When are parameters calibrated using one option type applicable to price other option types on the same underlying?

Question

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?

Answer 1

Short answer:

a. Ask yourself what distributions are implied by the options you have calibrated to?

b. Any other product priceable using at most a subset of those distributions can be safely priced by that model without a second thought. Also, any payoff statically replicable by the calibration instruments can be safely priced.

c. Any other product to be priced requires more careful understanding, as it's price is dependent on how rich the dynamics are as produced by your model, as noted in the long note below.

Long answer:

1. Any model has to get the value of the hedging strategy right. If the hedging strategy involves trading in the future, that means the model better provide reasonable prices to you in the future without frequent re calibration. A violation of this results in PnL leaks.

2. To imply from the market or to calibrate historically? If you're the sell side, you imply when you can. The only reason when it's not the obvious choice is that you believe that the derivative (usually complex) is not easily replicable so that supply and demand biases the price in a certain direction, not necessarily consistent with the market dynamics. For example, in rates, caplets are valued much lower than swaptions (assuming a reasonable correlation). Why? Because you can get away with it - there is no risk free strategy to take advantage of this "mispricing". Nevertheless, the sell side still implies when they can.

I'm not sure where the arbitrage argument that you speak of is coming from (I don't think it's well worded). But if there's a market with a certain price, your model needs to create that price if you're the sell side. There is nothing theoretically "correct" about historical calibration; one just hopes that it's an adequate representation of future dynamics.

Now, coming to your example, calibrating to vanillas and pricing a path dependent option -

A. make sure that among the set of vanillas, the ones that form your hedging strategy are a part of your calibration.

B. if the path dependent price is market observable, use the extra degree of freedom in your model (usually the parameter that determines autocorrelation, i.e. mean reversion in rates) to hit that price.

C. if the price is not observable, use whatever method you want that you believe will give you a calibration that best represents future market dynamics. That ultimately controls your PnL.