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.

One of the nice things about the Heston model for European option prices is that there is a semi-analytical closed-form solution once you have the characteristic function. In this video we will go through the mathematics of the characteristic funciton for European Option prices, implement this numerical integration in python and then calibrate the model to real world index option prices.

We will be using the S&P500 Index Options to calibrate the risk-neutral Heston Model. As a stochastic volatility model, the heston model can incorporate the real-world volatility smile within it's pricing dynamics.

Here I have drawn on a number of academic papers/online resources for reference:
- Heston Girsanov's Formula: https://quant.stackexchange.com/quest...
- Heston PDE : https://uwspace.uwaterloo.ca/bitstrea...
- Heston Characteristic Eq :https://www.maths.univ-evry.fr/pages_...
- Heston Implementation : https://hal.sorbonne-universite.fr/ha...
- Heston Calibration :   / build-the-heston-model-from-scratch-in-pyt...  

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00:00 Intro
00:45 The Mathematics Explained
09:05 Python Implementation of Semi-Analytical Solution
14:00 Real World Data - Options & Yield Curve
19:50 Heston Model Calibration

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