Stock Price Simulation under Jump-Diffusion Dynamics: A WGAN-Based Framework with Anomaly Detection

Abstract

Jump-diffusion path simulation is a popular topic in the finance area. In numerical path simulation, it is usually split into a diffusion part and a jump part. We propose a GAN-based framework that gives a general pattern for the jump-diffusion path simulation. The framework consists of two parts: 1) the diffusion learning part for the simulation of the diffusion part; 2) the jump detection part related to the jump simulation. The diffusion simulation is achieved by a conditional Wasserstein GAN with gradient penalty (called SDE-WGAN). The SDE-WGAN is an adapted model from a GAN-based SDEs simulation methodology, showing its advantages in stable training. The jump detection model is designed to detect the jump instances in a jump-diffusion path and estimate the jump parameters. The jump instances are detected by introducing a GAN-based anomaly detection method, as the jumps can be viewed as anomalies that are inconsistent with the non-jump data and rare to occur in the real market. The SDE-WGAN is well-combined since it can only generate non-jump states. The jumps are then recognized when the SDE-WGAN generated pattern significantly differs from the actual state. The maximum likelihood estimation is then applied to approximate the jump parameters based on the detected jump instances. We perform the proposed framework for simulating the Merton's model and obtain promising results. However, the framework may fail when the jump magnitude is small.