Nonlinear Option Pricing - Guyon Julien; Henry-Labordere Pierre | Libro Chapman And Hall/Crc 02/2014 -

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Nonlinear Option Pricing


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Lingua: Inglese
Pubblicazione: 02/2014
Edizione: 1° edizione

Note Editore

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 paradigm
Stochastic representation of solutions of linear PDEs

Monte Carlo
The Monte Carlo method
Euler discretization error
Romberg extrapolation

Some Excursions in Option Pricing
Complete market models
Beyond replication and super-replication

Nonlinear PDEs: A Bit of Theory
Nonlinear second order parabolic PDEs: some generalities
Why is a pricing equation a parabolic PDE?
Finite difference schemes
Stochastic control and the Hamilton-Jacobi-Bellman PDE
Viscosity solutions

Examples of Nonlinear Problems in Finance
American options
The uncertain volatility model
Transaction costs: Leland’s model
Illiquid markets
Super-replication under delta and gamma constraints
The 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 options
Case study: pricing and hedging of a multi-asset convertible bond
Introduction 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 BSDEs
Reflected first order BSDEs
Second 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 PDE
Numerical results using Monte Carlo

The Uncertain Volatility Model
The model
The parametric approach
Solving the UVM with BSDEs
Numerical experiments

McKean Nonlinear Stochastic Differential Equations
The particle method in a nutshell
Propagation of chaos and convergence of the particle method

Calibration of Local Stochastic Volatility Models to Market Smiles
The calibration condition
Existence of the calibrated local stochastic volatility model
The PDE method
The Markovian projection method
The particle method
Adding stochastic interest rates
The particle method: numerical tests

Calibration of Local Correlation Models to Market Smiles
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 correlations
Joint extrapolation of local volatilities
Price impact of correlation
The equity index smile calibration problem
Numerical experiments on the FX triangle problem
Generalization to stochastic volatility, stochastic interest rates, and stochastic dividend yield
Path-dependent volatility

Marked Branching Diffusions
Nonlinear Monte Carlo algorithms for some semilinear PDEs
Branching diffusions
Marked branching diffusions
Application: Credit valuation adjustment algorithm
System of semilinear PDEs
Nonlinear PDEs



Exercises appear at the end of each chapter.


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.

Altre Informazioni



Condizione: Nuovo
Collana: Chapman and Hall/CRC Financial Mathematics Series
Dimensioni: 9.25 x 6.125 in Ø 1.80 lb
Formato: Copertina rigida
Illustration Notes:55 b/w images and 55 tables
Pagine Arabe: 484

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