Stochastic Calculus & Mathematical Finance
This is a course for advanced undergraduate and graduate students with an aptitude and interest in quantitative methods. The students will be exposed to no-arbitrage-based deep and beautiful ideas underlying mathematical finance. For this they will get an adequate exposure to stochastic calculus. Stochastic calculus is also relevant for the increasingly mainstream diffusion-based generative AI.
Course Period: Monsoon 2025.
Schedule: Tuesday & Thursday, 10:10 – 11:40 AM
Mode: Hybrid (In-person + Online)
Duration: 40 hours of rigorous lectures + student-led projects
Course Content
- Review of Measure-Theoretic Probability
- Martingales & Brownian Motion
- Stochastic Calculus & Stochastic Integration
- Itô's Formula & Girsanov's Theorem
- Stochastic Differential Equations & PDE Connections
- Discrete & Continuous-Time Pricing Theory
- Stochastic Volatility Models & Interest Rate Models
- PDEs in Finance
Eligibility
- Advanced undergraduates & master's-level students
- Strong background in Probability, Linear Algebra, and Analysis
- Aptitude & interest in quantitative methods
Lectures
- Lecture 1: Introduction to Derivatives (August 26, 2025)
- Lecture 2: Basic Probability & Discrete Finance (August 28, 2025)
- Lecture 3: Risk Neutral Measure & Complete Markets (September 2, 2025)
- Lecture 4: Measure Theory Overview: Sigma Algebras (September 4, 2025)
- Lecture 5: Lebesgue Integral and Convergence (September 9, 2025)
- Lecture 6: Borel-Cantelli Lemma and Relationship Between Convergence Types (September 11, 2025)
- Lecture 7: Law of Large Numbers and Central Limit Theorem (September 16, 2025)
- Lecture 8: Independance and Conditional Expectations (September 18, 2025)
- Lecture 9: Conditional MCT, DCT and the Tower Property (September 23, 2025)
- Lecture 10: Properties of Martingale and Stopping Time (September 25, 2025)
- Lecture 11: Martingales, Submartingales and Brownian Motion (September 30, 2025)
- Lecture 12: Brownian Motion Continued and Reflection Principle (October 14, 2025)
- Lecture 13: Brownian Motion as Gaussian Process & Ito's Integral (October 16, 2025)
- Lecture 14: Ito's Integral Continued (October 23, 2025)
- Lecture 15: General Ito's Formula (October 28, 2025)
- Lecture 16: Ornstein-Uhlenbeck (OU) Process and the Black-Scholes Model (October 30, 2025)
- Lecture 17: Black-Scholes PDE and Girsanov's Theorem (November 4, 2025)
- Lecture 18: Levy's Theorem, Girsanov's Theorem and Ito's Multivariate Calculus (November 6, 2025)
- Guest Lecture 19: Financial Modelling via compound Poisson Processes (November 11, 2025)
- Guest Lecture 20: Introduction to Jump Processes (November 13, 2025)
- Lecture 21: Martingale Representation Theorem and Black-Scholes Price Derivation (November 18, 2025)
- Lecture 22: Multi-dimensional Market Model (November 20, 2025)
- Lecture 23: Introduction to Interest Rate Models (November 25, 2025)
- Lecture 24: Interest Rate Models Continued (November 27, 2025)
- Lecture 25: Derivatives Pricing (December 4, 2025)
Lecture Notes
Prof. Sandeep Juneja
- Professor of Computer Science at Ashoka University. Director, Safexpress Centre for Data, Learning and Decision Sciences
- Former Professor at TIFR and IIT Delhi. Former Head of Quantitative Ops (Bank of America, India)
- Taught at JP Morgan, Bank of America, Morgan Stanley, Nomura. Consulted with Research Wing of Reserve Bank of India
- Research in Applied Probability, Financial Mathematics, Learning Theory
Aman Foujdar
- Research Assistant under Prof. Juneja at SCDLDS
- Interested in Machine Learning, Deep RL, GenAI, Financial Markets
- Mathematics & Computer Science at Ashoka University. President of the Ashoka QuantFin Club (AQFC).