Master 2021/2022

Applied Quantitative Finance

Type: Elective course (Financial Economics)

Area of studies: Economics

Delivered by: International College of Economics and Finance

Where: International College of Economics and Finance

When: 2 year, 3 module

Mode of studies: offline

Open to: students of one campus

Instructors: Victor A Lapshin

Master’s programme: Financial Economics

Language: English

ECTS credits: 3

Contact hours: 40

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.

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.
• 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 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

• Home Assignment 1: Basic Monte-Carlo
  Basic Monte-Carlo
• Home Assignment 2: Pseudorandom number generators
  Pseudorandom number generators
• Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing
  Brownian Motion and Basic Monte-Carlo Pricing
• Home Assignment 4: Asian Options and Interest Rates
  Asian Options and Interest Rates
• Home Assignment 5: Variance reduction techniques
  Variance Reduction
• Home Assignment 6: Sensitivity analysis
  Sensitivity Analysis
• Home Assignment 8: Pricing American options
  American options
• Home Assignment 7: Monte Carlo in risk management
  Risk Management
• Home Assignment 9: Pricing via finite differences
  Finite Differences
• Home Assignment 10: Pricing via finite elements
  Finite Elements
• Final Examination

Interim Assessment

• 2021/2022 3rd module
  0.09 * Home Assignment 10: Pricing via finite elements + 0.09 * Home Assignment 7: Monte Carlo in risk management + 0.09 * Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing + 0.09 * Home Assignment 8: Pricing American options + 0.09 * Home Assignment 4: Asian Options and Interest Rates + 0.09 * Home Assignment 2: Pseudorandom number generators + 0.1 * Final Examination + 0.09 * Home Assignment 1: Basic Monte-Carlo + 0.09 * Home Assignment 5: Variance reduction techniques + 0.09 * Home Assignment 9: Pricing via finite differences + 0.09 * Home Assignment 6: Sensitivity analysis

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