Fall 2015 Qing Zhang: Mathematics of Trading Trading of securities in open marketplaces has been around for hundreds of years. It is only in recent years that mathematics has played an increasingly important role in market analysis and trading system design. High level mathematics such as stochastic calculus has become an indispensable tool in modern finance. Mathematically defined processes such as Brownian motion and Markov chains have become cornerstones for sophisticated market models including geometric Brownian motion models, mean reversion models, and models with regime switching. These models allow systematic analysis of security price movements and advance much human understanding of the marketplace. This proposed seminar course will give its participants unique research experiences in applications of advanced mathematics in stochastic analysis in the marketplace. A typical participant of the program will be an undergraduate student who has taken 3500 and 3510 or 4600 and has some programming skills. S/he will spend the first few weeks and learn related topics in stochastic processes, stochastic analysis, equity markets, and derivatives pricing. S/he will also be required to directly observe the US stock and stock option markets for first hand market knowledge. Traditionally, a geometric Brownian motion (GBM) is used to capture stock price movements. However, it has been noticed that such modeling only works in trending markets. For a nontrending (sideways) market, a mean reverting (MR) model appears to be suitable for characterizing price fluctuations. Given up to date historical prices, it is important to determine which model (GBM vs. MR) fits better the market. There is no systematic way of addressing this issue in the literature. The goal is to develop a meaningful criterion measuring the degree of fitness. A related research topic is the calibration of both models. Furthermore, in actual trading environments, it is more desirable to predict what type of models the market is more likely to follow in the near future. There are some preliminary results along this direction. The idea is to use the corresponding stock option prices to forecast future model type. The student will be guided to start with numerical experiments mainly using Monte Carlo simulations for hypothetical models to validate the approach. Then, s/he will apply these methods in the real marketplace. S/he will also learn basics in trading system design. The corresponding profit/loss function will be used to measure how successful the model selection and prediction. The objective of this program is to expose participants to real-world problems, teach them the necessary tools in mathematics, and guide them to carry out the research projects.