Lecture Series:  Numerical Methods in Finance

March 21, 2013

Middlesex College, room MC106

Western University (University of Western Ontario)

Dr. Alexey Kuznetsov

Dr. Sebastian Jaimungal

Numerical methods for Levy processes

Levy processes are great objects for mathematical modeling: they are relatively simple,

there are many of them, and they fit the data better than some other popular models. The major

problem is that it is hard to compute things numerically. For example, there are dozens of good

research papers on pricing barrier options in the Variance-Gamma or CGMY models, but there is

no clear winner among these algorithms. A new trend has emerged in the last five years:

instead of banging one’s head against the wall while trying to compute things in one of the standard

models (VG, CGMY, NIG, KoBoL, etc.), let us try to find similar Levy processes, where computations

would be simpler, more efficient (and more enjoyable). In these lectures we will discuss some

recent progress in this area. First, we will give a general introduction to Levy processes, including

some results from fluctuation theory. Second, we will discuss two popular numerical algorithms for

computing Fourier/inverse Laplace transform. Finally, we will discuss Levy processes with

hyper-exponential jumps and will examine in detail several examples, including finding the

distribution of the first passage time, pricing barrier options and Asian options.

Optimal Execution of Accelerated Share Repurchases

Accelerated share repurchases (ASRs) allow a corporation to repurchase a significant portion of its

shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade

to a broker. The broker must then purchase the shares from the market over several days or weeks.

Some contracts allow the broker to specify when the repurchase ends, at which point the corporation

and the broker exchange the difference between the arrival price and the TWAP or VWAP over the

trading period plus a spread. Hence, the broker effectively has an American option embedded within

an optimal execution problem. In this work, we address the broker’s optimal execution and exit

strategy taking into account the impact that trading has on the market. We demonstrate that it is

optimal to exercise when the average price (TWAP or VWAP) exceeds a time-dependent proportion

of the fundamental price. Moreover, we develop a dimensional reduction of the stochastic control

and stopping problem and show how the dynamic programming equations can be efficiently solved

numerically and explore the qualitative behavior of the optimal trading and exit strategies.

This Lecture Series has organized by SIAM Student Chapter of Western University, which is co-sponsored by the departments of Applied Mathematics and

Statistical & Actuarial Sciences. Registration fee is 10 CAD. Registration by email: siam@stats.uwo.ca. For additional info, please call 1-519-6612111 ext 86828

Dr. Kenneth Jackson

Computation of Loss Distribution Based on the Structural Model for Credit Portfolios

Credit risk analysis and management at the portfolio level is a challenging issue for financial institutions

due to their portfolios’ large size, heterogeneity and complex correlation structure. We propose several

new asymptotic methods and exact methods to compute the distribution of a loanportfolio’s loss in the

CreditMatrics framework. For asymptotic methods, we give an approximation based on the

Central Limit Theorem (CLT), which gives more accurate approximations to the conditional portfolio loss

probabilities compared with existing approximations. To further increase the accuracy of approximations

for lumpy portfolios, we introduce a hybrid method which combines an asymptotic approximation with

Monte Carlo simulation. For exact methods, we improve the efficiency by exploiting the sparsity that often

arises in the obligors’ conditional losses. A sparse convolution method and a sparse FFT method are proposed,

which enjoy significant speedups compared with the straightforward convolution method. We also construct

truncated versions of the sparse convolution method and the sparse FFT method to further improve their

efficiency. To balance the aliasing errors and roundoff errors incurred in the truncated sparse FFT method,

an optimal exponential windowing approach is developed as well.