Applied Quantitative Finance
- 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.
- 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.
- 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 MethodsPricing 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.
- Home Assignment 1: Basic Monte-CarloBasic Monte-Carlo
- Home Assignment 2: Pseudorandom number generatorsPseudorandom number generators
- Home Assignment 3: Brownian Motion and Basic Monte-Carlo PricingBrownian Motion and Basic Monte-Carlo Pricing
- Home Assignment 4: Asian Options and Interest RatesAsian Options and Interest Rates
- Home Assignment 5: Variance reduction techniquesVariance Reduction
- Home Assignment 6: Sensitivity analysisSensitivity Analysis
- Home Assignment 7: Pricing American optionsAmerican options
- Home Assignment 8: Monte Carlo in risk managementRisk Management
- Home Assignment 9: Pricing via finite differencesFinite Differences
- Home Assignment 10: Pricing via finite elementsFinite Elements
- Final Examination
- Interim assessment (3 module)0.09 * Final Examination + 0.091 * Home Assignment 10: Pricing via finite elements + 0.091 * Home Assignment 1: Basic Monte-Carlo + 0.091 * Home Assignment 2: Pseudorandom number generators + 0.091 * Home Assignment 3: Brownian Motion and Basic Monte-Carlo Pricing + 0.091 * Home Assignment 4: Asian Options and Interest Rates + 0.091 * Home Assignment 5: Variance reduction techniques + 0.091 * Home Assignment 6: Sensitivity analysis + 0.091 * Home Assignment 7: Pricing American options + 0.091 * Home Assignment 8: Monte Carlo in risk management + 0.091 * Home Assignment 9: Pricing via finite differences
- 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
- Искусство программирования. Т.2: Получисленные алгоритмы, Кнут, Д. Э., Козаченко, Ю. В., 2012