Numerical Integration Schemes
The calculation of the call price in the Heston model often requires the evaluation of an integral. This is true for most of the formulations of the call price we have encountered, that by Heston (1993), Lewis (2000, 2001), Carr and Madan (1999), or Attari (2004). Integration usually involves finding the anti-derivative of the integrand, and applying the Fundamental Theorem of Calculus, according to which the value of the integral can be expressed in terms of the anti-derivative evaluated at the endpoints of the integration domain. Unfortunately, in the Heston model, the anti-derivative of the integrals for the probabilities Pj cannot be found and the integrals must be approximated numerically.
Quadratures approximate an integral on [a, b] as the sum of functional values evaluated at discrete points along the integration domain, and multiplied by a weight
The points (x1,…, xN) are called nodes, points, or abscissas, and the points (w1,…, wN) are called weights or coefficients. In this chapter, we present two general classes of quadratures, Newton-Cotes formulas and Gaussian quadrature. Newton-Cotes formulas are easy to implement, but they assume equally spaced abscissas. This means that many abscissas are required in order for the approximation ...