PDE and Martingale Methods in Option Pricing

1. Front Matter

   Pages I-XVII

   PDF

2. Derivatives and arbitrage pricing

   • Andrea Pascucci

   Pages 1-13

3. Discrete market models

   • Andrea Pascucci

   Pages 15-96

4. Continuous-time stochastic processes

   • Andrea Pascucci

   Pages 97-137

5. Brownian integration

   • Andrea Pascucci

   Pages 139-166

6. Itô calculus

   • Andrea Pascucci

   Pages 167-201

7. Parabolic PDEs with variable coefficients: uniqueness

   • Andrea Pascucci

   Pages 203-217

8. Black-Scholes model

   • Andrea Pascucci

   Pages 219-256

9. Parabolic PDEs with variable coefficients: existence

   • Andrea Pascucci

   Pages 257-274

10. Stochastic differential equations

    • Andrea Pascucci

    Pages 275-328

11. Continuous market models

    • Andrea Pascucci

    Pages 329-387

12. American options

    • Andrea Pascucci

    Pages 389-401

13. Numerical methods

    • Andrea Pascucci

    Pages 403-428

14. Introduction to Lévy processes

    • Andrea Pascucci

    Pages 429-495

15. Stochastic calculus for jump processes

    • Andrea Pascucci

    Pages 497-540

16. Fourier methods

    • Andrea Pascucci

    Pages 541-576

17. Elements of Malliavin calculus

    • Andrea Pascucci

    Pages 577-597

18. Back Matter

    Pages 599-719

    PDF

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.

• Arbitrage theory
• Numerical methods in finance
• Option pricing
• Quantitative finance
• Stochastic calculus

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)

• Department of Mathematics, University of Bologna, Bologna

  Andrea Pascucci

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).

Policies and ethics