Using the Intermediate Value Theorem we demonstrate the rules of Trapeze and Simpson's. Demonstrations with this approach and its generalization to new formulas are less laborious than those resulting from methods such as polynomial interpolation or Gaussian quadrature. In addition, we extend the theory of approximate integration by finding new approximate integration formulas. The methodology we used to obtain this generalization was to use the definition of the integral defined by Riemann sums. Each Riemann sum provides an approximation of the result of an integral. With the help of the Intermediate Value Theorem and a detailed analysis of the Middle Point, Trapezoidal and Simpson Rules we note that these rules of numerical integration are Riemann sums. The results we obtain with this analysis allowed us to generalize each of the rules mentioned above and obtain new rules of approximation of integrals. Since each of the rules we obtained uses a point in the interval we have called them according to the point of the interval we take. In conclusion we can say that the method developed here allows us to give new formulas of numerical integration and generalizes those that already exist.

Este trabajo desarrolla un modelo de equilibrio parcial para valuar productos derivados cuando la volat ilidad del activo subyacente presenta volatilidad estocástica. El modelo considera una economía poblada por agentes racionales (consumidores-inversionistas) que toman decisiones sobre co nsumo e inversión en un a mbiente de riesgo de mercado. E l precio del der ivado y el coefici ente d e riesgo se caracterizan como las soluciones d e un sistema de ecuaciones diferencia les parciales. Varias formas específicas de la función d e ut ilida d son analizadas en el proceso de valuación. AbstractThis paper d evelops a model of partial equilibrium to estirnate derivative products when t he volatility of the underlying asset displays stochastic volatility. T he model considers a n economy populated by rational agents (consumer-investors) who make decisions on consu m ption and investment in an environment of risk market . T he price of the d erivative and the coefficient of risk are characterized as solutions of a system of partial differential equat ions. Severa! sp ecific forms of the ut ility function are analyzed in t he valuing process.