Simulating the Heston Model with Python | Stochastic Volatility Modelling

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 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.
Written Tutorial on Medium: / simulating-the-heston-...
★ ★ Code Available on GitHub ★ ★
GitHub: github.com/The...
Specific Tutorial Link: github.com/The...
Great resource for explanation here in how to complete the Euler Discretization:
- Euler and Milstein Discretization by Fabrice Douglas Rouah frouah.com/fin...
00:00 Intro
00:53 Heston Model Dynamics
02:15 Monte Carlo Simulation and SDE Discretization
05:03 Heston Model Simulation in Python
10:00 Visualising the asset price density and volatility smile
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[@bryan-9742]

I love these videos btw. So many times in grad school they are like, "here is a model, figure it out on your own. You will be tested on it but we won't tell you what it means or how to do it." These videos are very helpful.

[@prestonhanzely5322]

Great video and explanations. Stochastic calculus is a hard to open door that when opened, reveals a lot about markets

[@bryan-9742]

100% agree. You don't actually know what's happening until you go through it to understand what's actually going on and what the assumptions even mean.

[@PiyapartB]

Appreciate your content. You just saved my life!!! I am doing master in quantitative finance and just found your channel. Your code and explanation are well clarified and really help me with the coursework. I couldn't imagine If I didn't find your video at the first place.

[@mountaindrew_]

Thank you very much for all the content you create on your channel! And by the way, a video about Girsanov’s theorem would be really nice!

[@kevinmak1996]

Great video! Can’t wait for the calibration video coming up :)

[@pritamsarkar2075]

I would very much appreciate a video series discussing rigorously the stochastic picture in finance. I have a background in Statistical Physics from where I learned about Random Processes. Now I believe a very thorough analytics as well as real world stochastic modelling would be very beneficial for many enthusiasts like me. I will look forward to this. Moreover, out of many YT resources I have found your contents, most reasonable and analytical. Thank you for your contribution.

[@victorribeiro8174]

Can't wait for the next tutorial! Thank you, very helpful

[@kilocesar]

Yes finally an excelent chanel with deep content. please make a video about change in meaasure

[@jamesguan5225]

Really appreciating the content! For other audience who might potentially use milstein scheme from the reference, there is a typo in equation (18) ,in the "Fabrice Douglas Rouah" reference provided in the video description. I think it should be 0.5 * St * Vt * dt (Zs**2 - 1); The simulation would be just wrong using the original equation.

[@bryan-9742]

AHHHHH. We just finished up Derivatives in my MFE. This is awesome. For parameters are you guys going to do MLE on the historical data? Can't wait to see it!

[@hongkyulee9724]

Thank you for the good video. 😉 Your explanation and code are very intuitive and concise to me. ❤❤

[@evanhong1690]

thanks for your work! They help me a lot

[@caetanogarelii6657]

Can you explain what measures and the risk neutral measure are? It's mentioned a lot but they're hard to understand.

[@OpenQuant]

Great explanation!

[@quant-prep2843]

please start a discord server! its not a request , its an order. your channel is so great

[@QuantPy]

Thanks, have a discord server through patreon link in description

[@sachinmittal5308]

Thank you for the great video again. I just wanted to understand that how did you get parameter combination as initial value for heston and monte carlo? If I remember it correctly you might have answered it some video, but these parameter combinations can be found out using argmax(MLE) OR using machine learning NN and other non-linear machine learning models as output with input as market observable prices which then sampled over by uniform sampling method like Latin hypercube sampling? Is my understanding correct that black volatility surface can also be created using numerical iterative methods like newton rhapson method and also via Machine Learning NN and other classical machine learning models? Why we use MC method for simulating Stochastic Volatility Modelling equation rather than other numerical methods like Finite di erence, COS method and numerical integration? Why to use any of these classical models like heston to find option price when the prices can be sourced from bloomberg and other readily available tools easily and then later trained via machine learning ensemble models to get the model calibrated parameters for valuing an option in future date?

[@pearsonmudzingwa597]

How do you price Delta, American and European Options using Heston Model?

[@siamgangte2826]

Please make a content on Girsanov's theorem

[@user-wp5gw3jl4s]

Please tell me why there is dt associated with Wt in the integral form. Isn't it should be just Wt? In the code Wt = sqrt(t)*Zt looks good for me though.

[@virensangwan667]

Nice video. A question - how to estimate heston parameters using only underlying price data? Example: to compute IV of an option on xyz asset that don't have any options market. Would you suggest to look into any other model?

[@MrEo89]

Thank you for these great videos (btw I was being sarcastic/cheeky when i mentioned the pyvollib vectorized lib), but I was wondering what you did regarding the ValueError you get when you try to compute S_p, v_p/S_n, v_n via the heston function. ValueError: operands could not be broadcast together with shapes (252, ), (100_000, ) (underscore mine, for readability)

[@MrEo89]

nvm, I had a typo. used np.full(shape = (N + 1, N), fill_value = S0) instead of N+1, M

[@priyankaranjan2328]

Do you have a video or code for pricing put option using Heston model?

[@Med.El-amine]

Pleas I want python code for currency options pricing using Heston model

[@simonlove4527]

Just an FYI, you have a typo on the Heston model simulation page on your website. It says dS_i+1 instead of S_i+1 in the Euler Discretisation section

[@QuantPy]

Legend, thanks for update. Sometimes it's hard to pick these up when writing in latex $\Large S_{i+1} = S_i e^{(r-\frac{v_i}{2}) \Delta t + \sqrt{v_{i}}\Delta tW^\mathbb{Q}_{S,i+1}}$

[@user-kn1lf4dn9v]

please make a post with details

[@AndrewCB07]

Possible to make a video on GARCH?

[@QuantPy]

Sure can

[@Prof.OrtizRamirez]

Accurate and fast explanation. I got this when run first part of code "ModuleNotFoundError: No module named '_testcapi'", how can i fix this? thanks in advance

[@QuantPy]

github.com/vollib/py_vollib/issues/11

[@triloksinghbhati333]

Do you teach ?

[@mojancevskialeksandar6955]

calculations are a devils business

[@abdulmajeedasiri8383]

Can you help with quantitative finance jobs? I am looking for roles in quantitative finance and have a master in financial engineering but the problem is that I had a family emergency and now there is a gap since I graduated. No firm is giving me an interview now as the gap is 14 months now.