Naively, it seems that Bayesian modeling, structural models particularly, would be quite useful in finance because of their ability to incorporate market idiosyncrasies and produce accurate probabilistic estimates.
The down-side of course, is model-brittleness and extremely slow computational speed. Has the Quant community overcome these issues, and how common are these tools?
As far as I know MCMC and also (PMCMC) can be usefull for (bayesian) estimation of parameters of some Hidden process like in the Heston Model case based on observations of the Stock (filtering). But the problem here is that those estimates are not matching those based on calibration of vanilla options of the Risk Neutral measure. So as an econometric tool it has limited utility in my opinion for financial application.
As an example, let's say that thank's to MCMC methods you've got an estimate of the parameters of Heston Model on a given stock based on the observations of the Stock values. Then you can (I won't blame you for that) hedge a call option on this stock using Heston Model based on your estimates. Nevertheless if there is a market for call options on this stock then you shall observe that the calibration of the Heston Model based on the vanilla prices will give you another set of parameters. So what should we do then ? Please do not forget that when filtering you are under Real World Probabilities but when you are hedging and pricing you are under Risk Neutral Probabilities. Definitely I won't follow (blindly) the filtering estimates mainly for the following reason which can be summed up in rather provocative way "Market is always right". I say this because if you are Marking to Market you position (as every one does) then you must use the calibration estimate to value your portfolio, now those calibration estimates can evolves in a way that goes against your filtering estimates and there is nothing you can do about this. Finally if your stop loss is attained (you should better have one) then even though you believe you will make money out of your filtering strategy by holding the strategy till the end of the contract you have to realize that it is not an arbitrage strategy and that you are entered in a risky position not because of the filtering estimate that was badly calculated but because the market can evolve against the best past history estimates. I hope I made my point clear.
Nevertheless as a tool to sample difficult to simulate random variables, it can be used as a tool for pricing. I think that I have seen a paper on arXiv using MCMC techniques to price american options.
MCMC can be used for Bayesian inference of other models with hidden variables. Gibbs sampling, for example, is used in Hidden Markov Models. Here is a paper that discuss the differences between MCMC and the more classical approach using the EM algorithm.
The question is: Are HMMs a useful model in finance? Some academics argue that they have predictive power.
One can look at: Stock Market Forecasting Using Hidden Markov Model: A New Approach. I'm not convinced by the approach they use.
On the other hand, HMM can be used to build volatility filters for trend following strategies.
There are certainly other models parameters that can be inferred using MCMC. I personally find it very time consuming (this is only based on experience and not on convergence analysis). Furthermore, as stated in the first paper, if one wants to use Bayesian inference then the EM algorithm can be used for computing MAP parameters.
All in all I haven't found it very useful.
