Category: Papers

At Risk.net Cutting Edge’s section:

Liquidity stress-testing using optimal portfolio liquidation by Mike Weber, Bastien Baldacci and Iuliia Manziuk

Optimal turnover, liquidity and autocorrelation by Bastien Baldacci, Jerome Benveniste and Gordon Ritter

Deep calibration of the quadratic rough Heston model by Mathieu Rosenbaum and Jianfei Zhang

An approximate solution for options market-making in high dimension by Bastien Baldacci, Joffrey Derchu and Iuliia Manziuk

• Optimal auction duration: A price formation viewpoint by Paul Jusselin, Thibaut Mastrolia, Mathieu Rosenbaum
  • We consider an auction market in which market makers fill the order book during a given time period while some other investors send market orders. We define the clearing price of the auction as the price maximizing the exchanged volume at the clearing time according to the supply and demand of each market participants. Then we derive in a semi-explicit form the error made between this clearing price and the efficient price as a function of the auction duration. We study the impact of the behavior of market takers on this error. To do so we consider the case of naive market takers and that of rational market takers playing a Nash equilibrium to minimize their transaction costs. We compute the optimal duration of the auctions for 77 stocks traded on Euronext and compare the quality of price formation process under this optimal value to the case of a continuous limit order book. Continuous limit order books are found to be usually sub-optimal. However, in term of our metric, they only moderately impair the quality of price formation process. Order of magnitude of optimal auction durations is from 2 to 10 minutes.
• A note on Almgren-Chriss optimal execution problem with geometric Brownian motion by Bastien Baldacci, Jerome Benveniste
  • 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.
• Algorithmic market making for options by Bastien Baldacci, Philippe Bergault, Olivier Guéant  (published in Quantitative Finance)
  • In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model — e.g. the Heston model — the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.
• Adaptive trading strategies across liquidity pools by Bastien Baldacci, Iuliia Manziuk
  • In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model — e.g. the Heston model — the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.
• An approximate solution for options market-making in high dimension by Bastien Baldacci, Joffrey Derchu, Iuliia Manziuk
  • 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.
• AHEAD : Ad-Hoc Electronic Auction Design by Joffrey Derchu, Philippe Guillot, Thibaut Mastrolia, Mathieu Rosenbaum
  • We introduce a new matching design for financial transactions in an electronic market. In this mechanism, called ad-hoc electronic auction design (AHEAD), market participants can trade between themselves at a fixed price and trigger an auction when they are no longer satisfied with this fixed price. In this context, we prove that a Nash equilibrium is obtained between market participants. Furthermore, we are able to assess quantitatively the relevance of ad-hoc auctions and to compare them with periodic auctions and continuous limit order books. We show that from the investors’ viewpoint, the microstructure of the asset is usually significantly improved when using AHEAD.

Published on arxiv:

A note on Almgren-Chriss optimal execution problem with geometric Brownian motion

Abstract: 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.

Published on arxiv:

On bid and ask side-specific tick sizes

Abstract: The tick size, which is the smallest increment between two consecutive prices for a given asset, is a key parameter of market microstructure. In particular, the behavior of high frequency market makers is highly related to its value. We take the point of view of an exchange and investigate the relevance of having different tick sizes on the bid and ask sides of the order book. Using an approach based on the model with uncertainty zones, we show that when side-specific tick sizes are suitably chosen, it enables the exchange to improve the quality of liquidity provision.

The information content of high-frequency traders aggressive orders: recent evidence by Pamela Saliba

The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem (arXiv link) was published on the Cutting Edge section of Risk.net:

Risk – Cutting Edge – The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem

Authors: Jim Gatheral, Paul Jusselin, Mathieu Rosenbaum

Abstract: Fitting simultaneously SPX and VIX smiles is known to be one of the most challenging problems in volatility modeling. A long-standing conjecture due to Julien Guyon is that it may not be possible to calibrate jointly these two quantities with a model with continuous sample-paths. We present the quadratic rough Heston model as a counterexample to this conjecture. The key idea is the combination of rough volatility together with a price-feedback (Zumbach) effect.

Published on SSRN:

How to build a cross-impact model from first principles: Theoretical requirements and empirical results

Abstract: Cross-impact, namely the fact that on average buy (sell) trades on a financial instrument induce positive (negative) price changes in other correlated assets, can be measured from abundant, although noisy, market data. In this paper we propose a principled approach that allows to perform model selection for cross-impact models, showing that symmetries and consistency requirements are particularly effective in reducing the universe of possible models to a much smaller set of viable candidates, thus mitigating the effect of noise on the properties of the inferred model. We review the empirical performance of a large number of cross-impact models, comparing their strengths and weaknesses on a number of asset classes (futures, stocks, calendar spreads). Besides showing which models perform better, we argue that in presence of comparable statistical performance, which is often the case in a noisy world, it is relevant to favor models that provide ex-ante theoretical guarantees on their behavior in limit cases. From this perspective, we advocate that the empirical validation of universal properties (symmetries, invariances) should be regarded as holding a much deeper epistemological value than any measure of statistical performance on specific model instances.

Published on arxiv:

From microscopic price dynamics to multidimensional rough volatility models

Abstract: Rough volatility is a well-established statistical stylised fact of financial assets. This property has lead to the design and analysis of various new rough stochastic volatility models. However, most of these developments have been carried out in the mono-asset case. In this work, we show that some specific multivariate rough volatility models arise naturally from microstructural properties of the joint dynamics of asset prices. To do so, we use Hawkes processes to build microscopic models that reproduce accurately high frequency cross-asset interactions and investigate their long term scaling limits. We emphasize the relevance of our approach by providing insights on the role of microscopic features such as momentum and mean-reversion on the multidimensional price formation process. We in particular recover classical properties of high-dimensional stock correlation matrices.

Published on arxiv:

Optimal market making with persistent order flow

Abstract: We address the issue of market making on electronic markets when taking into account the self exciting property of market order flow. We consider a market with order flows driven by Hawkes processes where one market maker operates, aiming at optimizing its profit. We characterize an optimal control solving this problem by proving existence and uniqueness of a viscosity solution to the associated Hamilton Jacobi Bellman equation. Finally we propose a methodology to approximate the optimal strategy.