Monday, March 29, 2010

Computational workshop - first day (a week later)

As promised, here is a day-by-day lookback at the workshop on Computational Methods in Finance that took place at Fields last week.

The majority of the talks in the first days consisted of what I would call "traditional" numerical methods (for lack of a better name), namely, the thorough investigation of very clever ways to obtain numerical solutions to challenging problems.

Ralf Korn opened the workshop with a very elegant overview of binomial methods. To tackle the tricky problem of matching the tree parameters to a complicated correlation structure in high dimensional, he proposed a universal method of decoupling based on a diagonalization of the correlation matrix, which reduces the matching problem to several uncorrelated one-dimensional trees, for which well refined methods already exist.

Kumar Muthuraman followed with a moving boundary method to solve free-boundary problems: start with a blatantly suboptimal boundary (often suggested naturally by the problem at hand) and systematically improve it in the region where the associated variational inequality is violated until the true free-boundary is well approximated.

John Chadan gracefully replaced Garud Iyengar, who could not attend, and explained how integral equations methods can lead to very detailed results for the American put option problem, in particular the fact that the exercise boundary might fail to be convex when dividends are higher than the risk-free interest rate.

In the context of portfolio optimization, Lorenzo Garlappi showed how a carefull decompostion of the state variables into an observable component and a random error, combined with a Taylor expansion of the value function (expressed in monetary terms) can lead to very accurate numerical approximations.

After lunch, a panel consisting of Jim Gatheral, Chris Rogers, Ernst Eberlein and Jeremy Staum discussed current challenges in quantitative finance, especially in light of the crisis of 2008.

Jeremy Staum then followed with a comprehensive set of proposals for the use of repeated simulations in finance. It is hard to do justice to his far reaching talk in just one line, so I'll just mention the key concepts of Database Monte Carlo and Metamodeling, while hoping to have more chance to study this in the future.

Birgit Rudloff concluded with perhaps the most technically demanding talk of the day (for me at least), investigation the effect of transaction costs on risk measures. My mind went in to a swirl right at the begining of her talk, since many of the familiar concepts such as arbitrage, portfolio value, risk measure, immediately became a special "scalar" case of their mutli-dimensional generalizations (a consequence of not being permitted to calculate the value of a portfolio as a simple scalar product at each point in time due to the presence of transaction costs). I partially recovered some sanity with the 2-dimensional example at the end, where nice pictures can be drawn ont he board, but at the point everyone was rightly ready for a glass of wine in the reception that followed.

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