Stochastic Calculus and Derivative Pricing

📚 Independent Study: Stochastic Calculus and Derivative Pricing

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This independent study written by myself, conducted under the supervision of Professor Roberto Fernandez at NYU Shanghai, explores the mathematical foundations of stochastic calculus and its applications in modern quantitative finance. It traces the development of stochastic processes from Brownian motion and martingales to Itô integrals, Doeblin formulas, and the Black-Scholes-Merton model.

Key topics include:

• Construction of Itô integrals and the Doeblin formula
• Derivation and PDE analysis of the Black-Scholes equation
• Extensions to geometric Brownian motion, dividend-paying assets, and barrier options
• A deep dive into multivariate stochastic calculus and its role in modeling correlated assets
• The probabilistic foundation of exotic derivatives, such as up-and-out calls

🧠 Through rigorous proof-based development and applied financial modeling, this study demonstrates the power of stochastic differential equations (SDEs) in pricing and hedging a wide range of derivative products.