ABSTRACT
Since the introduction of the Black-Scholes paradigm, several alternative models which allow to better capture the risk of exotic options have emerged : local volatility models, stochastic volatility models, jump-diffusion models, mixed stochastic volatility-jump diffusion models, etc. With the growing complication of Exotic and Hybrid options that can involve many underlyings (equity assets, foreign currencies, interest rates), the Black-Scholes PDE, which suffers from the curse of dimensionality (the dimension should be strictly less than four in practice), cannot be solved by finite difference methods. We must rely on Monte-Carlo methods. This appendix is organized as follows: In the first section, we review basic features in Monte-Carlo simulation from a modern (algebraic) point of view: generation of random numbers and discretization of SDEs. A precise mathematical formulation involves the use of the Taylor-Stratonovich expansion (TSE) that we define carefully. Details can be found in the classical references [20], [28]. In the second section, we show that the TSE can be framed in the setting of Hopf algebras.1 In particular, TSEs define group-like elements and solutions of SDEs can be written as exponentials of primitive elements (i.e., elements of the universal Lie algebra associated to the Hopf algebra). This section is quite technical and can be skipped by the reader. This Hopf algebra structure allows to prove easily the Yamato theorem that we explained in the last section. As an application, we classify local volatility models that can be written as a functional of a Brownian motion and therefore can be simulated exactly. The use of Yamato’s theorem allows us to reproduce and extend the results found by P. Carr and D. Madan in [69].
