# Research articles for the 2021-01-06

A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak approximation scheme of SDEs with Malliavin weights
arXiv

The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a stochastic weighted minimization with stochastic gradient descent which is inspired by a high-order weak approximation scheme for stochastic differential equations (SDEs) with Malliavin weights. Then solutions to high-dimensional Kolmogorov PDEs or expectations of functionals of solutions to high-dimensional SDEs are accurately approximated without suffering from the curse of dimensionality. Numerical examples for PDEs and SDEs up to 100 dimensions are shown by using second and third-order discretization schemes in order to demonstrate the effectiveness of our method.

Deep learning for efficient frontier calculation in finance
Xavier Warin
arXiv

We propose deep neural network algorithms to calculate efficient frontier in some Mean-Variance and Mean-CVaR portfolio optimization problems. We show that we are able to deal with such problems when both the dimension of the state and the dimension of the control are high. Adding some additional constraints, we compare different formulations and show that a new projected feedforward network is able to deal with some global constraints on the weights of the portfolio while outperforming classical penalization methods. All developed formulations are compared in between. Depending on the problem and its dimension, some formulations may be preferred.

Driver Surge Pricing
arXiv

Ride-hailing marketplaces like Uber and Lyft use dynamic pricing, often called surge, to balance the supply of available drivers with the demand for rides. We study driver-side payment mechanisms for such marketplaces, presenting the theoretical foundation that has informed the design of Uber's new additive driver surge mechanism. We present a dynamic stochastic model to capture the impact of surge pricing on driver earnings and their strategies to maximize such earnings. In this setting, some time periods (surge) are more valuable than others (non-surge), and so trips of different time lengths vary in the induced driver opportunity cost.

First, we show that multiplicative surge, historically the standard on ride-hailing platforms, is not incentive compatible in a dynamic setting. We then propose a structured, incentive-compatible pricing mechanism. This closed-form mechanism has a simple form and is well-approximated by Uber's new additive surge mechanism. Finally, through both numerical analysis and real data from a ride-hailing marketplace, we show that additive surge is more incentive compatible in practice than is multiplicative surge.

Liquidity Stress Testing in Asset Management -- Part 1. Modeling the Liability Liquidity Risk
Thierry Roncalli,Fatma Karray-Meziou,François Pan,Margaux Regnault
arXiv

This article is part of a comprehensive research project on liquidity risk in asset management, which can be divided into three dimensions. The first dimension covers liability liquidity risk (or funding liquidity) modeling, the second dimension focuses on asset liquidity risk (or market liquidity) modeling, and the third dimension considers asset-liability liquidity risk management (or asset-liability matching). The purpose of this research is to propose a methodological and practical framework in order to perform liquidity stress testing programs, which comply with regulatory guidelines (ESMA, 2019) and are useful for fund managers. The review of the academic literature and professional research studies shows that there is a lack of standardized and analytical models. The aim of this research project is then to fill the gap with the goal to develop mathematical and statistical approaches, and provide appropriate answers.

In this first part that focuses on liability liquidity risk modeling, we propose several statistical models for estimating redemption shocks. The historical approach must be complemented by an analytical approach based on zero-inflated models if we want to understand the true parameters that influence the redemption shocks. Moreover, we must also distinguish aggregate population models and individual-based models if we want to develop behavioral approaches. Once these different statistical models are calibrated, the second big issue is the risk measure to assess normal and stressed redemption shocks. Finally, the last issue is to develop a factor model that can translate stress scenarios on market risk factors into stress scenarios on fund liabilities.

Order book dynamics with liquidity fluctuations: limit theorems and large deviations
Helder Rojas,Artem Logachov,Anatoly Yambartsev
arXiv

We propose a class of stochastic models for a dynamics of limit order book with different type of liquidities. Within this class of models we study the one where a spread decreases uniformly, belonging to the class of processes known as a population processes with uniform catastrophes. The law of large numbers (LLN), central limit theorem (CLT) and large deviations (LD) are proved for our model with uniform catastrophes. Our results allow us to satisfactorily explain the volatility and local trends in the prices, relevant empirical characteristics that are observed in this type of markets. Furthermore, it shows us how these local trends and volatility are determined by the typical values of the bid-ask spread. In addition, we use our model to show how large deviations occur in the spread and prices, such as those observed in flash crashes.

Perpetual American options with asset-dependent discounting
In this paper we consider the following optimal stopping problem $$V^{\omega}_{\rm A}(s) = \sup_{\tau\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)],$$ where the process $S_t$ is a jump-diffusion process, $\mathcal{T}$ is a family of stopping times while $g$ and $\omega$ are fixed payoff function and discount function, respectively. In a financial market context, if $g(s)=(K-s)^+$ or $g(s)=(s-K)^+$ and $\mathbb{E}$ is the expectation taken with respect to a martingale measure, $V^{\omega}_{\rm A}(s)$ describes the price of a perpetual American option with a discount rate depending on the value of the asset process $S_t$. If $\omega$ is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function $V^{\omega}_{\rm A}(s)$. This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of $V^{\omega}_{\rm A}(s)$. We also prove a put-call symmetry for American options with asset-dependent discounting. In the case when $S_t$ is a geometric L\'evy process we give exact expressions using the so-called omega scale functions introduced in Li and Palmowski (2018). We prove that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function $V^{\omega}_{\rm A}(s)$.