At a recent workshop at Banff International Research Station for Mathematical Innovation and Discovery, I presented my research on high-dimensional numerical problems and its applications in finance.
Many high-dimensional valuation problems (think of an option on a basket of securities or an Asian option, which is an option on a security monitored at several time points) can be solved only with numerical techniques. A common method is Monte Carlo simulation, where the random outcomes of security prices are simulated by drawing random numbers from a computer. Because the outcome of a Monte Carlo simulation is different each time it is repeated, the Monte Carlo estimator itself is a random variable and subject to variance! So-called variance-reduction techniques aim at developing more efficient Monte Carlo estimators. In other words, for the same computational budget (i.e., CPU time) you get a more accurate price estimate.
One such variance-reduction method is “Latin hypercube sampling” (LHS). For a high-dimensional estimation problem, the resulting random numbers are more evenly spread than one would typically obtain from a simple Monte Carlo simulation. LHS assumes that the random numbers across the dimensions are independent. But remember from your finance and risk management courses that dependence (correlation) is what matters for your portfolio risk!
Together with Wolfgang Schmidt, Professor of Quantitative Finance, I have extended LHS to incorporate dependence so that we can obtain efficient estimates of option prices on baskets of securities.
Man, was I excited when I received the invitation to attend the workshop “Approximation of High-Dimensional Numerical Problems – Algorithms, Analysis and Applications” at BIRS! Workshops at BIRS are by invitation-only and will typically host approximately 30 participants from all over the world. This is a great opportunity to share research and to learn what other researchers in your field are currently working on. And, as if that was not enough, BIRS is located in Banff, a beautiful resort in the Canadian Rocky Mountains. Luckily we got one afternoon off, and I was able to hike from Lake Louise to Lake Agnes and enjoy the picturesque scenery.
You can find more information on the workshop website.
Hiking at Lake Louise and Lake Agnes with Art Owen (Stanford) and his wonderful wife Patricia, Joseph Dick (UNSW) and Mike Giles (Oxford).
And in case you’re keen on understanding more on the technique developed, my talk was recorded and is available.
The related papers are available here and here.