Quantitative Finance

Managing a book of options on several underlying involves controlling positions of several thousands of financial assets. It is one of the most challenging financial problems involving both pricing and microstructural modeling. An options market maker has to manage both long- and short-dated options having very different dynamics. In particular, short-dated options inventories cannot be managed as a part of an aggregated inventory, which prevents the use of dimensionality reduction techniques such as a factorial approach or first-order Greeks approximation. In this paper, we show that a simple analytical approximation of the solution of the market maker's problem provides significantly higher flexibility than the existing algorithms designing options market making strategies.

Working on different aspects of algorithmic trading we empirically discovered a new market invariant. It links together the volatility of the instrument with its traded volume, the average spread and the volume in the order book. The invariant has been tested on different markets and different asset classes. In all cases we did not find significant violation of the invariant. The formula for the invariant was used for the volatility estimation, which we called the instantaneous volatility. Quantitative comparison showed that it reproduces realised volatility better than one-day-ahead GARCH(1,1) prediction. Because of the short-term prediction nature, the instantaneous volatility could be used by algo developers, volatility traders and other market professionals.

We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and on the intraday price, typically due to error in the prediction of wind power generation. Finally, we solve the problem when taking into account delay constraints in thermal power production.

We summarize the fundamental issues at stake in algorithmic trading, and the progress made in this field over the last twenty years. We first present the key problems of algorithmic trading, describing the concepts of optimal execution, optimal placement, and price impact. We then discuss the most recent advances in algorithmic trading through the use of Machine Learning, discussing the use of Deep Learning, Reinforcement Learning, and Generative Adversarial Networks.

We propose a strategy for automated trading, outline theoretical justification of the profitability of this strategy and overview the hypothetical results in application to currency pairs trading. The proposed methodology relies on the assumption that processes reflecting the dynamics of currency exchange rates are in a certain sense similar to the class of Ornstein-Uhlenbeck processes and exhibits the mean reverting property. In order to describe the quantitative characteristics of the projected return of the strategy, we derive the explicit expression for the running maximum of the Ornstein-Uhlenbeck process stopped at maximum drawdown and look at the correspondence between derived characteristics and the observed ones.

An agent-based model for financial markets has to incorporate two aspects: decision making and price formation. We introduce a simple decision model and consider its implications in two different pricing schemes. First, we study its parameter dependence within a supply-demand balance setting. We find realistic behavior in a wide parameter range. Second, we embed our decision model in an order book setting. Here we observe interesting features which are not present in the equilibrium pricing scheme. In particular, we find a nontrivial behavior of the order book volumes which reminds of a trend switching phenomenon. Thus, the decision making model alone does not realistically represent the trading and the stylized facts. The order book mechanism is crucial.

We introduce a new non parametric method that allows for a direct, fast and efficient estimation of the matrix of kernel norms of a multivariate Hawkes process, also called branching ratio matrix. We demonstrate the capabilities of this method by applying it to high-frequency order book data from the EUREX exchange. We show that it is able to uncover (or recover) various relationships between all the first level order book events associated with some asset when mapped to a 12-dimensional process. We then scale up the model so as to account for events on two assets simultaneously and we discuss the joint high-frequency dynamics.

Many empirical studies have discussed market liquidity, which is regarded as a measure of a booming financial market. Further, various indicators for objectively evaluating market liquidity have also been proposed and their merits have been discussed. In recent years, the impact of high-frequency traders (HFTs) on financial markets has been a focal concern, but no studies have systematically discussed their relationship with major market liquidity indicators, including volume, tightness, resiliency, and depth. In this study, we used agent-based simulations to compare the major liquidity indicators in an artificial market where an HFT participated was compared to one where no HFT participated. The results showed that all liquidity indicators in the market where an HFT participated improved more than those in the market where no HFT participated. Furthermore, as a result of investigating the correlations between the major liquidity indicators in our simulations and the extant empirical literature, we found that market liquidity can be measured not only by the major liquidity indicators but also by execution rate. Therefore, it is suggested that it could be appropriate to employ execution rate as a novel liquidity indicator in future studies.

Recently, most stock exchanges in the U.S. employ maker-taker fees, in which an exchange pays rebates to traders placing orders in the order book and charges fees to traders taking orders from the order book. Maker-taker fees encourage traders to place many orders that provide market liquidity to the exchange. However, it is not clear how maker-taker fees affect the total cost of a taking order, including all the charged fees and the market impact. In this study, we investigated the effect of maker-taker fees on the total cost of a taking order with our artificial market model, which is an agent-based model for financial markets. We found that maker-taker fees encourage market efficiency but increase the total costs of taking orders.

We study the problem of the execution of a moderate size order in an illiquid market within the framework of a solvable Markovian model. We suppose that in order to avoid impact costs, a trader decides to execute her order through a unique trade, waiting for enough liquidity to accumulate at the best quote. We find that despite the absence of a proper price impact, such trader faces an execution cost arising from a non-vanishing correlation among volume at the best quotes and price changes. We characterize analytically the statistics of the execution time and its cost by mapping the problem to the simpler one of calculating a set of first-passage probabilities on a semi-infinite strip. We finally argue that price impact cannot be completely avoided by conditioning the execution of an order to a more favorable liquidity scenario.