PDE and Martingale Methods in Option Pricing

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Springer Science & Business Media, Apr 15, 2011 - Mathematics - 721 pages
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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 LÚvy 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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A fantastic, in-depth, well written book. I am surprised it is not more popular! Gives equal weighting to the PDE and martingale approaches in finance; well spaced, glossy, well referenced, etc.

Contents

1 Derivatives and arbitrage pricing
1
2 Discrete market models
15
3 Continuoustime stochastic processes
97
4 Brownian integration
138
5 Itˆo calculus
167
uniqueness
203
7 BlackScholes model
218
existence
257
11 American options
389
12 Numerical methods
402
13 Introduction to Levy processes
429
14 Stochastic calculus for jump processes
497
15 Fourier methods
541
16 Elements of Malliavin calculus
577
a primer in probability and parabolic PDEs
598
References
691

9 Stochastic differential equations
275
10 Continuous market models
329

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About the author (2011)

Andrea Pascucci is Professor of Mathematics at the University of Bologna where he is director of a master program in Quantitative Finance. His research interests include second order parabolic partial differential equations and stochastic analysis with applications to finance (pricing of European, American and Asian options).

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