Past Mathematical and Computational Finance Internal Seminar

Whether one uses the sales, the number of employees or any other proxy for firm "size", it is well known that this quantity is power-law distributed, with important consequences to aggregate macroeconomic fluctuations. The Gibrat model explained this by proposing that firms grow multiplicatively, and much work has been done to study the statistics of their growth rates. Inspired by past work in the statistics of financial returns, I present a new framework to study these growth rates. In particular, I will show that they follow approximately Gaussian statistics, provided their heteroskedastic nature is taken into account. I will also elucidate the size/volatility scaling relation, and show that volatility may have a strong sectoral dependence. Finally, I will show how this framework can be used to study intra-firm and supply chain dynamics.

Joint work with JP Bouchaud and Angelo Secchi.

Abstract: In many situations, one needs to decide between acting to reveal data about a system and acting to generate profit; this is the trade-off between exploration and exploitation. A simple situation where we face this trade-off is a multiarmed bandit problem, where one has M ‘bandits’ which generate reward from an unknown distribution, and one must choose which bandit to play at each time. The key difficulty in the multi-armed bandit problem is that the action often affects the information obtained. Due to the curse of dimensionality, solving the bandit problem directly is often computationally intractable.

In this talk, we will formulate a general class of the multi-armed bandit problem as a relaxed stochastic control problem. By introducing an entropy premium, we obtain a smooth asymptotic approximation to the value function. This yields a novel semi-index approximation of the optimal decision process, obtained numerically by solving a fixed point problem, which can be interpreted as explicitly balancing an exploration–exploitation trade-off.  Performance of the resulting Asymptotic Randomised Control (ARC) algorithm compares favourably with other approaches to correlated multi-armed bandits.

As an application of the multi-armed bandit, we also consider a multi-armed bandit problem where the observation from each bandit arrive from a Generalised Linear Model. We then use such model to consider a dynamic online pricing problem. The numerical simulation shows that the ARC algorithm also performs well compared to others.
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We analyze the regularity of solutions and discrete-time approximations of extended mean-field control (extended MFC) problems, which seek optimal control of McKean-Vlasov dynamics with coefficients involving mean-field interactions both on the  state and actions, and where objectives are optimized over
open-loop strategies.

We show for a large class of extended MFC problems that the unique optimal open-loop control is 1/2-Hölder continuous in time. Based on the regularity of the solution, we prove that the value functions of such extended MFC problems can be approximated by those with piecewise constant controls and discrete-time state processes arising from Euler-Maruyama time stepping up to an order 1/2 error, which is optimal in our setting. Further, we show that any epsilon-optimal control of these discrete-time problems
converge to the optimal control of the original problems.

To establish the time regularity of optimal controls and the convergence of time discretizations, we extend the canonical path regularity results to general coupled 
McKean-Vlasov forward-backward stochastic differential equations, which are of independent interest.

This is based on join work joint work with C. Reisinger and Y. Zhang.

Agent-based models are an intuitive, interpretable way to model markets and give us a powerful mechanism to analyze counterfactual scenarios that might rarely occur in historical market data. However, building realistic agent-based models is challenging and requires that we (a) ensure that agent behaviors are realistic, and (b) calibrate the agent composition using real data. In this talk, we will present our work to build realistic agent-based models using a multi-agent reinforcement learning approach. Firstly, we show that we can learn a range of realistic behaviors for heterogeneous agents using a shared policy conditioned on agent parameters and analyze the game-theoretic implications of this approach. Secondly, we propose a new calibration algorithm (CALSHEQ) which can estimate the agent composition for which calibration targets are approximately matched, while simultaneously learning the shared policy for the agents. Our contributions make the building of realistic agent-based models more efficient and scalable.

The development of high frequency and algorithmic trading allowed to considerably reduce the bid ask spread by increasing liquidity in limit order books. Beyond the problem of optimal placement of market and limit orders, the possibility to cancel orders for free leaves room for price manipulations, one of such being spoofing. Detecting spoofing from a regulatory viewpoint is challenging due to the sheer amount of orders and difficulty to discriminate between legitimate and manipulative flows of orders. However, it is empirical evidence that volume imbalance reflecting offer and demand on both sides of the limit order book has an impact on subsequent price movements. Spoofers use this effect to artificially modify the imbalance by posting limit orders and then execute market orders at subsequent better prices while canceling at a high speed their previous limit orders. In this work we set up a model to determine where a spoofer would place its limit orders to maximize its gains as a function of the imbalance impact on the price movement. We study the solution of this non local optimization problem as a function of the imbalance. With this at hand, we calibrate on real data from TMX the imbalance impact (as a function of its depth) on the resulting price movement. Based on this calibration and theoretical results, we then provide some methods and numerical results as how to detect in real time some eventual spoofing behavior based on Wasserstein distances. Joint work with Tao Xuan (SJTU), Ling Lan (SJTU) and Andrew Day (Western University)
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I will present a simple model of market microstructure which explains the concavity of price impact. In the proposed model, 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 model 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 of the model, one of which can be tested using publicly available market data and does not require the (difficult to obtain) information about meta-orders. I will present the theoretical results and will support them with the empirical analysis.

We show how traders use immediate execution limit orders (IELOs) to liquidate a position when the time between a trade attempt and the outcome of the attempt is random, i.e., there is latency in the marketplace and latency is random. We frame our model as a delayed impulse control problem in which the trader controls the times and the price limit of the IELOs she sends to the exchange. The contribution of the paper is twofold: (i) Our paper is the first to study an optimal liquidation problem that accounts for random delays, price impact, and transaction costs. (ii) We introduce a new type of impulse control problem with stochastic delay, not previously studied in the literature. We characterise the value functions as the solution to a coupled system of a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) and a partial differential equation. We use a Feynman-Kac type representation to reduce the system of coupled value functions to a non-standard HJBQVI, and we prove existence and uniqueness of this HJBQVI in a viscosity sense. Finally, we implement the latency-optimal strategy and compare it with three benchmarks:  (i)  optimal execution with deterministic latency, (ii) optimal execution with zero latency, (iii) time-weighted average price strategy. We show that when trading in the EUR/USD currency pair, the latency-optimal strategy outperforms the benchmarks between ten USD per million EUR traded and ninety USD per million EUR traded.

We consider continuous time financial models with continuous paths, in a pathwise setting using functional Ito calculus. We look at applications of optimal transport duality in context of robust pricing and hedging and that of calibration. First, we explore exntesions of the discrete-time results in Aksamit et al. [Math. Fin. 29(3), 2019] to a continuous time setting. Second, we addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton--Jacobi--Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options and VIX futures simultaneously.

Based on joint works with Ivan Guo, Gregoire Loeper, Shiyi Wang.
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Abstract: We use a deep neural network to generate controllers for optimal trading on high frequency data. For the first time, a neural network learns the mapping between the preferences of the trader, i.e. risk aversion parameters, and the optimal controls. An important challenge in learning this mapping is that in intraday trading, trader's actions influence price dynamics in closed loop via the market impact. The exploration--exploitation tradeoff generated by the efficient execution is addressed by tuning the trader's preferences to ensure long enough trajectories are produced during the learning phase. The issue of scarcity of financial data is solved by transfer learning: the neural network is first trained on trajectories generated thanks to a Monte-Carlo scheme, leading to a good initialization before training on historical trajectories. Moreover, to answer to genuine requests of financial regulators on the explainability of machine learning generated controls, we project the obtained ``blackbox controls'' on the space usually spanned by the closed-form solution of the stylized optimal trading problem, leading to a transparent structure. For more realistic loss functions that have no closed-form solution, we show that the average distance between the generated controls and their explainable version remains small. This opens the door to the acceptance of ML-generated controls by financial regulators.
 

Multi-agent reinforcement learning (MARL) has enjoyed substantial successes in many applications including the game of Go, online Ad bidding systems, realtime resource allocation, and autonomous driving. Despite the empirical success of MARL, general theories behind MARL algorithms are less developed due to the intractability of interactions, complex information structure, and the curse of dimensionality. Instead of directly analyzing the multi-agent games, mean-field theory provides a powerful approach to approximate the games under various notions of equilibria. Moreover, the analytical feasible framework of mean-field theory leads to learning algorithms with theoretical guarantees. In this talk, we will demonstrate how mean-field theory can contribute to the simultaneous-learning-and-decision-making problems with unknown rewards and dynamics. 

To approximate Nash equilibrium, we first formulate a generalized mean-field game (MFG) and establish the existence and uniqueness of the MFG solution. Next we show the lack of stability in naive combination of the Q-learning algorithm and the three-step fixed-point approach in classical MFGs. We then propose both value-based and policy-based algorithms with smoothing and stabilizing techniques, and establish their convergence and complexity results. The numerical performance shows superior computational efficiency. This is based on joint work with Xin Guo (UC Berkeley), Anran Hu (UC Berkeley), and Junzi Zhang (Stanford).

If time allows, we will also discuss learning algorithms for multi-agent collaborative games using mean-field control. The key idea is to establish the time consistent property, i.e., the dynamic programming principle (DPP) on the lifted probability measure space. We then propose a kernel-based Q-learning algorithm. The convergence and complexity results are carried out accordingly. This is based on joint work with Haotian Gu, Xin Guo, and Xiaoli Wei (UC Berkeley).

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints.  We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we prove that LASSO leads to parameter shrinkage, propose measures to quantify robustness of neural networks to adversarial examples and compute sensitivities of optimised certainty equivalents in finance. We also propose extensions of this framework to a multiperiod setting. This talk is based on joint work with Daniel Bartl, Samuel Drapeau and Jan Obloj.

We study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. We will see that the frontier of the LOB (e.g., the best ask price) is the value function of a mean-field stochastic control problem, as the limiting version of a Bertrand-type competition among the liquidity providers.
With a detailed analysis on the N-seller static Bertrand game, we formulate a continuous time limiting mean-field control problem of the representative seller.
We then validate the dynamic programming principle (DPP) and show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation.
We argue that the value function can be used to obtain the equilibrium density function of the LOB. (Joint work with Jin Ma)

Abstract: In this talk we start by introducing a simple model for interbank default contagion in the vein of the  seminal clearing frameworks of Eisenberg & Noe (2001) and Rogers & Veraart (2013). The key feature, and main novelty, consists in combining stochastic dynamics of the external assets with a simple but realistic balance sheet methodology for determining early defaults. After first developing the model for a finite number of banks, we present a natural way of passing to the mean field limit such that the original network structure (of the interbank obligations) is maintained in a meaningful way. Thus, we provide a clear connection between the more classical network-based literature on systemic risk and the recent approaches rooted in stochastic particle systems and mean field theory.