Expanding a bit more, risk calculations are basically calculating the partial derivatives of the current value of your books with respect to inputs observable in the outside world ("observables", almost exclusively current market prices). In other words, it's calculating the Jacobian matrix for the total value of everything on your books.

I worked on a project to compile a declarative description of complex financial products into GPU code to perform Monte Carlo simulations to estimate the fair market price of those products. We found that having a function return its present value and its N-dimensional first-order linear approximation costs about twice as much computing power as just calculating the present value. Approximating the first-order linearization by bumping inputs is exponential in the number of dimensions.

In particular, with local stochastic volatility (LSV) models, you're using these linearizations to work out the values of some hidden variables (implied volatilities at various strike prices, and implied volatilities of those volatilities ("vol on vol") going forward). That gives you a smooth, self-consistent volatility surface that you can use to estimate the value of things that don't trade very often based on the current prices of things that do trade very often. A lot of financial modelling isn't about predicting the future, but instead "predicting the present" that you can't see based on the present that you can see.

On a side note, before I did work with financial models, I didn't understand why so many models assume Gaussian distributions even though the distributions have been known for decades to be non-Gaussian. It turns out that most non-Gaussian distributions don't give the same distribution (with different parameters) if you take a linear combination of them. (i.e. most distributions aren't stable distributions) The Levy distribution fits most financial data better than the Gaussian distribution, and is stable, but has infinite variance. So, in many financial modelling situations, the most efficient way to model behavior is to make piecewise Gaussian approximations of your actual distribution. So, it sounds crazy at first to model something as a Gaussian distribution with different variances at different points, there is a method to the madness.

Side note: did we work together in Hong Kong at an investment bank? My YAMS initials were the same as my HN username. (Though, when I started, my YAMS initials were KAM, as we were limited to 3 initials at that time.)

Err... sorry... bumping once in each of the directions is linear in the number of dimensions, not exponential. You don't bump all combinations, just one dimension at a time. In any case, the savings from automatic differentiation are often significant, but not exponential, unless you're doing something very wrong.

(I was FAB on YAMS, but based in London.)

Just to add to this that AD was also a game changer for the banks in terms of necessary computing power and the time it needs to do these calculations. Antoine Savine describes this well in "Modern Computational Finance: AAD and Parallel Simulations", if someone is interested in this topic.