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.
Table of Contents:
1 Introduction 3
2 A Brief History of Financial Mathematics 4
3 Stochastic Calculus for Finance 6
3.1 The Binomial Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Continuous-Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Basics in Probability Theory . . . . . . . . . . . . . . . . . . . . . . 13
3.2.2 Change of Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.3 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.4 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.5 Black-Scholes-Merton Model . . . . . . . . . . . . . . . . . . . . . . 28
3.2.6 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.7 Risk-Neutral Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.8 Martingale Representation Theorem . . . . . . . . . . . . . . . . . . 36
3.2.9 Risk-Neutral Valuation for Deriving the BSM Formula . . . . . . . 37
3.3 Section Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Convex Duality in Continuous Option Pricing Models 41
4.1 Model Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Assumptions and Propositions . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Novel Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Section Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Numerical Simulation and Machine Learning for Convex Duality in Continuous Option Pricing 52
5.1 Monte Carlo Simulation for Models’ Evaluation . . . . . . . . . . . . . . . 52
5.1.1 Model Comparison Examination . . . . . . . . . . . . . . . . . . . . 52
5.1.2 Simplicity Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1.3 Complexity Comparison . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Convex Duality and Convolutional Neural Network . . . . . . . . . 57
5.2.2 Logistic Regression and Logistic Model . . . . . . . . . . . . . . . . 58
5.3 Section Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 Observations and Comments 60
7 Acknowledgment 62
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