I am trying to solve this question:

Write down pseudocode to learn a local stochastic volatility for finitely many given option prices: assume a Heston stochastic variance and parametrize local volatility by a neural network and solve the pricing equation by an Euler scheme, then define a loss function and write down the optimization problem that one needs to solve such that model prices and market prices are close. Discuss mini-batching in this approach, does it work or rather not?

I am familiar with the Heston Model, the euler scheme and how neural networks work. However, I am not sure what I need to input into the first layer of the network, and how the network actually produces the local volatility as the output in the final layer.

So far, I would define the Loss Function to be the Mean Squared Error between the observed option prices in the market and option prices computed using the learned local volatility and the euler scheme as the numerical Scheme. This could work by MC-Sampling: simulating the underlying price process, computing the final value of the option and then taking the average discounted value of those. But I don't know how we can do this in a single neural network, where the network adjusts the weights and biases to directly produce the "correct" local volatility, calibrated to market prices.



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