Arbitrage-free neural-SDE market models of traded options

Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This thesis develops a nonparametric model for the European options book respecting underlying financial constraints while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE (stochastic differential equation) system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate \textit{neural-SDE} models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using both (i) synthetic data generated from a Heston stochastic local volatility model and (ii) real world data. In both cases, we assess how well the model captures option price dynamics, and therefore its capacity as a realistic option market simulator. However, the presence of arbitrage in the real world option price data causes difficulties in the calibration of our models, making pre-processing of the data to eliminate arbitrage necessary. We formulate the arbitrage repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimise prices' changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from price data by our repair method can improve model calibration with enhanced robustness and reduced calibration error. Finally, we explore the capacity of the model in two important applications for managing risks of option books, namely the Value-at-Risk (VaR) calculation and hedging. In terms of calculating VaR, we show that our models are more computationally efficient and accurate for evaluating risks of option portfolios, with better coverage performance and less procyclicality than standard filtered historical simulation approaches. In the hedging application, we derive sensitivity and minimum variance based hedging ratios under neural-SDE market models, and compare the performance of hedging various option portfolios with standard Greeks-based hedging under the Black--Scholes model, demonstrating favourable performance of our approach.