modeling - Why aren't econometric models used more in Quant Finance?

There is a big body of literature on econometric models like ARIMA, ARIMAX or VAR. Yet to the best of my knowledge practically nobody is making use of that in Quantitative Finance. Yes, there is a paper here and there and sometimes you find an example where stock prices are being used for illustrative purposes but this is far from the main stream.

My question
Is there a good reason for that? Is it just because of tradition and different schools of thought or is there a good technical explanation?

(By the way I was pleased to find an arima tag here... but this is again a case in point: only 8 out of nearly 7,000 questions (~ 0.1% !) use this tag! ...ok, make this 9 now ;-)

Answer

It's an interesting question.

I particularly agree with the $\mathbb{Q}-\mathbb{P}$ dichotomy mentioned by many.

I would add to the other answers that, come to think of it, the Black-Scholes postulated Geometric Brownian Motion could be interpreted as an AR(1) process on the logarithm of the stock price as you discretise the SDE from which it is a solution, which is exactly what you do when running Monte-Carlo simulations (same thing for the Ornstein-Uhlenbeck process as explained here and noted by @Richard).

Actually, when taking the continuous-time limit, many more econometric models can be shown to correspond to stochastic processes frequently used by $\Bbb{Q}$ quants (see this paper for instance and the comment of @Kiwiakos below and discussed here with interesting references).

So why do we, at least on the sell-side, tend to favour (jump-)diffusion models over econometric models, while the latter have the advantage that volatility/variance is an observable quantity and not a hidden variable, making them easier to calibrate on historical time series, that is, information observed under $\mathbb{P}$ ?

Well... essentially because derivatives pricing happens under a risk-neutral measure $\mathbb{Q}$ and not the physical measure $\mathbb{P}$.

When working under $\mathbb{Q}$, we do relative valuation. Voluntarily over-simplifying the situation, we appeal to the absence of arbitrage opportunity to claim that any financial instrument can be priced solely by looking at the prices of other securities (typically listed options) that can be combined to perfectly replicate the former instrument's behaviour (or used as a perfect hedge, which is equivalent).

Therefore, it is not important to have a model which can be easily calibrated to historical time series, hence under $\mathbb{P}$ (which is the key feature of econometric models IMHO) but essential to have a model that leads to nice closed form formulas for the price of simple instruments that could be used as a relative pricing basis under $\mathbb{Q}$ (which is the key feature of most jump-diffusion models used by quants IMHO).

Consider the GARCH pricing model proposed by Duan for instance. True, it is easily calibrable to historical time series, but:

1. 
   

   Is the past really useful to understand what will happen in the future, which is the crux of derivatives pricing? Not necessarily, especially since we are in a relative valuation framework: it is the evolution of the market prices at which we can trade the elementary replication blocks that matters, not the historical behaviour of the underlying asset.

   
   


2. 
   

   You need Monte-Carlo simulations to compute European option prices under this model: think about how much computational resources would be needed to calibrate such a model to 1000 vanilla prices several times a day in a live production environment (especially compared to something like Heston were Fast Fourier Transform techniques can be implemented).

   
   

Summarising (again, with a voluntary over-simplification)

• 
  

  Econometric models: easily calibrated under $\mathbb{P}$ (discrete time + observable volatility/variance), yet need simulation methods à la Monte Carlo to be able to compute option prices, even for the most basic types of options.

  
  


• 
  

  (Jump-)Diffusion models: painful to calibrate to time series (continuous time + hidden Markov models) but admittedly lead to (semi-)closed form formulas for many benchmark instruments (or at least the popular models are the ones that do...), making them easy to calibrate/use under $\mathbb{Q}$.