NC State Research Experience for Undergraduates Mathematics:
Summer 2005 Experience

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From left to right (back row) Anastasia Wong, Eamonn Tweed, Jennifer Geiss, Thomas Deems
(Front row) Loek Helminck, Hien Tran, Troy Tingey, Sandy Paur, Sarah Lynn Joyner, Robert Benim, Chris Vogle

Last summer's project descriptions:

Financial Math: Optimized Least Squares Approach for Valuation of American Options
Thomas Deems, Slippery Rock University
Jennifer Geis, Augsburg College
Troy Tingey, Arizona State University
Anastasia Wong, Mills College

Option pricing has been a very hot research topic in financial mathematics for long time. Unfortunately, only for few situations we can have explicit formulas. In most cases, numerical methods or simulations have to be used to obtain the fair prices for options. One such method is Monte Carlo simulation. The goal of this REU project is to further improve the efficiency of this method for some type of options, especially on American Options.

Four students, T. J. Deems, Jennifer Geis, Troy Tingey, Anastasia Wong, worked two months on this project and they got very promising results. One part of their work was to dramatically reduce the standard error (by 95%) of the simulation for Arithmetic Asian Option with floating strike price by introducing a control variate of Geometric Asian Option with floating strike price. The method of control variate was also tested on barrier options and the results are very satisfactory. The main part of their research was to combine the control variate technique with the recently developed least-square Monte Carlo method with application to American put option. Usually, the classic Monte Carlo Method cannot be used for American options due to the difficulty to evaluate the expect payoff for continuation. A recent paper by Longstaff and Schwartz was able to overcome the difficulty by integrating Monte Carlo simulations with least-squares regression. To improve the efficiency of the method, they introduced the control variate. European Put and Geometric Asian with fixed strike price were used to serve as control variate for American put option. The standard error improvement was about 30% to 50% for both cases. Their results also indicated that Geometric Asian is a better control variate when the stock is in the money and European Put is better when the stock is out-of-the-money.

These results were presented at the undergraduate poster presentation at the 2006 annual meeting of the AMS/MAA in San Antonio, TX.

Biological Project: Heart Rate Regulation during Postural Change from Sitting to Standing

Robert Benim, University of Portland
Sarah Joyner, Meredith College
Eamonn Tweedy, North Carolina State University
Chris Vogl, Illinois Wesleyan University

Four students Robert Benim, Sarah Lynn Joyner, Eamonn Tweedy and Chris Vogl worked on a project entitled "Modeling Heart Rate Regulation During Postural Change from Sitting to Standing in Healthy Young, Healthy Elderly, and Elderly Hypertensive People". The work by the REU students resulted in the development of a mathematical model that predicts concentration of noradrenaline, they made important changes to the heart-rate model in order to reflect realistic dynamics, and most importantly, the students discovered, by performing detailed model analysis, that a correct physiological response could not be obtained without including time-delay between sympathetic and parasympathetic nervous responses and an impulse function that accounts for the subject's physical preparation for standing. With these modifications the model developed could predict heart-rate changes observed for both healthy young, healthy elderly, and healthy hypertensive subjects. During the REU program the students performed model validation for six subjects and currently, two students C. Vogl and E. Tweedy are working to finish model validation and perform statistical analysis for approximately 60 datasets. Furthermore, E. Tweedy worked during the fall semester as part of his honor thesis to conduct sensitivity analysis and to write a manuscript that we plan to submit to a high-profile international journal such as the American Journal of Physiology. These important results obtained by the students were a result of an excellent teamwork among the students and a collaboration with our medical collaborator who visited NCSU during the REU program to discuss the final model design with the students.

These results received an honorable mention at the undergraduate poster presentation of the 2006 annual meeting of the AMS/MAA in San Antonio, TX.

In addition, they presented a poster at the 2005 annual fall meeting of the BMES in Baltimore, MD.

Publications

M.S. Olufsen, J.T. Ottesen, REU program, L.A. Lipsitz and V. Novak, “Modeling Baroreflex Regulation of Heart Rate during Orthostatic Stress”, Am. J. Physiol. Regulatory Integrative Comp Physiol, 2006, vol. 291, pp. R1355-1368. 

 

Comments and suggestions to loek@math.ncsu.edu