Three Bachelor's Degree Project Presentations in Mathematics

Time: 10.15 - 11.00
Student: Axel Olsson
Title: Extending the Least Squares Monte Carlo Method for American Options with Discrete Dividends

Abstarct: This thesis develops an extension of the Least Squares Monte Carlo algorithm to price American‐style options on assets with discrete dividends. Two extra exercise dates, immediately before and after each dividend, are inserted into the original time grid. Detailed mathematical derivations and a practical code solution are presented. Numerical experiments show pricing errors below 0.2 % for puts and 1 % for calls versus chosen benchmarks.

Optimization techniques are presented and implemented to reduce the algorithm’s runtime, but it still remains far slower than the competitive benchmark.

Time: 13.15 - 14.00
Studen: Max Callenmark
Title: Estimating PSA Growth Rates After Prostate Cancer Treatment

Abstract: This essay studied methods for estimating the rate of growth of prostate-specific antigen (PSA) levels in the blood following prostate cancer treatment, particularly in cases where only limited data are available. Monitoring the rate of PSA increase is clinically important, as it can indicate cancer recurrence or progression and a high growth rate is associated with higher risk of death from prostate cancer. The growth rate after surgery is particularly problematic to estimate since PSA values are typically below the detection limit (0.1 ng/ml) for a short or long period

To estimate the growth rate of PSA at the time when it has become measurable (two consecutive values above 0.1 ng/ml), we aimed to test and compare different estimation techniques by simulating a dataset that imitated PSA behavior over time in patients. The estimation methods evaluated included maximum likelihood estimation (MLE), complete case analysis (CCA) with lookback and deterministic imputation (DI). A simulation study was conducted on the generated dataset to assess how accurately each method could estimate the growth rate under varying choices of parameters. These results were then compared to analyses on a real-life dataset.

In the simulated datasets, MLE performed the best in estimating the PSA growth rate, followed by CCA using all available information. DI showed somewhat lower accuracy but still performed reasonably well, whereas CCA with limited information generally produced the least accurate estimates. In the real-world dataset, the relative performance of the methods differed from the theoretical results; it was unclear which method performed best.

These findings suggest that while MLE is generally the most effective for estimating the rate of growth of PSA, simpler methods like DI may be preferable when the number of available measurements is very limited.

Time: 14.15 - 15.00
Student: Pontus Johansson
Title: Heegaard Diagrams Through Morse Theory

Abstract: This thesis aims to study the concept of Heegaard diagrams of 3-manifolds at a level appropriate for an undergraduate student. This is done by describing Heegaard splittings using the handle decompositions detailed by Morse functions, concepts that only require a background in basic and differential topology. First, we define several core concepts in topology, such as smooth manifolds, homotopy, and retraction. Then we describe important aspects of Morse theory, such as Morse functions and Morse lemma. Using Morse lemma, one can find that a manifold can be decomposed into a union of handles, each associated with a critical point of the Morse function. Restricting our study to 3-manifolds, we find that we can separate the manifold into two similar handlebodies, consisting of a solid sphere with several 3-dimensional 1-handles attached, called a Heegaard splitting. It is then shown how one can construct the 3-manifold by detailing how the boundaries of the two handlebodies attach to the abstract Heegaard surface. The result is a description of the topological properties of the 3-manifold using a comparably simpler 2-manifold (Heegaard surface) and a collection of curves on that surface. Finally, the thesis ends with a set of examples, where we find the Heegaard diagrams of the product 3-manifolds of a circle and a genus g surface.