# Quantitative Modelling in Finance # Lectures

## Financial instruments and pricing (winter semester 2021/22)

### 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

1. 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, …)
2. 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….)
3. Financial markets (money/capital markets, primary/secondary markets, OTC markets vs. regulated markets, examples of main exchanges, basic trading and settlement rules, …)
4. 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, …)
5. 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 2021/22)

### 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

1. 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
2. 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
3. Statistical analysis and statistical inference
• Regression analysis (least-squares estimation, assumptions, properties, implications, interpretation, …)
• Basics of hypothesis testing
• Diagnostic checking (non-normality, heteroscedasticity, autocorrelation, …)
4. 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, ….
5. 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
6. 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, …)
7. 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
8. 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 2021/22)

### 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

1. Python basics
2. Basic numerical calculations, sources of errors, numerical complexity
3. Numerical differentiation, differentiation matrix
4. Exact methods for solving linear systems of algebraic equations
5. Iterative methods for solving linear systems of algebraic equations
6. Solving nonlinear equations with one variable
7. Solving nonlinear systems with many variables
8. Minimization (optimization)
9. Methods for solving ordinary differential equations
10. Pseudorandom number generators
11. Time sequence analysis
12. Monte Carlo algorithms (stochastic differential equations, risk measures)
13. Solving partial differential equations (diffusion, Black-Scholes)
14. Spectral methods
15. Data cleansing