Бакалавриат
2021/2022
Количественные финансы
Лучший по критерию «Новизна полученных знаний»
Статус:
Курс по выбору
Направление:
01.03.02. Прикладная математика и информатика
Где читается:
Факультет компьютерных наук
Когда читается:
4-й курс, 1, 2 модуль
Формат изучения:
без онлайн-курса
Охват аудитории:
для своего кампуса
Преподаватели:
Лапшин Виктор Александрович
Язык:
английский
Кредиты:
6
Контактные часы:
60
Course Syllabus
Abstract
Modern banks, investment companies and other financial institutions can’t be thought of without quantitative analysis. The people involved, quantitative analysts (quants), are often considered the ‘elite’ of financial analysts. This course provides an introduction to the exciting world of pricing derivative instruments via solving stochastic equations and other numerical procedures via a computer. You will learn how to find the price of a derivative instrument numerically, using a computer, and why modern banks buy supercomputers. You will learn to use mainstream modeling tools in derivative pricing, namely Monte-Carlo methods and numerical methods for partial differential equations (PDEs). Even though the course is focused on pricing financial instruments, the skills acquired may also be useful in other applications of computer simulation. Students are assumed to be knowledgeable in probability theory, calculus and basic financial instruments (stocks, bonds, futures and options). Familiarity with the basics of stochastic analysis is recommended, but not required. The computer part of the course will be using the Python language and will assume basic programming knowledge: variables, loops and functions. Familiarity with specialized packages like numpy and matplotlib is recommended, but not required. This is not a ‘push-this-button-to-get-the-answer’ course. Be ready to spend several hours in front of a computer each week (more if you are only learning programming at the same time).
Learning Objectives
- Understand Monte-Carlo approach and acquire practical experience in programming Monte Carlo simulations for pricing common derivatives and risk estimation.
- Understand tree-based and PDE-based approaches to pricing derivatives and acquire practical experience in coding the appropriate algorithms.
- Understand basic finanical instruments and their theoretical pricing mechanisms.
Expected Learning Outcomes
- Access stock and bond trading data from python code.
- Calculate sensitivities (delta, gamma, theta, rho and others) of prices obtained via Monte-Carlo via fixing the random seed, pathwise derivatives and the likelihood ratio method.
- Derive basic financial relations using no arbitrage arguments. Explain margin requirements for derivatives trading. Price basic derivative instruments using the binomial model and the Black-Scholes formula. Calculate sensitivities using the binomial model and the Black-Scholes formula.
- Implement a basic finite elements approach to solve the Black-Scholes-Merton PDE.
- Implement a basic Monte-Carlo simulation to solve a deterministic problem and assess its convergence.
- Implement sampling procedures for various distributions given a uniform random number generator.
- Implement variance reduction techniques for derivatives pricing: antithetic variables, control variates, stratified sampling, importance sampling, quasi Monte-Carlo.
- Perform basic financial calculations: stock returns and distributions, form bond cashflows, price bonds via discounted cashflows, calculate bond duration and yield to maturity, fit a term structure model to the data.
- Perform explicit discretization of the Black-Scholes-Merton PDE and understand the arising stability issues. Understand the implicit and Crank-Nicholson schemes and their drawbacks. Implement a numerical scheme to solve the Black-Scholes-Metron PDE. Discuss pricing American and barrier options via PDE's.
- Price American options via Monte-Carlo by solving the optimal stopping problem. Understand the dynamic programming approach and the execution boundary. Implement discrete dynamic programming, Longstaff-Schwartz method and some other numerical schemes to estimate the execution boundary.
- Price path-dependent options (e.g. Asian) and options with multiple underlying assets using Monte-Carlo. Implement stochastic interest rate models. Price basic interest rate derivatives using a stochastic model via Monte-Carlo. Fit the parameters of a stochastic interest rate model to observable instrument prices.
- Simulate trajectories of Brownian motion and geometric Brownian motion via exact solutions and Euler scheme. Simulate trajectories of a Brownian bridge. Implement an Euler scheme for a given SDE. Understand the difference between strong and weak convergence of numerical solutions to SDE's and numerically assess the corresponding orders of convergence. Price a simple stock option using Monte-Carlo.
- Understand pricing derivatives via solving the Black-Scholes-Merton PDE. Understand using numerical schemes to solve the Black-Scholes-Merton PDE. Understand boundary conditions for various derivative instruments. Reduce the Black-Scholes-Merton PDE to the heat equation.
- Understand the difference between using Monte-Carlo for pricing and risk management purposes. Estimate Value-at-Risk and Expected Shortfall using full revaluation, delta and delta-gamma approximations. Use variance reduction techniques in these calculations. Understand copulas and implement a basic credit risk model via Monte-Carlo.
Course Contents
- Quantitative Finance - Topic 3. Numerical Solutions of the Black-Scholes-Merton Partial Differential Equation.
- Quantfin - Topic 2. Basic derivative instruments - forwards, futures and options.
- Quantitative Finance - Topic 2. Advanced Monte-Carlo Methods
- Quantitative Finance - Topic 1. Basics of Monte-Carlo Pricing for Derivatives.
- QuantFin - Topic 1. Basic financial instruments: stocks and bonds
Assessment Elements
- Home Assignment 1 - Financial Arithmetics
- Home Assignment 2: Basic option pricing - Binomial and Black-Scholes.
- Home Assignment 3: Basic Monte-Carlo and Distribution Sampling
- Home Assignment 4: Pricing financial instruments via Monte-Carlo
- Home Assignment 5: Variance reduction techniquesVariance Reduction
- Home Assignment 6: Sensitivity analysisSensitivity Analysis
- Home Assignment 8: Pricing American optionsAmerican options
- Home Assignment 7: Monte Carlo in risk managementRisk Management
- Home Assignment 9: Pricing via finite differencesFinite Differences
- Home Assignment 10: Pricing via finite elementsFinite Elements
Interim Assessment
- 2021/2022 2nd module0.1 * Home Assignment 6: Sensitivity analysis + 0.1 * Home Assignment 9: Pricing via finite differences + 0.1 * Home Assignment 1 - Financial Arithmetics + 0.1 * Home Assignment 7: Monte Carlo in risk management + 0.1 * Home Assignment 5: Variance reduction techniques + 0.1 * Home Assignment 10: Pricing via finite elements + 0.1 * Home Assignment 3: Basic Monte-Carlo and Distribution Sampling + 0.1 * Home Assignment 2: Basic option pricing - Binomial and Black-Scholes. + 0.1 * Home Assignment 4: Pricing financial instruments via Monte-Carlo + 0.1 * Home Assignment 8: Pricing American options
Bibliography
Recommended Core Bibliography
- Brandimarte, P. (2014). Handbook in Monte Carlo Simulation : Applications in Financial Engineering, Risk Management, and Economics. Hoboken, New Jersey: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=800911
- Hull, J. C. (2017). Options, Futures, and Other Derivatives, Global Edition. [Place of publication not identified]: Pearson. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1538007
- Options, futures, and other derivatives, Hull, J. C., 2009
- Wang, H. (2012). Monte Carlo Simulation with Applications to Finance. [Place of publication not identified]: Chapman and Hall/CRC. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1763376
Recommended Additional Bibliography
- Искусство программирования. Т.2: Получисленные алгоритмы, Кнут, Д. Э., 2012