Heston Model numerical instabilities

Question

I hope that all is well,

I am working on creating a neural network to compute the implied volatilities of options using the Heston Model. However, I am coming across some issues with the numerical instabilities in the Heston model when creating the training data set.

I am using the following method to get the implied volatilities:

1. Compute option price using Lipton pricing formula (Guaranteed under the full dimensional and unrestricted parameter space): $ C_t = S_t - \frac{Ke^{-r(T-t)}}{\pi} \int_{0}^{\inf} Re [e^{(iu + 0.5)\hat{F}_{t,T}} \phi_{T-t}(u-i/2)]\frac{d}{u^2-1/4} $
2. Use py_vollib.black_scholes_merton.implied_volatility formula from the py_vollib method (Documentation), built from Jaeckel, Let's be rationale, 2015.

However, for some parameters combination, I am unable to compute the implied volatility. I thus wanted to know if there are any bounds recommended for Heston paramters to ensure that there won't be any numerical instabilities. For the moment, I am using the parameters bounds proposed by Asridi et al, 2023 (Differential Machine Learning).

Thank you very much in advance,

Best,

Answer 1

Are you meaning that you can’t calculate the IV from a specific option price generated from the Heston? If so, it’s most likely because the option is deep ITM. Normally OTM options are used to calculate IV due to the concave-convex nature of ITM options, which lead to numerical instabilities (it’s discussed in the let’s be rational and also by implication from Jackel.)

If it’s from the definite-integral, normally decreasing the $du$ (so increasing the number of points) will lead to more realistic results