Mathematical Models and Applications
In general, a mathematical model is a description of a system using mathematical concepts and language. It can take many forms such as dynamical systems, statistical models, differential equations, game theoretic models, or stochastic process. I am interesting in applications of mathematical models in finance, statistics, climate change, and actuarial sciences. Some of my favorite models are:
Hidden Markov Models (HMM)
In many real-world problems, the states of a system can be modeled as a Markov chain in which each state depends upon the previous state in a non-deterministic way. In hidden Markov models (HMMs), these states are invisible while observations (the inputs of the model), which depend on the states, are visible. I applied the model to predict economic regimes, stock prices, and weather conditions.
Variance Gamma Model
Variance gamma (VG) process was introduced by Madan and Seneta in 1990 as a model for option pricing. The improvement in the VG model is that there is no continuous martingale component, allowing it to overcome the well-known shortcomings of the Black-Scholes model.
Monte Carlo and Quasi Monte Carlo Method
The Monte Carlo simulation is a popular numerical method across sciences, engineering, statistics, and computational mathematics. In simple terms, the method involves solving a problem by simulating the underlying model using pseudorandom numbers, and then estimates the quantity of interest as a result of the simulation. In recent years, a deterministic version of the MC method, the so-called quasi-Monte Carlo (QMC) method, has been widely used by researchers. Different from the MC method which relies upon pseudorandom numbers, QMC method uses low-discrepancy sequences for the sampling procedure. The QMC method converges generally faster than the corresponding MC method. One example of my research in this field is
Currently, the main method for transforming low-discrepancy sequences to non-uniform distributions is the inverse transformation technique. However, this technique can be computationally expensive for complicated distributions. The acceptance-rejection technique was developed precisely for this reason for the pseudorandom sequences.
SOA individual research funding in finance: July 2016-June 2017, “Hidden Markov Model for Portfolio Management with Mortgage Backed Securities Exchange-Traded Funds”.
Nguyet (Moon) Nguyen