Option Pricing via Quadrature

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

3.1.1.1. Implementation via Trapezoid Rule

3.1.1.2. Implementation via Simpson Rule

3.1.2. The Carr-Madan Representation of Single Integration Formula

3.1.2.1. Implementation via Trapezoid Rule

3.1.2.2. Implementation via Simpson Rule

3.2 Explicit representations via Gauss Schemes

3.2.1. Quasi BS Formula

3.2.1.1. Implementation via Gauss Lobatto Rule

3.2.1.2. Implementation via Gauss Laguerre Rule

3.2.2. The Carr - Madan Representation of Single Integration Formula

3.2.2.1. Implementation via Gauss Lobatto Rule

3.2.2.2. 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

4.1.3.1. Newton - Cotes Schemes

4.1.3.2. Gauss Schemes

4.1.4. Stability: Summary Tables

4.2 Accuracy Assessment

4.2.1. The Choice of Optimal Alpha

5 PSEUDOCODES

6 References

Index