New Tools to Solve Your Option Pricing Problems For nonlinear PDEs encountered in quantitative finance, advanced probabilistic methods are needed to address dimensionality issues. Written by two leaders in quantitative research-including Risk magazine's 2013 Quant of the Year-Nonlinear Option Pricing compares various numerical methods for solving high-dimensional nonlinear problems arising in option pricing. Designed for practitioners, it is the first authored book to discuss nonlinear Black-Scholes PDEs and compare the efficiency of many different methods. Real-World Solutions for Quantitative Analysts The book helps quants develop both their analytical and numerical expertise. It focuses on general mathematical tools rather than specific financial questions so that readers can easily use the tools to solve their own nonlinear problems. The authors build intuition through numerous real-world examples of numerical implementation. Although the focus is on ideas and numerical examples, the authors introduce relevant mathematical notions and important results and proofs. The book also covers several original approaches, including regression methods and dual methods for pricing chooser options, Monte Carlo approaches for pricing in the uncertain volatility model and the uncertain lapse and mortality model, the Markovian projection method and the particle method for calibrating local stochastic volatility models to market prices of vanilla options with/without stochastic interest rates, the a + b technique for building local correlation models that calibrate to market prices of vanilla options on a basket, and a new stochastic representation of nonlinear PDE solutions based on marked branching diffusions.

In this book, I strove to propose a both precise and handy probabilistic modeling in some areas of finance and biology. Both fields are made of complex random systems, described by huge and noisy data, and call for expertise in probability theory and statistics. My work is a contribution to modeling interactions in such systems, numerical analysis of the models, and statistical analysis of experimental data. In finance, I first focus on analysis of weak error of the density of the Euler scheme for stochastic differential equations, deriving new expansions in Gaussian-like functional spaces. Then I study ergodic properties of stochastic volatility models, with extended numerical experiments. In biology, I work on cellular aging, suggesting a bifurcating autoregressive model to describe growth rates of cells and building statistical procedures to estimate parameters and test biological hypothesis. To this end, I introduce the concept of bifurcating Markov chains and prove that such stochastic processes satisfy original limit theorems. This book should be useful to academic researchers or PhD students in applied mathematics, as well as to practitioners in finance or biology.