Appendix

Miquel Noguer i Alonso, Daniel Bloch and David Pacheco Aznar

This appendix collects together advanced theoretical and practical details that extend the core ideas from the preceding chapters. Section A1 explores local risk minimisation, detailing portfolio evaluation, optimal hedging strategies and the construction of risk-averse prices, as well as the connection to the Black–Scholes limit. Section A2 covers diverse topics ranging from Black–Scholes expansions and the profit attribution analysis (PAA) methodology to Kolmogorov compatibility, providing deeper mathematical insights and tools. Next, Section A3 presents the parametric MixVol model, discussing both the model formulation and the computation of Greeks, alongside scenario analyses. Section A4 then revisits option pricing under daily price limits, offering an implementation-oriented perspective on geometric Brownian motion with boundaries. Sections A5 and A6 provide additional resources on reinforcement learning optimisation and an overview of Transformers (including multi-head attention and decoder structures), thus rounding out the book with further theoretical underpinnings and computational techniques.

A1 LOCAL RISK MINIMISATION

We briefly describe a local risk-minimisation