Nonlinear Option Pricing

Collecting many methods that have previously been scattered in the literature, this book presents advanced techniques for solving high-dimensional nonlinear problems. Designed for practitioners, it is one of the first books to discuss nonlinear Black-Scholes partial differential equations (PDEs). The authors explain regression and dual methods for chooser options, the Monte Carlo approach for pricing the uncertain volatility model and the uncertain lapse and mortality model, the Markovian projection/particle method to calibrate local stochastic volatility, hybrid models to market vanilla options, and stochastic representations based on marked branching diffusions.

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.

Option Pricing in a Nutshell The super-replication paradigmStochastic representation of solutions of linear PDEs Monte Carlo The Monte Carlo method Euler discretization error Romberg extrapolation Some Excursions in Option Pricing Complete market modelsBeyond replication and super-replication Nonlinear PDEs: A Bit of Theory Nonlinear second order parabolic PDEs: some generalitiesWhy is a pricing equation a parabolic PDE? Finite difference schemesStochastic control and the Hamilton-Jacobi-Bellman PDEViscosity solutions Examples of Nonlinear Problems in Finance American options The uncertain volatility modelTransaction costs: Leland’s model Illiquid markets Super-replication under delta and gamma constraintsThe uncertain mortality model for reinsurance deals Credit valuation adjustment The passport option Early Exercise Problems Super-replication of American options American options and semilinear PDEs The dual method for American options On the ownership of the exercise right On the finiteness of exercise dates On the accounting of multiple coupons Finite difference methods for American options Monte Carlo methods for American optionsCase study: pricing and hedging of a multi-asset convertible bondIntroduction to chooser options Regression methods for chooser options The dual algorithm for chooser options Numerical examples of pricing of chooser options Backward Stochastic Differential Equations First order BSDEsReflected first order BSDEsSecond order BSDEs The Uncertain Lapse and Mortality Model Reinsurance deals The deterministic lapse and mortality model The uncertain lapse and mortality model Path-dependent payoffs Pricing the option on the up-and-out barrier An example of PDE implementation Monte Carlo pricing Monte Carlo pricing of the option on the up-and-out barrier Link with first order BSDEs Numerical results using PDENumerical results using Monte Carlo The Uncertain Volatility Model Introduction The model The parametric approachSolving the UVM with BSDEs Numerical experiments McKean Nonlinear Stochastic Differential Equations Definition The particle method in a nutshell Propagation of chaos and convergence of the particle method Calibration of Local Stochastic Volatility Models to Market Smiles Introduction The calibration condition Existence of the calibrated local stochastic volatility model The PDE method The Markovian projection method The particle methodAdding stochastic interest ratesThe particle method: numerical tests Calibration of Local Correlation Models to Market Smiles Introduction The FX triangle smile calibration problem A new representation of admissible correlations The particle method for local correlation Some examples of pairs of functions (a, b) Some links between local correlationsJoint extrapolation of local volatilities Price impact of correlationThe equity index smile calibration problem Numerical experiments on the FX triangle problemGeneralization to stochastic volatility, stochastic interest rates, and stochastic dividend yieldPath-dependent volatility Marked Branching Diffusions Nonlinear Monte Carlo algorithms for some semilinear PDEs Branching diffusions Marked branching diffusions Application: Credit valuation adjustment algorithmSystem of semilinear PDEs Nonlinear PDEs References Index Exercises appear at the end of each chapter.