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Introduction

This question mainly arises out of the context of [Quasi Monte Carlo integration][1].

Which uses "quasi-random" numbers, (i.e. deterministic) with low discrepancy to reduce the variance in Monte Carlo integration. This reduction in variance is more prevalent in higher dimensions. And has thus found use in financial mathematics where they require numerical solutions to very high dimension integrals (>10^2).

Question

I have been unable to find any explicit examples of these very high dimension integrals anywhere online. Many articles reference these to arise from financial mathematics but are quite vague about its precise origins.

I would like to know how I could construct a integral of this manner and what its implications would be? Unfortunately I am very clueless on what goes on in financial mathematics.

Even better would be if anyone knows of an explicitly stated example of such a higher dimensional integral, but that seems unlikely.

My main purpose is really more of a showcase of Quasi Monte Carlo, but at the same time I want to avoid simply constructing an elementary integral like: $$\int_{\Omega^d} \cos(x)^d\,dx$$

Thanks for any and all help!

Bhoris Dhanjal
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    Probably the simplest example would be some sort of basket or rainbow option depending explicitly on a set of many underlying prices. But in practice high-dimensional integrals can arise rather easily from path dependencies in a derivative contract referring to just one price. – Ali Sep 04 '21 at 08:57
  • Computing the value at risk or similar statistics for a large portfolio could be another example. For example CDO:s might consist of thousands of mortgages so the value at risk would be an integral over a very high dimensional joint distribution. However, because this is complicated one often resorts to simplifying assumptions. – fes Sep 04 '21 at 14:13

1 Answers1

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Here are two examples for natural high-dimensional integrals. I assume you have heard about risk-neutral valuation, which is the reason why expectations (=integrals) are important in the first place.

Asian options

Asian options are derivatives mostly on a single asset. But their value depends on the average of the whole path of a stochastic process. Since there are no closed form solutions, you discretize the process and then simulate according to the law of the process. The high dimension is caused by the large number of discrete time steps required to properly discretize the process. Ballpark figure for dimension: 3 month tenor of the option, daily time-steps ~ 3*30 = 90.

Explicit function definition for an Asian with strike $K$: Just apply the cash flow rule to the discretization $s_1,\ldots,s_T$: $$ f(s_1,\ldots,s_T)= \max \left(\frac 1T\sum_{t=1}^T s_t - K,0\right)$$

Enterprise valuation models

To calculate the value of an life insurance portfolio you have to project the relevant cash-flows over many years, say for 60 years. In addition the policyholder benefits are financed by a large and diverse pool of assets, each having its own distribution. Ballpark figure for dimension: 60 for time * 10 asset (classes) = 600.

Explicit models for those tend to be more complicated and proprietory. One example is found in Appendix C of this PhD thesis:

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  • Replacing the sum with an integral in your Asian option payoff formula, it seems this can be dealt with by a PDE in just two dimensions. https://wayback.archive-it.org/all/20090320135332/http://people.math.jussieu.fr/~boka/enseignement/isifar/refs/Ref_Asiatiques_Rogers_Shi_95.pdf – Ali Sep 04 '21 at 13:16