Numerical approximation for an impulse control problem arising in portfolio selection under liquidity risk

18 pagesWe investigate numerical aspects of a portfolio selection problem studied in [10], in which we suggest a model of liquidity risk and price impact and formulate the problem as an impulse control problem under state constraint. We show that our impulse control problem could be reduced to an iterative sequence of optimal stopping problems. Given the dimension of our problem and the complexity of its solvency region, we use Monte Carlo methods instead of finite difference methods to calculate the value function, the transaction and no-transaction regions. We provide a numerical approximation algorithm as well as numerical results for the optimal transaction strategy

The Self-Financing Equation in High Frequency Markets

High Frequency Trading (HFT) represents an ever growing proportion of all financial transactions as most markets have now switched to electronic order book systems. The main goal of the paper is to propose continuous time equations which generalize the self-financing relationships of frictionless markets to electronic markets with limit order books. We use NASDAQ ITCH data to identify significant empirical features such as price impact and recovery, rough paths of inventories and vanishing bid-ask spreads. Starting from these features, we identify microscopic identities holding on the trade clock, and through a diffusion limit argument, derive continuous time equations which provide a macroscopic description of properties of the order book. These equations naturally differentiate between trading via limit and market orders. We give several applications (including hedging European options with limit orders, market maker optimal spread choice, and toxicity indexes) to illustrate their impact and how they can be used to the benefit of Low Frequency Traders (LFTs)

Essays on Liquidity Risk and Modern Market Microstructure

Liquidity, often defined as the ability of markets to absorb large transactions without much effect on prices, plays a central role in the functioning of financial markets. This dissertation aims to investigate the implications of liquidity from several different perspectives, and can help to close the gap between theoretical modeling and practice. In the first part of the thesis, we study the implication of liquidity costs for systemic risks in markets cleared by multiple central counterparties (CCPs). Recent regulatory changes are trans- forming the multi-trillion dollar swaps market from a network of bilateral contracts to one in which swaps are cleared through central counterparties (CCPs). The stability of the new framework de- pends on the resilience of CCPs. Margin requirements are a CCP’s first line of defense against the default of a counterparty. To capture liquidity costs at default, margin requirements need to increase superlinearly in position size. However, convex margin requirements create an incentive for a swaps dealer to split its positions across multiple CCPs, effectively “hiding” potential liquidation costs from each CCP. To compensate, each CCP needs to set higher margin requirements than it would in isolation. In a model with two CCPs, we define an equilibrium as a pair of margin schedules through which both CCPs collect sufficient margin under a dealer’s optimal allocation of trades. In the case of linear price impact, we show that a necessary and sufficient condition for the existence of an equilibrium is that the two CCPs agree on liquidity costs, and we characterize all equilibria when this holds. A difference in views can lead to a race to the bottom. We provide extensions of this result and discuss its implications for CCP oversight and risk management. In the second part of the thesis, we provide a framework to estimate liquidity costs at a portfolio level. Traditionally, liquidity costs are estimated by means of single-asset models. Yet such an approach ignores the fact that, fundamentally, liquidity is a portfolio problem: asset prices are correlated. We develop a model to estimate portfolio liquidity costs through a multi-dimensional generalization of the optimal execution model of Almgren and Chriss (1999). Our model allows for the trading of standardized liquid bundles of assets (e.g., ETFs or indices). We show that the benefits of hedging when trading with many assets significantly reduce cost when liquidating a large position. In a “large-universe” asymptotic limit, where the correlations across a large number of assets arise from a relatively few underlying common factors, the liquidity cost of a portfolio is essentially driven by its idiosyncratic risk. Moreover, the additional benefit from trading standardized bundles is roughly equivalent to increasing the liquidity of individual assets. Our method is tractable and can be easily calibrated from market data. In the third part of the thesis, we look at liquidity from the perspective of market microstructure, we analyze the value of limit orders at different queue positions of the limit order book. Many modern financial markets are organized as electronic limit order books operating under a price- time priority rule. In such a setup, among all resting orders awaiting trade at a given price, earlier orders are prioritized for matching with contra-side liquidity takers. In practice, this creates a technological arms race among high-frequency traders and other automated market participants to establish early (and hence advantageous) positions in the resulting first-in-first-out (FIFO) queue. We develop a model for valuing orders based on their relative queue position. Our model identifies two important components of positional value. First, there is a static component that relates to the trade-off at an instant of trade execution between earning a spread and incurring adverse selection costs, and incorporates the fact that adverse selection costs are increasing with queue position. Second, there is also a dynamic component, that captures the optionality associated with the future value that accrues by locking in a given queue position. Our model offers predictions of order value at different positions in the queue as a function of market primitives, and can be empirically calibrated. We validate our model by comparing it with estimates of queue value realized in backtesting simulations using marker-by-order data, and find the predictions to be accurate. Moreover, for some large tick-size stocks, we find that queue value can be of the same order of magnitude as the bid-ask spread. This suggests that accurate valuation of queue position is a necessary and important ingredient in considering optimal execution or market-making strategies for such assets

Optimal Execution with Dynamic Order Flow Imbalance

We examine optimal execution models that take into account both market microstructure impact and informational costs. Informational footprint is related to order flow and is represented by the trader's influence on the flow imbalance process, while microstructure influence is captured by instantaneous price impact. We propose a continuous-time stochastic control problem that balances between these two costs. Incorporating order flow imbalance leads to the consideration of the current market state and specifically whether one's orders lean with or against the prevailing order flow, key components often ignored by execution models in the literature. In particular, to react to changing order flow, we endogenize the trading horizon $T$. After developing the general indefinite-horizon formulation, we investigate several tractable approximations that sequentially optimize over price impact and over $T$. These approximations, especially a dynamic version based on receding horizon control, are shown to be very accurate and connect to the prevailing Almgren-Chriss framework. We also discuss features of empirical order flow and links between our model and "Optimal Execution Horizon" by Easley et al (Mathematical Finance, 2013).Comment: 31 pages, 8 figure