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Regular version of the site

Quantitative Finance

2019/2020
Academic Year
ENG
Instruction in English
5
ECTS credits
Delivered at:
School of Finance
Course type:
Elective course
When:
2 year, 1, 2 module

Instructor

Программа дисциплины

Аннотация

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. Most of the methods considered will be Monte-Carlo methods, which is one of the main modeling tools in derivative pricing. Even though the course is focused on pricing financial instruments, the skills acquired may also be useful in other applications of computer simulation. The theoretical part of the course will assume that the student is knowledgeable in probability theory, calculus and basic financial instruments (stocks, bonds, futures and options). Taking the ‘Derivatives II’ course prior to this one is recommended, but not required. The computer part of the course will be using the Python language or the Matlab software (at students’ choice) and will assume either basic programming knowledge (a high-school-level course will suffice: you need to know the notions of variables, loops and functions) or the readiness to acquire it. 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).
Цель освоения дисциплины

Цель освоения дисциплины

  • 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 Bayesian approach to model parameter estimation and acquire practical experience in Bayesian inference using specialized software packages.
Результаты освоения дисциплины

Результаты освоения дисциплины

  • 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.
  • Implement variance reduction techniques for derivatives pricing: antithetic variables, control variates, stratified sampling, importance sampling, quasi Monte-Carlo.
  • 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.
  • 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.
  • 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.
  • Understand the Bayesian approach to parameter estimation. Understand basic Markov Chain Monte-Carlo concepts: Gibbs sampling, Metropolis-Hastings, Metropolis-within-Gibbs. Estimate the posterior distribution of the parameters of a simple model and perform Bayesian data augmentation.
  • Price stock options via trees. Price interest rate derivatives via trinomial trees. Determine tree parameters from observed instrument prices.
  • 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.
  • 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.
  • Implement a basic finite elements approach to solve the Black-Scholes-Merton PDE.
Содержание учебной дисциплины

Содержание учебной дисциплины

  • Quantitative Finance - Topic 1. Basics of Monte-Carlo Pricing for Derivatives.
    The general idea of Monte-Carlo methods. Calculating definite integrals via Monte-Carlo. The notion of randomness. Random and pseudorandom numbers. Pseudorandom number generators. Sampling from various distributions. Simulating Brownian motion. Basic Monte-Carlo derivative pricing.
  • Quantitative Finance - Topic 2. Advanced Monte-Carlo Methods
    Pricing path-dependent instruments. Incorporating stochastic interest rates. Pricing interest rate based derivatives. Fitting model parameters to instrument prices. Variance reduction techniques. Sensitivity analysis. Pricing American options via Monte-Carlo and optimal stopping Risk management applications.
  • Quantitative Finance - Topic 3. Numerical Solutions of the Black-Scholes-Merton Partial Differential Equation.
    Tree-based derivatives pricing. Pricing via finite differences. Pricing via finite elements.
Элементы контроля

Элементы контроля

  • неблокирующий Created with Sketch. Home Assignment 1: Basic Monte-Carlo
    Basic Monte-Carlo
  • неблокирующий Created with Sketch. Home Assignment 2: Pseudorandom number generators
    Pseudorandom number generators
  • неблокирующий Created with Sketch. Home Assignment 3: Sampling from various distributions
    Sampling from various distributions
  • неблокирующий Created with Sketch. Home Assignment 4: Brownian Motion and Basic Monte-Carlo Pricing
    Brownian Motion and Basic Monte-Carlo Pricing
  • неблокирующий Created with Sketch. Home Assignment 5: Interest Rates
    Interest rates
  • неблокирующий Created with Sketch. Home Assignment 6: Variance reduction techniques
    Variance Reduction
  • неблокирующий Created with Sketch. Home Assignment 7: Sensitivity analysis
    Sensitivity Analysis
  • неблокирующий Created with Sketch. Home Assignment 8: Pricing American options
    American options
  • неблокирующий Created with Sketch. Home Assignment 9: Monte Carlo in risk management
    Risk Management
  • неблокирующий Created with Sketch. Home assignment 10: Bayesian methods
    Bayesian methods
  • неблокирующий Created with Sketch. Home Assignment 11: Tree-based pricing
    Trees
  • неблокирующий Created with Sketch. Home Assignment 12: Pricing via finite differences
    Finite Differences
  • неблокирующий Created with Sketch. Home Assignment 13: Pricing via finite elements
    Finite Elements
Промежуточная аттестация

Промежуточная аттестация

  • Промежуточная аттестация (1 модуль)
    0.167 * Home Assignment 1: Basic Monte-Carlo + 0.167 * Home Assignment 2: Pseudorandom number generators + 0.167 * Home Assignment 3: Sampling from various distributions + 0.167 * Home Assignment 4: Brownian Motion and Basic Monte-Carlo Pricing + 0.166 * Home Assignment 5: Interest Rates + 0.166 * Home Assignment 6: Variance reduction techniques
  • Промежуточная аттестация (2 модуль)
    0.077 * Home assignment 10: Bayesian methods + 0.077 * Home Assignment 11: Tree-based pricing + 0.077 * Home Assignment 12: Pricing via finite differences + 0.076 * Home Assignment 13: Pricing via finite elements + 0.077 * Home Assignment 1: Basic Monte-Carlo + 0.077 * Home Assignment 2: Pseudorandom number generators + 0.077 * Home Assignment 3: Sampling from various distributions + 0.077 * Home Assignment 4: Brownian Motion and Basic Monte-Carlo Pricing + 0.077 * Home Assignment 5: Interest Rates + 0.077 * Home Assignment 6: Variance reduction techniques + 0.077 * Home Assignment 7: Sensitivity analysis + 0.077 * Home Assignment 8: Pricing American options + 0.077 * Home Assignment 9: Monte Carlo in risk management
Список литературы

Список литературы

Рекомендуемая основная литература

  • 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
  • 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

Рекомендуемая дополнительная литература

  • Искусство программирования. Т.2: Получисленные алгоритмы, Кнут Д. Э., Козаченко Ю. В., 2012