Backward Stochastic Differential Equation (BSDE)

Unsupervised learning-based calibration scheme for Rough Bergomi model ArXiv ID: 2412.02135 “View on arXiv” Authors: Unknown Abstract Current deep learning-based calibration schemes for rough volatility models are based on the supervised learning framework, which can be costly due to a large amount of training data being generated. In this work, we propose a novel unsupervised learning-based scheme for the rough Bergomi (rBergomi) model which does not require accessing training data. The main idea is to use the backward stochastic differential equation (BSDE) derived in [“Bayer, Qiu and Yao, {“SIAM J. Financial Math.”}, 2022”] and simultaneously learn the BSDE solutions with the model parameters. We establish that the mean squares error between the option prices under the learned model parameters and the historical data is bounded by the loss function. Moreover, the loss can be made arbitrarily small under suitable conditions on the fitting ability of the rBergomi model to the market and the universal approximation capability of neural networks. Numerical experiments for both simulated and historical data confirm the efficiency of scheme. ...

Mean field equilibrium asset pricing model with habit formation ArXiv ID: 2406.02155 “View on arXiv” Authors: Unknown Abstract This paper presents an asset pricing model in an incomplete market involving a large number of heterogeneous agents based on the mean field game theory. In the model, we incorporate habit formation in consumption preferences, which has been widely used to explain various phenomena in financial economics. In order to characterize the market-clearing equilibrium, we derive a quadratic-growth mean field backward stochastic differential equation (BSDE) and study its well-posedness and asymptotic behavior in the large population limit. Additionally, we introduce an exponential quadratic Gaussian reformulation of the asset pricing model, in which the solution is obtained in a semi-analytic form. ...

Residual U-net with Self-Attention to Solve Multi-Agent Time-Consistent Optimal Trade Execution ArXiv ID: 2312.09353 “View on arXiv” Authors: Unknown Abstract In this paper, we explore the use of a deep residual U-net with self-attention to solve the the continuous time time-consistent mean variance optimal trade execution problem for multiple agents and assets. Given a finite horizon we formulate the time-consistent mean-variance optimal trade execution problem following the Almgren-Chriss model as a Hamilton-Jacobi-Bellman (HJB) equation. The HJB formulation is known to have a viscosity solution to the unknown value function. We reformulate the HJB to a backward stochastic differential equation (BSDE) to extend the problem to multiple agents and assets. We utilize a residual U-net with self-attention to numerically approximate the value function for multiple agents and assets which can be used to determine the time-consistent optimal control. In this paper, we show that the proposed neural network approach overcomes the limitations of finite difference methods. We validate our results and study parameter sensitivity. With our framework we study how an agent with significant price impact interacts with an agent without any price impact and the optimal strategies used by both types of agents. We also study the performance of multiple sellers and buyers and how they compare to a holding strategy under different economic conditions. ...