I am coding up some basic models to show prospective employers, but I am forced to guess "what is done in practice" since I don't yet work in the industry. I am implementing various algorithms for handling the Heston-Hull-White model.
One thing that has confused me, that has been pointed out in other questions, is that the parameters for the Heston model are often calibrated using option prices.
For stochastic volatility models like Heston, it seems like the standard approach is to calibrate the models from option prices. This seems a bit like a chicken and an egg problem -- wouldn't we prefer a model, based only on historical data, that we can use to price options? I don't see that as frequently.
I agree, and I don't quite understand from the responses what the benefit of pricing using historical data is. One user says:
First, we model prices have to be arbitrate free. If you estimate the parameters based on history they do not necessarily have to agree with the current market prices. But then that's arbitrage. Or is it your historic estimation wrong? Instead of pondering whichever it is you can just fit your model to the market and voila - the market prices are consistent with the model.
I would think that if there was an arbitrage argument to be made for using implied data, then that could be formalized using mathematics. I haven't seen any such argument.
I am wanting to code up an algorithm to calibrate a Heston model. If I calibrate the parameters using vanilla European option data, are those parameters usable to price other types of options such as path-dependent or early exercise? Does the arbitrage argument from the second quote above still apply? What if I assume that there is no arbitrage in the market?