Research

Honors Senior Thesis: 

Stochastic Calculus for Option Pricing 

with Convex Duality, Logistic Model, and Numerical Examination

• Supervised by Professor Zhen-Qing Chen

• Abstract:

This thesis explores the historical progression and theoretical constructs of financial mathematics, with an in-depth exploration of Stochastic Calculus as showcased in the Binomial Asset Pricing Model and the Continuous-Time Models. A comprehensive survey of stochastic calculus principles applied to option pricing is offered, highlighting insights from Peter Carr and Lorenzo Torricelli’s “Convex Duality in Continuous Option Pricing Models”. This manuscript adopts techniques such as Monte-Carlo Simulation and machine learning algorithms to examine the propositions of Carr and Torricelli, drawing comparisons between the Logistic and Bachelier models. Additionally, it suggests directions for potential future research on option pricing methods.

• Submitted on June 9, 2023

• Currently being furnished and further researched.

Elevator Optimization: Application of Spatial Process and Gibbs Random Field Approaches for Dumbwaiter Modeling and Multi-Dumbwaiter Systems

• Coauthored with Benjamin Lu Davis, Wanchaloem Wunkaew, & Xinyu Chang

• Abstract:

This research investigates analytical and quantitative methods for simulating elevator optimizations. To maximize overall elevator usage, we concentrate on creating a multiple-user positive-sum system that is inspired by agent-based game theory. We define and create basic "Dumbwaiter" models by attempting both the Spatial Process Approach and the Gibbs Random Field Approach. These two mathematical techniques approach the problem from different points of view: the spatial process can give an analytical solution in continuous space and the Gibbs Random Field provides a discrete framework to flexibly model the problem on a computer. Starting from the simplest case, we target the assumptions to provide concrete solutions to the models and develop a "Multi-Dumbwaiter System". This paper examines, evaluates, and proves the ultimate success of such implemented strategies to design the basic elevator's optimal policy; consequently, not only do we believe in the results' practicality for industry, but also their potential for application.

• Submitted on September 26, 2022, last revised on December 23, 2022.

Optimizing Stock Option Forecasting with the Assembly of Machine Learning Models and Improved Trading Strategies

• • Coauthored with Raymond Guo, Wenyu Du, Jiayi Gao, & Kirill V. Golubnichiy

  • Abstract:

This paper introduced key aspects of applying Machine Learning (ML) models, improved trading strategies, and the Quasi-Reversibility Method (QRM) to optimize stock option forecasting and trading results. It presented the findings of the follow-up project of the research "Application of Convolutional Neural Networks with Quasi-Reversibility Method Results for Option Forecasting". First, the project included an application of Recurrent Neural Networks (RNN) and Long Short-Term Memory (LSTM) networks to provide a novel way of predicting stock option trends. Additionally, it examined the dependence of the ML models by evaluating the experimental method of combining multiple ML models to improve prediction results and decision-making. Lastly, two improved trading strategies and simulated investing results were presented. The Binomial Asset Pricing Model with discrete time stochastic process analysis and portfolio hedging was applied and suggested an optimized investment expectation. These results can be utilized in real-life trading strategies to optimize stock option investment results based on historical data.

• • Submitted on November 29, 2022.

Application of Convolutional Neural Networks with Quasi-Reversibility Method Results for Option Forecasting

• • Coauthored with Wenyu Du & Kirill V. Golubnichiy

  • Abstract: 

This paper presents a novel way to apply mathematical finance and machine learning (ML) to forecast stock options prices. Following results from the paper "Quasi-Reversibility Method and Neural Network Machine Learning to Solution of Black-Scholes Equations" (appeared on the AMS Contemporary Mathematics journal), we create and evaluate new empirical mathematical models for the Black-Scholes equation to analyze data for 92,846 companies. We solve the Black-Scholes (BS) equation forwards in time as an ill-posed inverse problem, using the Quasi-Reversibility Method (QRM), to predict option price for the future one day. For each company, we have 13 elements including stock and option daily prices, volatility, minimizer, etc. Because the market is so complicated that there exists no perfect model, we apply ML to train algorithms to make the best prediction. The current stage of research combines QRM with Convolutional Neural Networks (CNN), which learn information across a large number of data points simultaneously. We implement CNN to generate new results by validating and testing on sample market data. We test different ways of applying CNN and compare our CNN models with previous models to see if achieving a higher profit rate is possible.

• • Submitted on August 25, 2022, last revised December 12, 2022.