• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Quantitative Finance

2021/2022
Academic Year
ENG
Instruction in English
6
ECTS credits
Delivered at:
School of Finance
Course type:
Elective course
When:
2 year, 1, 2 module

Instructor


Lapshin, Victor A.

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. 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).
Learning Objectives

Learning Objectives

  • Understand Monte-Carlo approach and acquire practical experience in programming Monte Carlo simulations for pricing common derivatives and risk estimation.
  • Understand PDE-based approaches to pricing derivatives and acquire practical experience in coding the appropriate algorithms.
  • 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.
  • Understand numerical methods for assessing Monte-Carlo estimate sensitivities and acquire practical experience in using them for risk management purposes.
Expected Learning Outcomes

Expected Learning Outcomes

  • 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.
  • 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.
  • Know the difference between random and pseudorandom numbers. Implement a simple pseudorandom number generator and test its quality.
  • 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.
  • Price stock options via trees. Price interest rate derivatives via trinomial trees. Determine tree parameters from observed 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 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.
  • 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

Course Contents

  • Quantitative Finance - Topic 2. Advanced Monte-Carlo Methods
  • Quantitative Finance - Topic 1. Basics of Monte-Carlo Pricing for Derivatives.
  • Quantitative Finance - Topic 3. Numerical Solutions of the Black-Scholes-Merton Partial Differential Equation.
Assessment Elements

Assessment Elements

  • non-blocking Home Assignment 1: Basic Monte-Carlo
    Basic Monte-Carlo
  • non-blocking Home Assignment 2: Sampling from various distributions
    Sampling from various distributions
  • non-blocking Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing
    Brownian Motion and Basic Monte-Carlo Pricing
  • non-blocking Home Assignment 4: Interest Rates
    Interest rates
  • non-blocking Home Assignment 5: Variance reduction techniques
    Variance Reduction
  • non-blocking Home Assignment 6: Sensitivity analysis
    Sensitivity Analysis
  • non-blocking Home Assignment 8: Pricing American options
    American options
  • non-blocking Home Assignment 7: Monte Carlo in risk management
    Risk Management
  • non-blocking Home assignment 10: Bayesian methods
    Bayesian methods
  • non-blocking Home Assignment 11: Tree-based pricing
    Trees
  • non-blocking Home Assignment 9: Pricing via finite differences
    Finite Differences
  • non-blocking Home Assignment 10: Pricing via finite elements
    Finite Elements
Interim Assessment

Interim Assessment

  • 2021/2022 1st module
    0.166 * Home Assignment 5: Variance reduction techniques + 0.167 * Home Assignment 1: Basic Monte-Carlo + 0.167 * Home Assignment 2: Sampling from various distributions + 0.166 * Home Assignment 4: Interest Rates + 0.167 * Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing
  • 2021/2022 2nd module
    0.1 * Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing + 0.1 * Home Assignment 10: Pricing via finite elements + 0.1 * Home Assignment 5: Variance reduction techniques + 0.1 * Home Assignment 4: Interest Rates + 0.1 * Home Assignment 9: Pricing via finite differences + 0.1 * Home Assignment 8: Pricing American options + 0.1 * Home Assignment 2: Sampling from various distributions + 0.1 * Home Assignment 1: Basic Monte-Carlo + 0.1 * Home Assignment 6: Sensitivity analysis + 0.1 * Home Assignment 7: Monte Carlo in risk management
Bibliography

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
  • 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