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# Quantitative Finance

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

### 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

• 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 numerical methods for assessing Monte-Carlo estimate sensitivities and acquire practical experience in using them for risk management purposes.

#### Expected Learning Outcomes

• Implement a basic Monte-Carlo simulation to solve a deterministic problem and assess its convergence.
• Know the difference between random and pseudorandom numbers. Implement a simple pseudorandom number generator and test its quality.
• Implement sampling procedures for various distributions given a uniform random number generator.
• 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.
• 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 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.

#### Course Contents

• 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 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 3. Numerical Solutions of the Black-Scholes-Merton Partial Differential Equation.
Tree-based derivatives pricing. Pricing via finite differences. Pricing via finite elements.

#### Assessment Elements

• Home Assignment 1: Basic Monte-Carlo
Basic Monte-Carlo
• Home Assignment 2: Sampling from various distributions
Sampling from various distributions
• Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing
Brownian Motion and Basic Monte-Carlo Pricing
• Home Assignment 4: Interest Rates
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 10: Bayesian methods
Bayesian methods
• Home Assignment 11: Tree-based pricing
Trees
• Home Assignment 9: Pricing via finite differences
Finite Differences
• Home Assignment 10: Pricing via finite elements
Finite Elements

#### Interim Assessment

• Interim assessment (1 module)
0.167 * Home Assignment 1: Basic Monte-Carlo + 0.167 * Home Assignment 2: Pseudorandom number generators + 0.167 * Home Assignment 2: Sampling from various distributions + 0.167 * Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing + 0.166 * Home Assignment 4: Interest Rates + 0.166 * Home Assignment 5: Variance reduction techniques
• Interim assessment (2 module)
0.1 * Home Assignment 10: Pricing via finite elements + 0.1 * Home Assignment 1: Basic Monte-Carlo + 0.1 * Home Assignment 2: Sampling from various distributions + 0.1 * Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing + 0.1 * Home Assignment 4: Interest Rates + 0.1 * Home Assignment 5: Variance reduction techniques + 0.1 * Home Assignment 6: Sensitivity analysis + 0.1 * Home Assignment 7: Monte Carlo in risk management + 0.1 * Home Assignment 8: Pricing American options + 0.1 * Home Assignment 9: Pricing via finite differences

#### 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