ARCH Models and Option Pricing: the Continuous-Time Connection

To implement continuous time option pricing models in which ARCH models can be used as direct or indirect approximators of stochastic volatility, we construct continuous time economies exhibiting equilibrium dynamics to which most asymmetric ARCH models converge in distribution as the sample frequency becomes indefinitely large. In the candidate economies, volatility is a diffusion that allows for the "leverage effect," and has a variance that is proportional to the square of the volatility itself. Such characteristics introduce non-linearities in the resulting pricing models; models of the term structure of interest rates, for instance, are not affine, and are treated following two approaches. The first uses a variant of the Crank-Nicholson scheme numerically to integrate the pde followed by the equilibrium price of a bond. We also implement a second, less traditional approach that is based on a method of iterated approximations. In all cases, we provide parameter estimates that are based on indirect inference procedures in which the previous convergence results are used to exploit ARCH as auxiliary models. The convergence foundations of the pricing models considered here are based on a class of ARCH models that is large enough to make these pricing models incorporate realistic patterns of volatility of the Markovian type.