The probabilistic representation
4.8 Appendix: proof of Theorem 4.3
5 Long memory patterns in high-frequency trading 
5.1 From the mid-price to the fair one
5.2 The dynamic of the fair price
5.2.1 The tick times
5.2.2 The marks of the stock price
5.3 Execution via limit orders
5.4 The optimal trading problem
5.4.1 The agent strategy and the portfolio dynamic
5.4.2 The HJB equation
5.5 Numerical results
5.6 Appendix: proof of Theorem 5.1
5.7 Appendix: proof of Proposition 5.1


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