Option Pricing via Quadrature - Risk Books
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Option Pricing via Quadrature

By Marcello Minenna and Paolo Verzella


Aimed at advanced users of option pricing models, this technical report is a step-by-step guide for financial engineers looking for new quantitative techniques for assessing and pricing options.

Publish date: 1 Feb 2008

Availability: In stock


Book description

Most option pricing models and techniques employed by today’s analysts are rooted in the Black-Scholes model, but analysts are now moving beyond this established model to quadrature mathematics: numerical calculation under a curve or, more generally, using numerical integration to calculate a definite integral.

Whilst assuming a solid mathematical background, the report is easy to use and contains information that is not available anywhere else in literature, such as:

  • A complete theoretical overview of the cutting-edge methods available
  • A detailed performance comparison of the various cutting-edge quadrature schemes - covering both stability and accuracy issues
  • A clear breakdown of the complex quantitative formulae and equations used.

Readers will gain a clear idea of the pros and cons of every single method discussed. You will be guided through the implementation of the preferred pricing formula knowing exactly how this formula performs and why.

This pioneering report will enable you to go beyond Black-Scholes models to the application of the latest quadrature schemes now implemented at the likes of Deutsche Bank and Morgan Stanley.

Recommended for anyone involved in pricing options such as derivative modellers, financial analysts, financial engineers, fixed income researchers, model developers, quantitative analysts, risk managers and traders.

Book details

Publish date
1 Feb 2008
Technical report

Author biography

Marcello Minenna and Paolo Verzella

Marcello Minenna is the Head of the Quantitative Analysis Unit at CONSOB (the Italian Securities and Exchange Commission). In charge of what Risk magazine addressed as the "quant enforcement", he analyses and develops quantitative models for surveillance and supports the enforcement units in their activities.Marcello has taught mathematical models for finance in several Italian and foreign universities and is presently teaching financial mathematics at the universities of Milano Bicocca and Bocconi. He received his PhD in applied mathematics for social sciences from the State University of Brescia and his MA in mathematics in finance from Columbia University. He is the author of several publications including the bestselling book A Guide to Quantitative Finance also published by Risk Books.

Paolo Verzella is a Senior Analyst at the CONSOB Quantitative Analysis Unit. He was Assistant Professor in Mathematical Finance at Milano Bicocca University and has taught courses in mathematics and finance in Italian Universities namely Bocconi and Politecnico of Milano.Paolo received his Phd in Mathematics for Financial markets from Milano Bicocca University.His research interests focus mainly on Numerical Methods for Option Pricing, Optimisation Problems and Applied Harmonic Analysis and also includes more general areas of finance such as Structured Products.

Table of contents

1 Pricing Formulas for General Models

1.1 Quasi BSM Formula

1.2 The Single Integration Formula

1.2.1. The Carr - Madan Representation

2 Quadrature Algorithms

2.1 Lagrange Interpolation

2.2 Orthogonal Polynomials

2.2.1. Legendre Polynomials

2.2.2. Laguerre Polynomials

2.3 Newton - Cotes Schemes

2.3.1. The basic formula

2.3.2. The composite formula

2.3.3. The Trapezoid Rule

2.3.4. The Simpson Rule

2.3.5. A Derivation of NC Schemes via Lagrange Interpolation

2.4 Gauss Schemes

2.4.1. The Fundamental Theorem of Gaussian Quadrature

2.4.2. The Gauss - Lobatto Rule

2.4.3. The Gauss - Laguerre Rule

2.4.4. A Derivation of Gauss Schemes via Lagrange Interpolation

3 Pricing via Quadrature: Formulas Implementation

3.1 Explicit representations via Newton - Cotes Schemes

3.1.1. Quasi BS Formula Implementation via Trapezoid Rule Implementation via Simpson Rule

3.1.2. The Carr-Madan Representation of Single Integration Formula Implementation via Trapezoid Rule Implementation via Simpson Rule

3.2 Explicit representations via Gauss Schemes

3.2.1. Quasi BS Formula Implementation via Gauss Lobatto Rule Implementation via Gauss Laguerre Rule

3.2.2. The Carr - Madan Representation of Single Integration Formula Implementation via Gauss Lobatto Rule Implementation via Gauss Laguerre Rule

4 Pricing via Quadrature: Empirical Performances

4.1 Stability Assessment

4.1.1. Oscillations of the Characteristic Functions

4.1.2. The Stability Impact of Alpha in Single Integration Formula

4.1.3. Optimal Choice of the Sampling Grids Newton - Cotes Schemes Gauss Schemes

4.1.4. Stability: Summary Tables

4.2 Accuracy Assessment

4.2.1. The Choice of Optimal Alpha


6 References


NB - This table of contents is provisional until final publication of the book. Small changes to chapter titles and order may occur.

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