Бакалавриат
2025/2026
Финансовая математика
Статус:
Курс обязательный (Прикладной анализ данных)
Где читается:
Факультет компьютерных наук
Когда читается:
3-й курс, 1, 2 модуль
Охват аудитории:
для своего кампуса
Преподаватели:
Башминова Дарья Александровна
Язык:
английский
Контактные часы:
56
Course Syllabus
Abstract
Methods of financial mathematics have found wide application in all areas of economic and financial activities of enterprises, banks and other financial organizations. A deep understanding of foundations of mathematical modeling of finances is necessary in order to offer new financial products, control financial risks and provide realistic assessments of market conditions. The roots of financial mathematics came from Bachelier (early 20th century) and Samuelson (mid-20th century). However, the true development of modern financial mathematics, based on the profound results of probability theory and modelling market dynamics, takes its origin from the famous work of Black-Scholes (1973) on estimating the fair price of European options. The course will trace the main stages in the development of financial mathematics, from classical stationary models (Markowitz’s portfolios) to the most modern problematics (limit order book).
Learning Objectives
- This course provides a rigorous, practice-oriented introduction to financial derivatives and the mathematical principles behind their pricing and risk management. Students will learn to model asset dynamics with the Wiener process, apply no-arbitrage and risk-neutral valuation, and use binomial and Black–Scholes frameworks to price and hedge standard and path-dependent options
Expected Learning Outcomes
- Define major derivative types; draw/interpret payoff diagrams; explain hedging vs. speculation; apply no-arbitrage and put–call parity to detect simple arbitrage.
- State core properties of Brownian motion; compute basic moments/variations; relate martingale intuition to asset-price modeling.
- Compute BS prices for calls/puts; calculate and interpret key Greeks; design simple delta/vega hedges.
- Evaluate hedging error under discrete rebalancing; apply Leland’s volatility adjustment; discuss RL-style improvements at a conceptual level.
- Formulate pricing as a (free-)boundary PDE; explain early-exercise logic; outline numerical treatment for American options.
- Implement explicit/implicit/Crank–Nicolson schemes; set BC/IC correctly; assess stability/consistency and use basic iterative solvers.
- Choose suitable methods (tree/MC/PDE) for path-dependent or barrier payoffs; compute benchmark prices and interpret features. Compute implied volatility; interpret smile/surface; perform a basic Heston-style calibration and validate fit qualitatively.
- Simulate paths and estimate option values; construct confidence intervals; use simple variance-reduction.
Course Contents
- Introduction to derivatives & payoffs
- GBM and Itô’s lemma; deriving Black–Schole
- Black–Scholes pricing and Greeks
- Discrete hedging, transaction costs
- PDE formulation & American options
- Finite-difference methods
- Monte Carlo valuation & variance reduction
- Path-dependent and barrier options. Volatility smile
Assessment Elements
- ExamExam. A written form of control will be held at the session week. It will conduct theoretical and practical questions. In cases of suspected academic dishonesty, lecturer may require the student to undergo an oral defense of the exam. The final grade will then be determined based on both the written and oral components.
- MidtermA written midterm exam will be held at the first week of second module. In cases of suspected academic dishonesty, lecturer may require the student to undergo an oral defense of the exam. The final grade will then be determined based on both the written and oral components.
- HW
Bibliography
Recommended Core Bibliography
- Bernt Øksendal. (2010). Stochastic Differential Equations : An Introduction with Applications (Vol. 6th ed. 2003). Springer.
- Escobar, M., Ferrando, S., & Rubtsov, A. (2018). Dynamic derivative strategies with stochastic interest rates and model uncertainty. Journal of Economic Dynamics and Control, (C), 49. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.a.eee.dyncon.v86y2018icp49.71
- Financial mathematics : theory and problems for multi-period models, Pascucci, A., 2012
- Financial mathematics, Brusov, P., 2024
- Ikeda, N., & Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North Holland.
- Introduction to stochastic calculus applied to finance, Lamberton, D. M., 2008
- Основы стохастической финансовой математики. Т. 2: Теория, Ширяев, А. Н., 2016
Recommended Additional Bibliography
- Pierre Vernimmen, Pascal Quiry, Maurizio Dallocchio, Yann Le Fur, & Antonio Salvi. (2018). Corporate Finance : Theory and Practice. Post-Print. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.p.hal.journl.hal.02298273
- Основы стохастической финансовой математики. Т. 1: Факты. Модели, Ширяев, А. Н., 2016