PDE and Martingale Methods in Option Pricing

Name: PDE and Martingale Methods in Option Pricing
Brand: Springer
Price: 128.39 EUR
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Andrea Pascucci(Author)

Springer (Publisher)

1st Edition

Published on 15. April 2011

XVII, 721 pages

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978-88-470-1781-8 (ISBN)

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This book offers an introduction to the mathematical, probabilistic and numerical methods used in the modern theory of option pricing. The text is designed for readers with a basic mathematical background. The first part contains a presentation of the arbitrage theory in discrete time. In the second part, the theories of stochastic calculus and parabolic PDEs are developed in detail and the classical arbitrage theory is analyzed in a Markovian setting by means of of PDEs techniques. After the martingale representation theorems and the Girsanov theory have been presented, arbitrage pricing is revisited in the martingale theory optics. General tools from PDE and martingale theories are also used in the analysis of volatility modeling. The book also contains an Introduction to Lévy processes and Malliavin calculus. The last part is devoted to the description of the numerical methods used in option pricing: Monte Carlo, binomial trees, finite differences and Fourier transform.

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From the reviews:

"The author provides an excellent overview of methods from the theory of partial differential equations and stochastic processes used in mathematical finance. . The book is well written, the mathematical level is quite sophisticated and a broad range of material is covered. At the same time, there is a clear focus on applications. The book is therefore warmly recommended for graduate students as well as for professionals in the financial industry." (Johan Tysk, Mathematical Reviews, Issue 2012 i)

"The book is written for graduate and advanced undergraduate students and gives an introduction to the modern theory of option pricing. . this book covers a wide range of topics with good motivations on a rigorous mathematical level." (Sören Christensen, Zentralblatt MATH, Vol. 1214, 2011)

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Content

Intro
Title Page
Copyright Page
Preface
Table of Contents
General notations
Shortenings
Function spaces
Spaces of processes
1 Derivatives and arbitrage pricing
1.1 Options
1.1.1 Main purposes
1.1.2 Main problems
1.1.3 Rules of compounding
1.1.4 Arbitrage opportunities and Put-Call parity formula
1.2 Risk-neutral price and arbitrage pricing
1.2.1 Risk-neutral price
1.2.2 Risk-neutral probability
1.2.3 Arbitrage price
1.2.4 A generalization of the Put-Call parity
1.2.5 Incomplete markets
2 Discrete market models
2.1 Discrete markets and arbitrage strategies
2.1.1 Self-financing and predictable strategies
2.1.2 Normalized market
2.1.3 Arbitrage opportunities and admissible strategies
2.1.4 Equivalent martingale measure
2.1.5 Change of numeraire
2.2 European derivatives
2.2.1 Pricing in an arbitrage-free market
2.2.2 Completeness
2.2.3 Fundamental theorems of asset pricing
2.2.4 Markov property
2.3 Binomial model
2.3.1 Martingale measure and arbitrage price
2.3.2 Hedging strategies
2.3.3 Binomial algorithm
2.3.4 Calibration
2.3.5 Binomial model and Black-Scholes formula
2.3.6 Black-Scholes differential equation
2.4 Trinomial model
2.4.1 Pricing and hedging in an incomplete market
2.5 American derivatives
2.5.1 Arbitrage price
2.5.2 Optimal exercise strategies
2.5.3 Pricing and hedging algorithms
2.5.4 Relations with European options
2.5.5 Free-boundary problem for American options
2.5.6 American and European options in the binomial model
3 Continuous-time stochastic processes
3.1 Stochastic processes and real Brownian motion
3.1.1 Markov property
3.1.2 Brownian motion and the heat equation
3.2 Uniqueness
3.2.1 Law of a continuous process
3.2.2 Equivalence of processes
3.2.3 Modifications and indistinguishable processes
3.2.4 Adapted and progressively measurable processes
3.3 Martingales
3.3.1 Doob's inequality
3.3.2 Martingale spaces: M2 and M2
3.3.3 The usual hypotheses
3.3.4 Stopping times and martingales
3.4 Riemann-Stieltjes integral
3.4.1 Bounded-variation functions
3.4.2 Riemann-Stieltjes integral and Ito formula
3.4.3 Regularity of the paths of a Brownian motion
4 Brownian integration
4.1 Stochastic integral of deterministic functions
4.2 Stochastic integral of simple processes
4.3 Integral of L2-processes
4.3.1 It^o and Riemann-Stieltjes integral
4.3.2 It^o integral and stopping times
4.3.3 Quadratic variation process
4.3.4 Martingales with bounded variation
4.3.5 Co-variation process
4.4 Integral of L2loc-processes
4.4.1 Local martingales
4.4.2 Localization and quadratic variation
5 It^o calculus
5.1 It^o processes
5.1.1 It^o formula for Brownian motion
5.1.2 General formulation
5.1.3 Martingales+and parabolic equations
5.1.4 Geometric Brownian motion
5.2 Multi-dimensional It^o processes
5.2.1 Multi-dimensional It^o formula
5.2.2 Correlated Brownian motion+and martingales
5.3 Generalized It^o formulas
5.3.1 It^o formula and+weak derivatives
5.3.2 Tanaka formula+and local times
5.3.3 Tanaka+formula for It^o processes
5.3.4 Local+time and Black-Scholes formula
6 Parabolic PDEs with variable coefficients: uniqueness
6.1 Maximum principle and Cauchy-Dirichlet problem
6.2 Maximum principle and Cauchy problem
6.3 Non-negative solutions of the Cauchy problem
7 Black-Scholes model
7.1 Self-financing strategies
7.2 Markovian strategies and Black-Scholes equation
7.3 Pricing
7.3.1 Dividends and time-dependent parameters
7.3.2 Admissibility and absence of arbitrage
7.3.3 Black-Scholes analysis: heuristic approaches
7.3.4 Market price of risk
7.4 Hedging
7.4.1 The Greeks
7.4.2 Robustness of the model
7.4.3 Gamma and Vega-hedging
7.5 Implied volatility
7.6 Asian options
7.6.1 Arithmetic average
7.6.2 Geometric average
8 Parabolic PDEs with variable coefficients: existence
8.1 Cauchy problem and fundamental solution
8.1.1 Levi's parametrix method
8.1.2 Gaussian estimates and adjoint operator
8.2 Obstacle problem
8.2.1 Strong solutions
8.2.2 Penalization method
9 Stochastic differential equations
9.1 Strong solutions
9.1.1 Uniqueness
9.1.2 Existence
9.1.3 Properties of solutions
9.2 Weak solutions
9.2.1 Tanaka's example
9.2.2 Existence: the martingale problem
9.2.3 Uniqueness
9.3 Maximal estimates
9.3.1 Maximal estimates for martingales
9.3.2 Maximal estimates for diffusions
9.4 Feynman-Ka?c representation formulas
9.4.1 Exit time from a bounded domain
9.4.2 Elliptic-parabolic equations and Dirichlet problem
9.4.3 Evolution equations and Cauchy-Dirichlet problem
9.4.4 Fundamental solution and transition density
9.4.5 Obstacle problem and optimal stopping
9.5 Linear equations
9.5.1 Kalman condition
9.5.2 Kolmogorov equations and Hormander condition
9.5.3 Examples
10 Continuous market models
10.1 Change of measure
10.1.1 Exponential martingales
10.1.2 Girsanov's theorem
10.1.3 Representation of Brownian martingales
10.1.4 Change of drift
10.2 Arbitrage theory
10.2.1 Change of drift with correlation
10.2.2 Martingale measures and market prices of risk
10.2.3 Examples
10.2.4 Admissible strategies and arbitrage opportunities
10.2.5 Arbitrage pricing
10.2.6 Complete markets
10.2.7 Parity formulas
10.3 Markovian models: the PDE approach
10.3.1 Martingale models for the short rate
10.3.2 Pricing and hedging in a complete model
10.4 Change of numeraire
10.4.1 LIBOR market model
10.4.2 Change of numeraire for It^o processes
10.4.3 Pricing with stochastic interest rate
10.5 Diffusion-based volatility models
10.5.1 Local and path-dependent volatility
10.5.2 CEV model
10.5.3 Stochastic volatility and the SABR model
11 American options
11.1 Pricing and hedging in the Black-Scholes model
11.2 American Call and Put options in the Black-Scholes model
11.3 Pricing and hedging in a complete market
12 Numerical methods
12.1 Euler method for ordinary equations
12.1.1 Higher order schemes
12.2 Euler method for stochastic differential equations
12.2.1 Milstein scheme
12.3 Finite-difference methods for parabolic equations
12.3.1 Localization
12.3.2 ?-schemes for the Cauchy-Dirichlet problem
12.3.3 Free-boundary problem
12.4 Monte Carlo methods
12.4.1 Simulation
12.4.2 Computation of the Greeks
12.4.3 Error analysis
13 Introduction to Levy processes
13.1 Beyond Brownian motion
13.2 Poisson process
13.3 Levy processes
13.3.1 Infinite divisibility and characteristic function
13.3.2 Jump measures of compound Poisson processes
13.3.3 Levy-It^o decomposition
13.3.4 Levy-Khintchine representation
13.3.5 Cumulants and Levy martingales
13.4 Examples of Levy processes
13.4.1 Jump-diffusion processes
13.4.2 Stable processes
13.4.3 Tempered stable processes
13.4.4 Subordination
13.4.5 Hyperbolic processes
13.5 Option pricing under exponential Levy processes
13.5.1 Martingale modeling in Levy markets
13.5.2 Incompleteness and choice of an EMM
13.5.3 Esscher transform
13.5.4 Exotic option pricing
13.5.5 Beyond Levy processes
14 Stochastic calculus for jump processes
14.1 Stochastic integrals
14.1.1 Predictable processes
14.1.2 Semimartingales
14.1.3 Integrals with respect to jump measures
14.1.4 Levy-type stochastic integrals
14.2 Stochastic differentials
14.2.1 It^o formula for discontinuous functions
14.2.2 Quadratic variation
14.2.3 It^o formula for semimartingales
14.2.4 It^o formula for Levy processes
14.2.5 SDEs with jumps and It^o formula
14.2.6 PIDEs and Feynman-Ka?c representation
14.2.7 Linear SDEs with jumps
14.3 Levy models with stochastic volatility
14.3.1 Levy-driven models and pricing PIDEs
14.3.2 Bates model
14.3.3 Barndorff-Nielsen and Shephard model
15 Fourier methods
15.1 Characteristic functions and branch cut
15.2 Integral pricing formulas
15.2.1 Damping method
15.2.2 Pricing formulas
15.2.3 Implementation
15.2.4 Choice of the damping parameter
15.3 Fourier-cosine series expansions
15.3.1 Implementation
16 Elements of Malliavin calculus
16.1 Stochastic derivative
16.1.1 Examples
16.1.2 Chain rule
16.2 Duality
16.2.1 Clark-Ocone formula
16.2.2 Integration by parts and computation of the Greeks
16.2.3 Examples
Appendix: a primer in probability and parabolic PDEs
A.1 Probability spaces
A.1.1 Dynkin's theorems
A.1.2 Distributions
A.1.3 Random variables
A.1.4 Integration
A.1.5 Mean and variance
A.1.6 s-algebras and information
A.1.7 Independence
A.1.8 Product measure and joint distribution
A.1.9 Markov inequality
A.2 Fourier transform
A.3 Parabolic equations with constant coefficients
A.3.1 A special case
A.3.2 General case
A.3.3 Locally integrable initial datum
A.3.4 Non-homogeneous Cauchy problem
A.3.5 Adjoint operator
A.4 Characteristic function and normal distribution
A.4.1 Multi-normal distribution
A.5 Conditional expectation
A.5.1 Radon-Nikodym theorem
A.5.2 Conditional expectation
A.5.3 Conditional expectation and discrete random variables
A.5.4 Properties of the conditional expectation
A.5.5 Conditional expectation in L2
A.5.6 Change of measure
A.6 Stochastic processes in discrete time
A.6.1 Doob's decomposition
A.6.2 Stopping times
A.6.3 Doob's maximal inequality
A.7 Convergence of random variables
A.7.1 Characteristic function and convergence of variables
A.7.2 Uniform integrability
A.8 Topologies and s-algebras
A.9 Generalized derivatives
A.9.1 Weak derivatives in R
A.9.2 Sobolev spaces and embedding theorems
A.9.3 Distributions
A.9.4 Mollifiers
A.10 Separation of convex sets
References
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PDE and Martingale Methods in Option Pricing

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