When Percentile Mapping Gives Way to Monte Carlo and Quasi-Monte Carlo
Summary
The document considers replacing random scenario generation with equally spaced probability points when estimating a terminal payoff distribution, using a European call as an example. In a single-variable setting, mapping quantiles through a payoff can be sensible when the underlying cumulative distribution is available. Such a grid is essentially a simple quasi-Monte Carlo integration scheme, but quantile mapping is impractical when the cumulative distribution is unknown or expensive to calculate.
The discussion also explains why the approach becomes harder for multiple variables. The number of grid points needed can grow rapidly with dimension, and knowing each variable’s marginal distribution does not specify their joint distribution or dependence. A copula is one way to impose dependence, though it may not represent it accurately. Low-discrepancy sequences, such as Sobol sequences, can help with high-dimensional integration. If a characteristic function is known, direct numerical inversion for an option price may be more efficient than first constructing scenarios. These are general numerical considerations, not a guarantee that any one method will be most accurate for a given model.
Key ideas
- Equally spaced probability points are a simple quasi-Monte Carlo approach to integrating a one-dimensional payoff.
- Quantile mapping requires a usable cumulative distribution and can be costly when that distribution is unavailable in closed form.
- Multivariate grids face rapidly increasing sample requirements as dimension grows.
- Marginal distributions do not determine the joint distribution or its dependence structure.
- Low-discrepancy sequences can help with high-dimensional integration, while characteristic functions may enable direct pricing inversion.
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# Stochastic Simulation vs percentile-to-percentile map
# Stochastic Simulation vs percentile-to-percentile map
I was wondering why someone would go to the trouble to generate random variables in scenarios that are not path dependent. Let me provide a simple (although somewhat contrived) example. Lets say that we want the terminal distribution of a vanilla European call option. We could generate terminal values for the underlying, then plug it into $\mathrm{max}(0;S_T - K)$, and there we have our terminal distribution of the call...
So here's the question. Why not simple do a percentile-to-percentile map from the underlying distribution to the derivatives distribution using equally spaced points?
I do not know if the following is correct, but someone I know told me that we do this because things occasionally fall apart when we use equally spaced points in a multivariate context... I however do not understand why this would happen?
I was hoping someone could shed some light on this issue for me.
## Answer by Kiwiakos (score 2, accepted)
https://quant.stackexchange.com/a/25883
What you describe is a very simple quasi monte carlo, where the 'random' points are equally spaced in probability space. Like numerical integration.
Sometimes you can use it, but in general you will need the cumulative distribution to do percentile mapping. This very frequently is not known in closed form, and can be very expensive to compute numerically.
In fact, if you have the cdf in closed form then you will also have the option price in closed form and there is no need to evaluate numerically (Baksi and Madan have shown this amongst others).
If you have the characteristic function in closed form, then it is much more efficient to numerically invert for the option price directly rather than going through probabilities.
## Answer by user29970 (score 0)
https://quant.stackexchange.com/a/25919
Your approach is sensible for a single variate case if the cdf is available, but does (as your friend said) break down for more variates.
One issue with multivariate case is the "curse of dimensionality" - as the number of variates increases your number of samples will get infeasibly large very quickly. To address this, one can use a low discrepancy sequence (eg Sobol) rather than a uniform grid.
Another issue is that even if the terminal marginal distributions are all known, the terminal joint distribution may not be. Of course, one can pick a copula, but this may not be sufficiently accurate.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.