Computational methods in finance 1000-135MOF
1. Computational methods for pricing financial instruments: tree algorithms, Monte Carlo methods, solutions to the Black-Scholes PDE.
2. Introduction to Monte Carlo methods. Generation of pseudo-random numbers. Low discrepancy sequences (Halton, Sobol) and Quasi Monte Carlo methods. Uniform generators. Generators from other distributions: inverse transform method, acceptance-rejection method, transformations of random variables.
3. Numerical solutions of SDE of Ito type: Euler and Milstein schemes. Week and strong convergence. Order of convergence. Proof of convergence for the Euler scheme. Brief description of the Milstein scheme convergence.
4. Variance reduction: antithetic variates, control variates, importance sampling. Application of variance reduction: VaR of an investment portfolio.
5. Applications of Monte Carlo methods: Asian options, barrier options – Brownian bridge, computation of Greeks.
6. Black-Scholes PDE and its mathematical properties. Linear elliptic and parabolic operators. Maximum principle. First, second and third boundary problem for parabolic equations. Existence and uniqueness for first boundary problem. Existence of nonnegative solutions. Comparison theorems.
7. Finite difference method for linear parabolic PDE. Explicit, implicit and the Crank-Nicolson schemes. Stability – CFL condition. Convergence proofs. Order of convergence. Oscillation of solutions due to first derivatives in equations, upwind scheme. Solution of the Black-Scholes equation.
8. Computations for Black-Scholes PDE: universal scheme for vanilla options, barrier options – boundary conditions, Asian and lookback options.
9. Brief information on other approaches. American options: free boundary problem and penalty method. Finite element method.
Type of course
Course coordinators
Term 2023L: | Term 2024L: |
Learning outcomes
Knowledge ans skills:
1. apply Monte Carlo methods to compute prices of financial instruments, generate samples from most popular distributions (discrete, uniform, normal, t-Student), generate a sample from a given distribution using uniform samples, generate samples of low discrepancy;
2. solve a simple SDE using Euler or Milstein schemes, understand the concept of scheme order of convergence and draw links between order of convergence and computational error;
3. know variance reduction methods and can apply them in MC computations, use antithetic variates in MC simulations, use Monte Carlo methods to compute VaR, use Monte Carlo methods to price vanilla options and a number of exotic options, and their Greeks;
4. understand solutions of parabolic equations and their properties, understand the concept of well-posedness of BVP for these equations, know basic existence theorems for these equations, use the mainstream finite difference schemes for parabolic equations, understand convergence of these schemes and apply this concept in computations, use finite difference methods to compute prices of vanilla and many exotic options;
5. draw links between pricing of American options and free boundary problems, solve simple examples of such problems.
Competence:
1. understand the role of numerical methods on financial markets,
2. understand the importance of numerical knowledge in practical computations.
Bibliography
1. Y. Achdou, O. Pironneau. Computational Methods for Option Pricing, SIAM 2005.
2. S. Asmussen, P. Glynn. Stochastic Simulation: Algorithms and Analysis, Springer 2007
3. L. C. Evans. Równania Różniczkowe Cząstkowe, PWN, Warszawa 2002.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: