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:

– Heston PDE :

– Heston Characteristic Eq :

– Heston Implementation :

– Heston Calibration :

Code on my website:

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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Amazing! thank you so much!

what a great video so useful to me thanks.

Can you give me a lecture on SABR model ?

How does the Heston model actually play out when used for options trading? I think it is certainly beautiful as a math person, but I have heard many say the fine-tuning of the hyper parameters make the Heston model rather cumbersome to use in practice, so many just stick with Black-Scholes, or a modified Black-Scholes or use local volatility model. Also I have wondered since we are calibrating the Heston model from a price surface which was calculated from the Black-Scholes model, how can it be a better predictor? These are just some naive questions as a beginner quant. Great video thank you for your time.

This is great

Could you compare the Heston vs the Bachelier model at some point because it sounds like the Pros (aside from the Brownian vs random process between them) of Heston is similar to Bach so trying to understand the use case for model selection here, do you select the model based on the properties of the underlying instrument. Thanks.

I have a very important question: when the rates are negative like in Europe; can the Heston model help in pricing a European type of options?

The blackscholes or Heston model does not make the assumption of positive interest rates r > 0. So you can definitely use negative interest rates, just be aware your borrowing rate is likely different from your lending rate

A bates extension on this would be amazing and a bit easy since it only adds the jump component!

Why this all looks like formulas in Geographic Mapping, Map projection have same formulas.

I’ll be doing SOA (Society of Actuary)’s QFI Quantitative Finance exam in a few weeks. this video connected the dots for me. Thanks so much!

This was amazing!! Could you go over how Heston model calibration for American options would look like, but with JUMPS added to the model, please?

Thank you so much for the tutorial! I have a question if you don’t mind: Can the data you used be found free in yahoo finance?

Options data can be tricky, but if you find a free resource feel free to use the data of course