Real-world exposure and CVA simulationThe risk-neutral approach assumes that asset prices follow a stochastic process with a drift coinciding with the short rate r(t) being a risk-free interest rate. dS(t)=S(t)[rdt+σ(S(t),t) 〖dW〗^Q (t)]Instead, in real-world measurement, they follow a more complex process, which embodies time and investors' risk aversion, namely: dS(t)=S(t)[μ(S(t),t)dt+σ(S(t),t) 〖dW〗^R (t)] or, equivalently, a process with real-world stochastic discount factors that depend on risk-free interest rates but also on the prices of the assets themselves. The form of this process with the almost arbitrary drift term of the process complicates the implementation; for example, it makes it difficult in practice to simulate asset prices by standard analytical or quasi-analytical approaches to measurement transformation via Girsanov's theorem. Not to mention the case of imperfect replication with an infinite number of consistent measurements without arbitrage, the lack of an unambiguous common real-world measurement, as opposed to risk-neutral measurements, makes real-world simulation much more difficult to implement. implemented. 1. With a limited number of analytical or quasi-analytical shortcuts and a single, consistent measure without arbitrage, practitioners often choose a brute force approach for risk management purposes, such as PFE and CVA simulations of real world, both of which require full simulation of exposure distributions. for any instrument other than vanilla. This brute force approach, known as Monte Carlo nested on Monte Carlo, includes the following two steps repeated in a loop: drawing actual yield curves and other relevant market factors at all stages of the advanced model using explicit model assumptions (equivalently, assumptions about drift terms) for a subset of stochastic paths, ...... middle of paper ...... k-neutral market models. The nested Monte Carlo scheme requires the calculation of the instrument. There are a few shortcuts for vanilla instruments. We can omit the next step of nested Monte Carlo and simply evaluate the vanilla instruments from the forward curves at each step of the t_i model. Likewise, we can reduce the pricing of vanilla caps or swaptions to a mechanism for interpolating volatilities/futures prices, which is an effective but valid way to avoid building a nested evolution model. For weakly path-dependent instruments, such as Bermuda callable instruments, which allow the representation of conditional exposures as a non-linear function of certain market states or underlying non-callable exposures, there is a way to avoid by reusing risk-neutral American Monte Carlo results.