Stochastic Calculus and Derivatives

Introduction

Please note that this is a preliminary course description. The final version will be published in June 2026.

This course provides a rigorous yet practical introduction to derivative securities and the stochastic calculus tools used to price and hedge them. Students will explore key concepts such as no-arbitrage pricing, the binomial and Black-Scholes models, and the use of stochastic differential equations in financial modeling. Topics include delta hedging, exotic options, credit risk, and numerical methods for option valuation. Emphasis is placed on both theoretical foundations and real-world applications in modern financial markets.

Course content

1. Foundations

• Review of probability theory and deterministic calculus
• Discrete-time financial models
• Martingale processes
• Wiener processes (Brownian motion)

2. Stochastic Calculus and Continuous-Time Finance

• Stochastic integrals and stochastic differential equations (SDEs)
• Itô’s lemma
• Major models of SDEs
• Analytical and simulation methods for solving SDEs
• Girsanov’s theorem
• Equivalent martingale measure
• Fundamental theorems of asset pricing
• Models with jumps

3. Derivatives Pricing Theory

• Overview of options and derivatives markets
• Model-free no-arbitrage bounds
• Trading strategies with options
• Binomial tree models
• Black–Scholes–Merton model
• Partial differential equations in option pricing
• Option Greeks and risk management

4. Numerical Methods and Applications

• Numerical techniques for derivatives pricing
• Exotic options
• Volatility smiles and model calibration
• Real options analysis
• Credit risk modeling
• International derivatives