I am doing my master thesis on deep calibration in the Heston Model, and after reading a few academic paper (eg. Horvath et al. 2019) on the subject I understand pretty well the procedure and the motivation, however I do not understand how the authors compute the implied volatility. Do they start by computing the option prices using Heston closed form solution and than use the Black Scholes formula to obtain the implied volatility from these prices ? Also if this is the case how can you train a neural network to do so ? Do you need to compute the prices and implied volatilities numerically first to input them to train the neural networks ? Thanks in advance
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$\begingroup$ Equity? If so, thee are price quoted. You usually compute a vol surface using black scholes (applying techniques like de-americanization) and calibrate your heston model to IV, not prices. What you are calibrating are the parameters that model the shape of the volatility. Once you fit vanilla options you can use Heston to price exotic derivatives (often in combination with local vol). $\endgroup$– user70573Commented Mar 16 at 10:01
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$\begingroup$ Thanks, so for you there is no need to generate synthetic data to train the neural network ? I can directly train it using the market implied volatilities ? $\endgroup$– sxminhoCommented Mar 16 at 10:32
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$\begingroup$ @sxminho yes. Most papers use synthetic data to show it’s possible, rather than having to deal with robustness and the inconsistencies of real-time data. $\endgroup$– THATS MY QUANT MY QUANTITATIVECommented Mar 17 at 6:44
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In Horvath et al. (2019), the authors split the procedure into two parts:
- Training the neural network (NN)
- Calibrating the network to the current IV surface. This step 'merely' determines the input parameters, and doesn't affect the NN itself.
In step 1, a NN is trained to approximate the mapping from model parameters to IV. The training data consists of IV grids, across a range of model parameters (in their case 80 000 parameter settings). These IV's must be computed the 'classical' way, e.g. by Heston's formula, or for more complex models, MC simulation.
AFAICT, the authors do not specify how they choose parameter settings for the training data, but a common method is to sample parameters from some probability distribution. You can check out Rømer (2022) for a in depth example, he also has a github with the trained models.
Once the NN has been trained, model parameters can be efficiently estimated, by calibrating to the current IV.
