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Financial Mathematics: Lecture Notes

These lecture notes, still a work in progress, are for a course taught at Shanghai Advanced Institute for Finance, Shanghai Jiao Tong University, for graduate students.

Course Objective

Stochastics as a mathematical field evolved in parallel with the development of the finance industry, starting with insurance, followed by stock markets, derivatives, and more. This lecture serves as an introduction to the mathematical theory underpinning modern finance. The course aims to introduce mathematical concepts in finance through the following topics:

  • One-period financial markets: Financial assets, self-financing strategies, arbitrage, the fundamental theorem of asset pricing, and option pricing.
    From a mathematical perspective, this introduces probability spaces, expectations, pricing measures, and measure changes.

  • Modern risk management and quantification: Value at Risk (V@R), Expected Shortfall (ES), and systemic risk. From a mathematical perspective, this introduces the concept of probability distribution (CDF, PDF, quantile), joint distributions and tail risk.

  • Multi-period financial markets: Concepts of information, the CRR model, pricing and hedging, exotic options, stopping times, and American options.
    This includes mathematical concepts such as filtrations, conditional expectations, martingales, and stopping times.

  • Basics of ruin theory and default pricing.

  • Continuous-time financial markets: Introduction to the Black-Scholes framework.

Concrete Approach

The course combines blackboard lectures with practical applications in Python. Lecture notes will be provided and updated during the course. Simple homework exercises (not graded but corrected and discussed by the TA) will be assigned. Additionally, students will complete two group projects (5-6 members per group), alongside a midterm and final exam.

For further reading, we recommend Shreve1 for an introduction to mathematical finance in discrete time and Föllmer and Schied2 for a more advanced treatment.

References

  1. Steven E. Shreve. Stochastic Calculus for Finance. Volume I of Springer Finance. Springer-Verlag, New York, 2004. ISBN 0-387-40100-8. The binomial asset pricing model. ↩

  2. Hans Föllmer and Alexander Schied. Stochastic Finance. An Introduction in Discrete Time. De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin, New York, 3rd edition, 2011. ↩