We introduce a novel multi-asset market simulator for generating sequences of implied volatility (IV) surfaces and the corresponding equity price paths that is faithful to historical data. We do so using a combination of functional data analysis and neural stochastic differential equations (SDEs) combined with a probability integral transform penalty to reduce model misspecification.

The simulator leverages functional principal component (FPC) analysis to decompose the implied volatility surfaces into a set of basis functions that capture the maximum variability. By representing the volatility surface as a linear combination of these basis functions, FuNVol achieves a dimensionality reduction, enabling efficient simulation and analysis. Furthermore, FuNVol employs neural SDEs to capture the temporal dynamics of the implied volatility surfaces. The neural SDEs model the stochastic evolution of the basis function coefficients over time, allowing for realistic and flexible volatility simulation. The neural network architecture embedded within the SDE framework is trained on historical data, enabling it to learn and replicate complex patterns.

The performance of FuNVol is extensively evaluated using real-world financial data. We demonstrate that learning the joint dynamics of IV surfaces and prices produces market scenarios that are consistent with historical features and lie within the sub-manifold of surfaces that are essentially free of static arbitrage, without the need to explicitly impose such constraints. Moreover, the simulator showcases its versatility by simulating multi-asset scenarios, allowing for comprehensive analysis of inter-asset volatility dependencies. Finally, we demonstrate that delta hedging using the simulated scenarios generates profit and loss (P&L) distributions that are consistent with realized P&Ls.

The practical applications of FuNVol are numerous. It can serve as a valuable tool for option pricing, risk management, and portfolio optimization. Moreover, its ability to generate realistic implied volatility surfaces opens up new possibilities for volatility forecasting, derivative trading strategy development, and market simulation studies.