The Heston Model and its Extensions in Matlab and C#, + Website

Time-Dependent Heston Models

Abstract

The Heston model is sometimes unable to provide a good fit to short-maturity market implied volatilities. One common remedy to this problem is to enrich the model with additional parameters by specifying a more realistic volatility process, which is the approach of the double Heston model covered in Chapter 12. Another approach is to allow the parameters to be time-dependent. This latter approach is the one adopted by Mikhailov and Nögel (2003), Elices (2009), Benhamou, Gobet, and Miri (2010) and others. In this chapter, we present these time-dependent models. First, we introduce a generalization of the Riccati equation from Chapter 1 that allows for non-zero initial conditions. Then we introduce the bivariate characteristic function, and we show that the generalization of the Riccati equation arises as a special case. We then present the models with time-dependent parameters and show how to estimate these parameters using loss functions.

GENERALIZATION OF THE RICCATI EQUATION

Recall from Chapter 1 that the general solution to the Riccati equation for is

Recall also that the initial condition at expiry τ = 0 produced and the solution for D