# Heston Model Integration Oscillations

Is there a way to reduce oscillations for the numerical integration when evaluating the Heston model. I am pricing a series of 5000 options scattered over the Heston model parameter space and I find that for some parameters, often deep-out-of-the-money options I get negative option prices. I am using 32 Gauss-Laguerre integration, so the integration grid is rather fine, also I have tried extending the maturities to say 10 years, but this only reduces the frequency.

If not I guess Monte-Carlo is the only way to make sure I get no negative prices.

Thanks Sam

• For anyone interested I discuss this issue with GL in a little more detail in my thesis discovery.ucl.ac.uk/id/eprint/10068568. On pg.148 (pdf page 176) e.q 4.22 I give a condition that helps reduce oscillations in GL integration. Mar 9 '20 at 15:01

## 5 Answers

In SV model, it is well-known that the integrand for the call price can sometimes show high oscillation, can dampen very slowly along the integration axis, and can show discontinuities.

Remedy

• The ‘‘Little Trap’’ formulation of Albrecher et al.

Also , you can use Fourier transforms

• I currently use little trap, my understanding was Little Trap removed discontinuities rather than oscillations from the integration. I shall try FT methods and see if that helps Jul 14 '16 at 13:28

There has been a huge amount of work on this. Generally a Fourier transform approach is used.

First, be careful to use the form of the characteristic function that does not wind about zero in order to avoid having to count the normal of windings.

Second, using contour shifts can make the integral much better behaved. eg integrate along the line with $0.5$ imaginary part to price a covered call.

Third, use a Black--Scholes call with the same strike as a control. This removes poles and makes the integrand much nicer.

For details, see my book More Mathematical Finance Chapter 17 and/or my paper http://ssrn.com/abstract=1941464 Fourier Transforms, Option pricing and controls.

I'd use FFT or similar rather than direct integration. Here is an old paper with Heston example:

Option pricing using fractional FFT

This is a well known issue. There are three possible tricks:

1. I am surprised that none of the answers so far mention the work of Lord and Kahl Optimal Fourier Inversion in Semi-Analytical Option Pricing. They study this oscillation problem and propose an optimal contour for the integration. The challenge is to write a small algorithm to obtain the optimal $$\alpha$$. I believe it can be found in an article from Mike Staunton in a recent Wilmott magazine issue.

2. A different trick is to use the Black-Scholes model as control variate in the integration (its characteristic function). This is detailed in Andersen and Piterbarg book "Interest Rate Modeling, Volume I: Foundations and Vanilla Models", as well as in @MarkJoshi and Chan paper.

3. Use a quadrature that takes care of oscillations naturally. This is the approach described in An adaptive Filon quadrature for stochastic volatility models.

Use Fourier-Cosine expansions and you will never look back. Very easy to programme, maths is more intuitive also. Fang & Oosterlee, 2008 (edit: corrected the name)

• The link given is not accessible for people outside of TUDelft. And it's Fang and Oosterlee, not Wang. Mar 22 '19 at 9:59
• @jherek I'm not part of Delft and I can access. My bad with the misrembered name Mar 22 '19 at 11:59