Filtrations in Option Pricing and Backtesting

# How do practitioners make actual use of filtrations?


# How do practitioners make actual use of filtrations?












To clarify with an easy example: we know that, in the case of a self financing portfolio $V_t=\eta_t B_t+\xi_t S_t$ with just a risk free asset $B_t$ and a risky asset $S_t$, the price of a contingent claim payoff $C$ can be calculated via martingale methods simply as

$$\pi_t(C)=e^{-(T-t){r}}\mathbb E^*[C | \mathcal F_t] , $$

what would you take as $\mathcal F$ in practice for computing price? Does this question make sense?

Remark: I am perfectly fine with the abstract concept of filtration, I am asking about their actual usage (if there is one).

Thank you in advance

## Answer by Frido (score 11, accepted)

https://quant.stackexchange.com/a/82478

To elaborate on @Jesper Tidblom's comment:

$\mathcal F_t =$ market prices of tradable instruments at time $t$ $+$ model.

What do I mean by this: the universe of traded instruments is usually not sufficient to fully know the (risk-neutral) process generating prices; a particular choice of a pricing model is required. But the model is calibrated to vanilla options observed at $t$ to at least price vanillas consistently. If other liquid instruments are available then ideally the model should price those consistently as well.

One could even argue that $\mathcal F_t$ is defined by the current market prices of traded instruments only as the model is not unique.

## Answer by user129707 (score 4)

https://quant.stackexchange.com/a/82479

As far as I've seen, a Filtration for a given time is (like the comment by @Jan Stuller above says) a formal and rigorous way of saying "all the information you can have up to a given point in time", but have never seen applications beyond that in options or derivate pricing.

I have always considered the concept of a Filtration and its use in practice to be analogous to those little boxes in Riemannian integration (the ones that get vanishingly small in the end): They're how you define the integral, and when you want to prove that an integral exists (or otherwise theoretically show certain properties of the integral), you may need to consider them, but when analytically computing the primitive of a given function, you usually use different methods. This analogy of course breaks down when looking at the numerical evaluation of a definite integral, where those boxes may come into play again while I have not yet seen an practical use for the Filtration.

## Answer by Rylan (score 4)

https://quant.stackexchange.com/a/82480

As others have noted, it very rarely explicitly comes up. @Jan Stuller's phrase "all the information known at time t" is likely how most of us have built our intuition around filtrations. So if you're pricing an option, you never think "should I use information that's going to be revealed to me tomorrow?", and you would also likely not choose to "forget" past information entirely!

Within your model, it can certainly come up. Consider a Monte-Carlo pricer for an American option. A naive approach would be to simulate stock paths, find the best exercise along each path, and average those. However, that induces lookahead -- your best payoff might have been at time $s$, but at time $s$ you woudn't necessarily know that and might hold off from exercising. In order to price these options properly, you need to consider the filtration within your model. Longstaff-Schwartz provides what I think is a good explanation and overview of this.

Also, this less often comes up for pricing, but for backtesting it's important. Simple example would be you have a variable you want to predict such as stock returns, and you want to predict it using something like GDP estimates. If those estimates are revised, they don't represent what you would "know" at the time you have to make your investment decision, and might be biased towards having more predictive power than the "unrevised" estimate you had to trade with.