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

The Heston Greeks

Abstract

In this chapter, we present the option sensitivities—the Greeks—from the Heston model. We first derive analytic expressions for the most popular Greeks. We illustrate the Heston Greeks by comparing them to Greeks from Black–Scholes prices that are close to the Heston prices. We show that finite differences produce very good approximations to analytic Greeks, at the expense of increased computation time. We do this for Greeks obtained from the original Heston (1993) model, but also with Greeks from the Attari (2004), Lewis (2000, 2001), and Carr and Madan (1999) formulations. We show that fast Fourier transform (FFT) of Carr and Madan (1999) and that the fractional FFT of Chourdakis (2005), both covered in Chapter 5, are able to very quickly produce a set of Greeks across a wide range of strikes, in the same way that these methods produce prices. Finally, we show that Greeks of American options can be obtained from simulation methods presented in Chapter 7, from the Medvedev and Scaillet (2010) expansion covered in Chapter 8, and from the explicit method covered in Chapter 10.

ANALYTIC EXPRESSIONS FOR EUROPEAN GREEKS

The prices of European calls and puts in the Heston model are available in closed form. It is, therefore, possible to differentiate the call or put price and obtain expressions for most of the Greeks in closed form also. Recall that the call ...