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

ArXiv ID: 2408.05672 “View on arXiv”

Authors: Unknown

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.

Keywords: Stochastic Calculus, Option Pricing, Binomial Asset Pricing, Monte-Carlo Simulation, Convex Duality

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The thesis covers advanced topics in stochastic calculus, convex duality, and includes numerous proofs and derivations, indicating high mathematical density. However, it relies on conceptual simulations (Monte Carlo) and machine learning references without providing code, datasets, or backtesting results, making it more theoretical than implementation-ready.

  flowchart TD
    A["Research Goal: Examine Stochastic Duality & Option Pricing"] --> B["Data & Models"]
    B --> B1["Binomial & Continuous Models"]
    B --> B2["Carr & Torricelli's Duality Theory"]
    B --> B3["Logistic & Bachelier Models"]
    B1 & B2 & B3 --> C{"Key Methodology"}
    C --> C1["Monte-Carlo Simulation"]
    C --> C2["Machine Learning Algorithms"]
    C1 & C2 --> D["Computational Analysis"]
    D --> E["Key Findings & Outcomes"]
    E --> E1["Theoretical Proposition Validation"]
    E --> E2["Logistic vs. Bachelier Model Comparison"]
    E --> E3["Future Research Directions"]