Stochastic Calculus for Option Pricing with Convex Duality, Logistic Model, and Numerical Examination

TL;DR

This paper reviews stochastic calculus in option pricing, compares Logistic and Bachelier models using Monte Carlo and machine learning, and discusses convex duality and future research directions.

Contribution

It provides a comprehensive survey of stochastic calculus in option pricing, integrating convex duality, Logistic models, and numerical methods with comparative analysis.

Findings

Logistic model offers a viable alternative to Bachelier in option pricing.

Monte Carlo and machine learning techniques effectively examine model propositions.

Insights into convex duality enhance understanding of continuous option pricing models.

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.