Cambridge Mathematical Tripos Supervisions

I supervised third year undergraduates in the Part II courses ‘Stochastic Financial Models’ and ‘Applied Probability’ during Michaelmas 2024 and Lent 2025 respectively. I had the responsibility of discussing students’ submissions to problem sets (can be found here and here ) and more general questions regarding the material.

Stochastic Financial Models Michaelmas 2024 - Indicative syllabus

• Standing Assumptions and Notation: Assumptions about financial markets and asset prices; definitions of market setup.
• One-Period Model: Investor wealth and random asset prices; mean-variance analysis and portfolio optimization.
• Market Portfolio and Mutual Fund Theorem: Market portfolio definition; efficient frontier formulation.
• Capital Asset Pricing Model (CAPM): Assumptions and derivation; risk-return relationship.
• Expected Utility Hypothesis: Utility functions, preferences, and risk aversion measures.
• State Price Densities: Definition, derivation, and relationship to utility maximization.
• Risk Neutral Measures: Concept, applications in pricing, and equivalent measures.
• Arbitrage and Fundamental Theorem of Asset Pricing: Arbitrage definition and implications; fundamental theorem proof.
• Utility Maximization in Binomial Models: Optimal portfolios and contingent claim pricing.
• Multi-Period Models: Filtrations, adapted processes, and conditional expectations.
• Stopping Times and Optional Stopping Theorem: Definitions, applications, and proofs.
• Martingale Theory: Martingale transforms, examples, and properties.
• Pricing and Hedging of Derivatives: Attainable claims, no-arbitrage prices, and examples.
• Advanced Topics: Filtrations, sigma-algebras, and conditional probability.

Applied Probability Lent 2025 - Indicative syllabus

• Basic Aspects of Continuous Time Markov Chains: Markov Property, Regular Jump Chain, Holding Times, Poisson Process, Birth Process, Construction of Continuous Time Markov Chains, Kolmogorov’s Forward and Backward Equations, Non-minimal Chains.
• Qualitative Properties of Continuous Time Markov Chains: Class Structure, Hitting Times, Recurrence and Transience, Invariant Distributions, Convergence to Equilibrium, Reversibility, Ergodic Theorem.
• Queueing Theory: Introduction, M/M/1 Queue, M/M/∞ Queue, Burke’s Theorem, Queues in Tandem, Jackson Networks, Non-Markov Queues (M/G/1 Queue).
• Renewal Processes: Introduction, Elementary Renewal Theorem, Size Biased Picking, Equilibrium Theory of Renewal Processes, Renewal-Reward Processes, Example (Alternating Renewal Process, Busy Periods of M/G/1 Queue), Little’s Formula, G/G/1 Queue.
• Spatial Poisson pProcesses: Definition and superposition, Conditioning, Renyi's Theorem.