Derivatives pricing under rough and path-dependent volatility: Analytics, High-Performance Computing and Machine Learning

This dissertation delves into a traditional yet pivotal branch in Quantitative Finance, especially dealing with continuous-time volatility modeling and derivatives pricing. In particular, we focus on the rough and PDV (path-dependent volatility) families for joint SPX/VIX calibration. The numerical techniques we develop depend on the nature of the specific model at hand. As long as affine models are concerned, we treat pricing of SPX options by means of what we call the SINC approach. We derive Fourier Transform formulas for European payoffs and substantiate them with a rigorous proof of convergence. Extensive tests demonstrate their superiority with respect to traditional techniques in the literature. The method is extremely accurate and extension to an FFT-version makes it also very efficient in view of large scale-problems. However, many recent models prioritize physical soundness over mathematical tractability. As so, they are only amenable of slow Monte Carlo integration. In such cases, we propose one resorts to a neural approximation of the true pricing function. We construct an interpolation-free neural network pricer which results from pointwise learning on random grids. The presence of entire smiles in the training set conveys relevant information about skewness and kurtosis; this is fundamental for the quality of the approximation. The deep neural approach is obviously extended to VIX derivatives as well. Joint SPX/VIX calibration is therefore a cooperation of two networks. Crucially, nested (or Least Squares) simulation can be replaced with simple matrix-vector products that evaluate in milliseconds. The consequence is that optimization converges in real-time. Application to (an inflated version of) the 4-factor Markov PDV model supports our claims. Nonetheless, generation of samples is a critical issue that requires access to a high-performance computing (HPC) infrastructure. An additional note of realism is achieved when coupling path-dependence with an exogenous source of randomness. We do that with the EWMA two-factor exponential Ornstein-Uhlenbeck model. The idea is that the (log-)volatility factors mean-revert to (a function of) an exponentially weighted moving average of past asset returns. The model reproduces many well-known stylized facts under the empirical measure, and the joint dynamics of the underlying asset and the volatility seem to be flexible enough for joint calibration to be successful. Finally, we try to reconcile the PDV framework with a certain ease of computation by proposing a polynomial model that guarantees semi-closed pricing of VIX derivatives as a trivial corollary of the moment formula. The model is a natural generalization of several known instances in the mathematical finance literature, and it comes with very decent reconstruction of the historical volatility – as measured by the level of the VIX. Pricing and calibration are sensible but we find that the left tail of the distribution of the underlying is too fat to be compatible with certain market regimes. This is a consequence of the quadratic component in the trend feature to be symmetric, and it explains why a semi-parabolic inflation of the 4-factor Markov PDV model provides a good agreement with the market. We prove well-posedness and absence of explosion, and derive sufficient conditions for the variance process to stay non-negative/positive. The forward variance of the model is also easy (but very tedious) to derive.