Notes of past lectures

First lecture, September 19th

Speakers: 

Ilya Tolstukhin -  Introductory lecture: main problems and approaches to financial mathematics. Historical interlude. Overview of mathematics we will study and use.

Nikita Naumov -  Introduction to Risk Management: Risk Aversion, Risk Premium and Utility Functions. 

September 26th

Speakers: 

Ilya Tolstukhin - Review of probability theory. Stochastic processes and filtrations, basic definitions. Stochastic basis satisfying usual conditions. Indistinguishability and modifications with examples. Adapted, predictable, optional, progressively measurable, jointly measurable processes, relationships between these classes of processes. Additionally, we discussed motivations for considering Brownian motion and briefly mentioned Kolmogorov extension theorem and Kolmogorov continuity theorem in the context of the existence of Brownian motion. We solved one exercise on Brownian motion: find the average time W_t is greater than t^a when a=1; for which values of a does this average time exist? 

Artem Belsky - Characteristic function of a measure on (R^n, B(R^n)). Consistency conditions for measures on Euclidean spaces in terms of characteristic functions. Criterion for the existence of a process with independent increments. Existence of Brownian motion, second definition of a Wiener process through Gaussian processes. Haar and Schauder functions, construction of a continuous Wiener process on [0,1], and then on [0,∞). Non-differentiability of almost all trajectories of the Wiener process at any point. Markov property of the Wiener process. Markov moments, Début theorem. Strong Markov property of the Wiener process. Maximum of the Wiener process and its density function. Reflection principle. 

Speakers: 

Nikita Naumov -  Introduction to martingales.

Ilya Tolstukhin -  Introduction to stochastic integration. Discrete case: martingale transform, calculation of PnL as a motivation. Simple processes. Hilbert space of square-integrable predictable processes, Hilbert space of continuous L^2-bounded martingales. Ito isometry. Definition of Ito integral with respect to Brownian motion. Ito processes. Definition of stochastic integral with respect to Ito process. Ito formula. Examples and applications, geometric Brownian motion.

Speakers: 

Ilya Tolstukhin -  Quadratic variation, calculation for Ito processes. Ito formula, idea of the proof, multidimensional version. Stochastic integration by parts formula. Stochastic differential equations. Strong and weak solutions. Stochastic exponential. Ornstein-Uhlenbeck process. Linear SDEs. Existence and uniqueness conditions for SDEs.

Speakers: 

Ilya Bykov -  Martingale Representation Theorem and its relation to Fourier Analysis; Girsanov Theorem and its application to The First Fundamental Theorem of Asset Pricing; briefly reviewed Abstract Bayes Theorem and Levy’s characterisation of Brownian motion.

Speakers: 

Lisa Brill -  Contingent claims. Futures and European options. Payoff functions. Definition of arbitrage. Derivation of Black-Scholes PDE from no-arbitrage principle. Second fundamental theorem of asset pricing and complete markets. Numéraire. Price of a European contract in a complete market as discounted expected value under risk-neutral measure . Derivation of Black-Scholes formula. Replicating portfolios and hedging. 

Speakers: 

Vsevolod Tarsamaev -  Markov processes, the infinitesimal generator of Markov process. Kolmogorov backward equation and its generalizations. Feynman-Kac formula. Time-homogeneous processes, Markov semigroup. Kolmogorov forward equation. Applications of Brownian motion to the Dirichlet problem. 

References:

A.Aggazi, J.Mattingly. Introduction to Stochastic Calculus. Lecture notes. Math454, Duke University. Chapter 7

Speakers: 

Artur Sidorenko-  Arbitrage Theory, Fundamental Theorems, Incomplete Markets

Kirill Sokolov-  Introduction to Optimal Transport with Applications in Finance 

Speakers: 

Mauro Mariani-  Portfolio Hedging and Martingale Representation

Speakers: 

Vsevolod Tarsamaev -  Pricing in a Markovian environment