# Statistical estimation vs Stochastic calibration of models

I have never been able to deduce the precise differences between model building from the statistical perspective and the stochastic processes/calibration perspective. I can only infer that these are two disparate schools of thought because of how different they are in their language (p-tests, t-distributions, and regression for statistics, and random processes, calibration, and Q-measures for stochastic processes) despite doing the same exact thing, i.e., just fitting models on real world data.

Is there a way to understand their differences, and especially gain insights into the latter for someone coming from the statistics tradition?

I am sorry ahead of time for the vagueness of this almost philosophical question, and want to thank everyone ahead of time for any help.

When you build a model based on a stochastic process you de-facto provide a description of the world which can ensure certain mathematical properties are true. To name just one: abscence of arbitrage.

For example suppose you are looking at the vanilla options market for a give underlying security and you want to “fit a model” to it (A reason for doing this could be because you want to price non-vanilla instruments consistent with this market).

If you calibrate succesfully a local volatility model to this vanilla market you are guaranteed that there will be no arbitrage between any set of instruments priced using this model.

While a statistical model might achieve similar precision of the fit as a stochastic model for the vanilla instruments it will not be able to perform the same level of inference regarding the more generic instruments because it does not incorporate the intrinsic relationship that exist between all of them (absence of arbitrage which is a very non linear relationship).

On the other hand if all you care is the vanilla instruments themselves and trying to do some signal research for the purpose of trading only those specific instruments then it is possible that a statistical model is sufficient or superior to a stochastic model which is not designed to make time inference.

Hope this sheds a bit of light!

• Thanks a lot for your insight. The explicit inclusion of Financial relationships (such as no arbitrage here) making stochastic modelling superior to simple statistical modelling gives one a lot to think about. However, I never would have thought that such "facts" of the financial world couldn't be baked into statistical models, though. Are there any more resources that more mathematically go into why statistics is, I guess, lacking in this area? – Coolio2654 Dec 4 '18 at 19:04

Almost all statistical tests start with an underlying distribution, and then find the most likely parameters. You first set up a "stochastic assumption" (we can use central limit theorem if sample size is large so lets use a normal distribution, the relationship is linear so lets use a linear regression). After making those assumptions, you go into the statistical process of getting maximum likelihood estimates, p-values etc. You check the values, reexamine the assumptions, and repeat until you are convinced.

The distinction comes from where you spend most of your time thinking about. And oftentimes where you spend most of your time comes from the use case of your model. Let me clarify through some examples.

Suppose there is a coffee shop. If you are a simple investor, you might get statistical estimates for revenue/profit, growth, expenses etc. and get your rough estimates for its future cash flows and put a price on it. Those simple statistics could be enough for you, because you are less interested in how to model the actual business, and more interested in getting some reliable estimates. On the other hand if you are a manager, you might think about constructing a more sophisticated stochastic model like an MMC queue, and then get estimated arrival times, service times, revenue per costumer, costumer loss per waiting time, etc. Look at marginal differences and take action accordingly.

In financial world, sometimes a small correlation could be a good buy/sell signal. For high frequency trading you might heavily use statistical approach and don't justify the underlying economics much, as long as it does generate revenue. A good example could be momentum trading where you trade on the signal, but it doesn't have a strong economical explanation aside from slow diffusion of news. Whereas for a deep prepayment model or equity research you might want to build a stochastic model, come up with random variables for every effect you think. Then use statistics to get the stochastic model values.

In short often times they go together, with stochastic models you "declare" your model of the world and use statistics to get estimates for your model. Depending on which part you put more focus on, we say stochastic model or statistical model to identify loosely.

Hope it helps

• Hi: see the topic of "rational expectations" for how econometrics can be used to model behavior. I come from statistics and this was a real eye opener for me. So, this is another where statistics differs. It differs "stochastically" but it also can differ in the sense that statistical models generally don't consider behavior. – mark leeds Dec 4 '18 at 22:51