## Financial instruments and pricing

(winter semester 2023/24)

### 30h lectures + 30h tutorials

6 ECTS points

#### Basic idea

Provide students with background in science the basic knowledge of financial instruments and mathematical tools used in finance and pricing of financial instruments.

#### Prerequisites

Proven mathematical skills in analysis, algebra and probability calculus, e.g. candidates should finish Mathematical Methods in Physics or a similar course. Tutorials (problems solving) will require basic programming skills in e.g. Mathematica/Matlab/Maple/Python/…

#### Agenda

- Basics of financial mathematics
- Time value of money (cash flows, compounding, discounting, …)
- Effective interest rate (nominal rate, inflation, real rate, simple interest, compound interest, continuous rate, annuities, day count conventions, IRR …)
- Time structure of interest rates (zero coupon rate, bootstrapping, yield curve, …)

- Definitions and examples of basic financial instruments
- Basic spot instruments: equities and commodities (shares, commodities, stock exchange indices, …), fixed income and foreign exchange (deposits, loans, mortgages, bonds, T-bills, zero-coupon bonds, floating/indexed bonds, LIBOR curves, currencies,…), credit risk (corporate/municipal bonds, rating, CDOs, CDS, …)
- Derivative instruments (forwards, futures, FRA, swaps, IRS, CIRS, options, European/American vanilla options, cap, floor, collar, swaption, examples of exotic options, e.g. Bermuda, Asian, lookback, barrier, binary, compound, basket, rainbow, quanto options, examples of structured products….)
- Financial markets (money/capital markets, primary/secondary markets, OTC markets vs. regulated markets, examples of main exchanges, basic trading and settlement rules, …)
- Basic pricing models
- Calculations related to deposits and loans
- Bond pricing (clean/dirty price, accrued interest, YTM, duration, convexity, pricing using zero-coupon yield curves, …)
- Pricing of basic derivative instrument (forwards, forward-spot parity, swaps, hedging and arbitrage-free pricing idea, bands for option prices, put-call parity, …)

- Option pricing
- Stochastic calculus (stochastic processes, Binomial process, Wiener process, martingales, Ito’s integral, Ito’s lemma, Randon-Nikodym derivative, Girsanov theorem, martingale representation theorem, Feynman-Kac formula)
- Binomial model (derivation for European/American options using hedging/arbitrage-free argument, concept of risk neutral pricing, …)
- Black-Scholes model (geometric Wiener process, derivation of Black-Scholes PDE using hedging/arbitrage-free argument, Black-Scholes formula for European options, relations to Binomial model, …)
- Monte-Carlo pricing (concept of risk neutral pricing, examples for exotic options, …)
- Discussion of option hedging strategies (delta-hedging, implied volatility, Greeks, hedging option portfolios, tests of hedging strategies using Monte-Carlo techniques, …)

## Risk management

(summer semester 2023/24)

### 30h lectures + 30h tutorials

6 ECTS points

#### Basic idea

Provide students with background in science the basic knowledge of risk measures and risk management techniques, especially in the context of market risk.

#### Prerequisites

Proven mathematical skills in analysis, algebra and probability calculus, e.g. candidates should finish Mathematical Methods in Physics or a similar course. Tutorials (problems solving) will require basic programming skills in e.g. Mathematica/Matlab/Maple/Python/… The students should preferably finish "Financial instruments and pricing" lecture first.

#### Agenda

- General introduction to risk management
- Basic taxonomy of risk
- Importance of risk management in finance and banking
- Main regulatory frameworks: BASEL, FRTB, DFA, …
- Non-modellable risk factors, liquidity risk
- Theory vs. reality

- Reminder / introduction to the probability theory
- Probability distributions (PMF, PDF, CDF, examples of most important distributions …)
- Median, mode, quantiles, …
- Moments, generating function, …
- Central limit theorem, normal distribution
- Extreme value statistics (Gumbel, Frechet, Weibull distributions)
- Multivariate distributions

- Statistical analysis and statistical inference
- Regression analysis (least-squares estimation, assumptions, properties, implications, interpretation, …)
- Basics of hypothesis testing
- Diagnostic checking (non-normality, heteroscedasticity, autocorrelation, …)

- Basic market risk measures
- Volatility (historical, implied, …)
- VaR (historical, parametric, Monte-Carlo, relations to Volatility, relations to Extreme Value Statistics)
- Marginal VaR, Incremental VaR, Expected shortfall, ….

- Introduction to credit risk
- Probability of Default (PD), Loss Given Default (LGD), Exposure at Default (EAD)
- X-Value Adjustment (CVA, DVA, FVA)
- Rating based models
- Asset based models (e.g. Merton’s model)
- Portfolio models
- Expected vs. unexpected loss
- Modelling correlated defaults

- Classical risk-return models
- Idea of portfolio selection and diversification
- Markovitz model (expected return and risk, role of correlations, optimal portfolio, optimal barrier, analytical and numerical solutions, inclusion of risk-free assets)
- CAPM model (CML, SML, Beta, risk premium, systematic and specific risk, diversification)
- Efficiency measures (Alpha, Beta, Sharp ratio, Jensen ratio, Treynor ratio, …)

- Introduction to financial time series modelling
- Introduction to stochastic processes (definitions, stationarity, unconditional vs. conditional distributions, heteroscedasticity, …)
- AR, MA, ARMA, ARCH, GARCH models, …
- Multivariate time series

- Correlation matrix
- Problems with naïve approaches, spurious correlations
- Principal component analysis
- Random matrices (an introduction, eigenvalues spectrum, Wigner semi-circle…)
- Eigenvalues spectrum for correlation matrices (Wishart ensemble, testing of real correlations, …)
- Non-Gaussian correlated variables

## Numerical methods in financial physics

(winter semester 2023/24)

### 30h lectures + 30h tutorials

6 ECTS points

#### Basic idea

Provide students with background in science the basic knowledge of numerical methods used in finance.

#### Prerequisites

Proven mathematical skills in analysis, algebra and probability calculus, e.g. candidates should finish Mathematical Methods in Physics or a similar course. Tutorials (problems solving) will require basic programming skills in Python. The students should preferably finish "Financial instruments and pricing" and "Risk management" lectures first.

#### Agenda

- Python basics
- Basic numerical calculations, sources of errors, numerical complexity
- Numerical differentiation, differentiation matrix
- Exact methods for solving linear systems of algebraic equations
- Iterative methods for solving linear systems of algebraic equations
- Solving nonlinear equations with one variable
- Solving nonlinear systems with many variables
- Minimization (optimization)
- Methods for solving ordinary differential equations
- Pseudorandom number generators
- Time sequence analysis
- Monte Carlo algorithms (stochastic differential equations, risk measures)
- Solving partial differential equations (diffusion, Black-Scholes)
- Spectral methods
- Data cleansing