Stochastic Volatility Models

Heston Model Calibration to option prices

The Heston model is a useful model for simulating stochastic volatility and its effect on the potential paths an asset can take over the life of an option. It’s popular because of:
– easy closed-form solution for European option pricing
– no risk of negative variances
– incorporation of leverage effect
This allows for more effective modeling than the Black-Scholes formula allows due to its restrictive assumption of constant volatility.

Simulating Heston Model in Python

One of the nice things about the Heston model for European option prices is that there is a closed-form solution once you have the characteristic function. So, discretisation of the SDE is not required for valuing a European option, however if you would like to value other option types with complex features using the Heston model than you can use the following code:

Modelling Stock Volatility with Ornstein-Uhlenbeck process

Understanding and modelling volatility accurately is of utmost importance in financial mathematics. The emergence of volatility clustering in financial markets can make estimating volatility very difficult.

Here we explain how to use a stochastic model called Ornstein-Uhlenbeck process to model volatility. We explain the mathematics of using a method called Maximum Likelihood Estimation (MLE) to estimate the parameters of the Ornstein-Uhlenbeck process based on S&P500 historical/realised volatility.