Quantitative Finance

We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the conditional diffusion limit of the price process is the so-called Brownian meander.

We consider a Nash equilibrium between two high-frequency traders in a simple market impact model with transient price impact and additional quadratic transaction costs. Extending a result by Schöneborn (2008), we prove existence and uniqueness of the Nash equilibrium and show that for small transaction costs the high-frequency traders engage in a "hot-potato game", in which the same asset position is sold back and forth. We then identify a critical value for the size of the transaction costs above which all oscillations disappear and strategies become buy-only or sell-only. Numerical simulations show that for both traders the expected costs can be lower with transaction costs than without. Moreover, the costs can increase with the trading frequency when there are no transaction costs, but decrease with the trading frequency when transaction costs are sufficiently high. We argue that these effects occur due to the need of protection against predatory trading in the regime of low transaction costs.

We describe a simple model for speculative trading based on adaptive behavior of economic agents.The adaptive behavior is expressed through a feedback mechanism for changing agents' stock-to-bond ratios, depending on the past performance of their portfolios.The stock price is set according to the demand-supply for the asset derived from the agents' target risk levels. Using the methodology of agent-based modeling we show that agents, acting endogenously and adaptively, create a persistent price bubble. The price dynamics generated by the trading process does not reveal any singularities, however the process is accompanied by growing aggregated risk that indicates increasing likelihood of a crash.

We introduce a new Self-Organized Criticality (SOC) model for simulating price evolution in an artificial financial market, based on a multilayer network of traders. The model also implements, in a quite realistic way with respect to previous studies, the order book dy- namics, by considering two assets with variable fundamental prices. Fat tails in the probability distributions of normalized returns are observed, together with other features of real financial markets.

We solve explicitly the Almgren-Chriss optimal liquidation problem where the stock price process follows a geometric Brownian motion. Our technique is to work in terms of cash and to use functional analysis tools. We show that this framework extends readily to the case of a stochastic drift for the price process and the liquidation of a portfolio.

One popular approach to model the limit order books dynamics of the best bid and ask at level-1 is to use the reduced-form diffusion approximations. It is well known that the biggest contributing factor to the price movement is the imbalance of the best bid and ask. We investigate the data of the level-1 limit order books of a basket of stocks and study the numerical evidence of drift, correlation, volatility and their dependence on the imbalance. Based on the numerical discoveries, we develop a nonparametric discrete model for the dynamics of the best bid and ask, which can be approximated by a reduced-form model with analytical tractability that can fit the empirical data of correlation, volatilities and probability of price movement simultaneously.

Reinforcement learning is explored as a candidate machine learning technique to enhance existing analytical solutions for optimal trade execution with elements from the market microstructure. Given a volume-to-trade, fixed time horizon and discrete trading periods, the aim is to adapt a given volume trajectory such that it is dynamic with respect to favourable/unfavourable conditions during realtime execution, thereby improving overall cost of trading. We consider the standard Almgren-Chriss model with linear price impact as a candidate base model. This model is popular amongst sell-side institutions as a basis for arrival price benchmark execution algorithms. By training a learning agent to modify a volume trajectory based on the market's prevailing spread and volume dynamics, we are able to improve post-trade implementation shortfall by up to 10.3% on average compared to the base model, based on a sample of stocks and trade sizes in the South African equity market.

In today's economy, selling a new zero-marginal cost product is a real challenge, as it is difficult to determine a product's "correct" sales price based on its profit and dissemination. As an example, think of the price of a new app or video game. New sales mechanisms for selling this type of product need to be designed, in particular ones that consider consumer preferences and reality. Current auction mechanisms establish a time deadline for the auction to take place. This deadline is set to increase the number of bidders and thus the final offering price. Consumers want to obtain the product as quickly as possible from the moment they become interested in it, and this time does not always coincide with the seller's deadline. Naturally, consumers also want to pay a price they consider "fair". Here we introduce an auction model where buyers continuously place bids and the challenge is to decide quickly whether or not to accept them. The model does not include a deadline for placing bids, and exhibits self-organized criticality; it presents a critical price from which a bid is accepted with probability one, and avalanches of sales above this value are observed. This model is of particular interest for startup companies interested in profit as well as making the product known on the market.

This article provides a simple explanation of the asymptotic concavity of the price impact of a meta-order via the microstructural properties of the market. This explanation is made more precise by a model in which the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the proposed explanation is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions, one of which can be tested, at least for large-tick stocks, using publicly available market data.

Optimal trading strategies for pairs trading have been studied by models that try to find either optimal shares of stocks by assuming no transaction costs or optimal timing of trading fixed numbers of shares of stocks with transaction costs. To find optimal strategies which determine optimally both trade times and number of shares in pairs trading process, we use a singular stochastic control approach to study an optimal pairs trading problem with proportional transaction costs. Assuming a cointegrated relationship for a pair of stock log-prices, we consider a portfolio optimization problem which involves dynamic trading strategies with proportional transaction costs. We show that the value function of the control problem is the unique viscosity solution of a nonlinear quasi-variational inequality, which is equivalent to a free boundary problem for the singular stochastic control value function. We then develop a discrete time dynamic programming algorithm to compute the transaction regions, and show the convergence of the discretization scheme. We illustrate our approach with numerical examples and discuss the impact of different parameters on transaction regions. We study the out-of-sample performance in an empirical study that consists of six pairs of U.S. stocks selected from different industry sectors, and demonstrate the efficiency of the optimal strategy.