|Building Blocks of Finance
- The random nature of prices: Examination of data, unpredictability, the need for probabilistic models, drift and volatility.
- Probability preliminaries: Review of discrete and continuous random variables, transition density functions, moments and important distributions, the Central Limit Theorem.
- Fokker-Planck and Kolmogorov equations: similarity solutions.
- Products and strategies: examination of different asset classes, derivatives products and common trading strategies.
- Applied Itô calculus: Discrete-time random walks, continuous Wiener processes via rescaling and passing to the limit, quadratic variation, Itô integrals and Itô’s lemma.
- Simulating and manipulating stochastic differential equations.
- Martingale fundamentals: Conditional expectations, change of measure, stochastic processes as a martingale and tools of the trade.
- The binomial model: Up and down moves, delta hedging and self-financing replication, no arbitrage, a pricing model and risk-neutral probabilities.
- Visual Basic for Applications: VBA techniques and tricks for quant finance.
|Risk and Return
- Modern Portfolio Theory: Expected returns, variances and covariances, benefits of diversification, the opportunity set and the efficient frontier, the Sharpe ratio, utility functions and the Black-Litterman Model.
- Capital Asset Pricing Model: Single-index model, beta, diversification, optimal portfolios, the multi-index model.
- Portfolio Optimization: Formulation, implementation and use of calculus to solve constrained optimization.
- Value at risk: Profit and loss for simple portfolios, tails of distributions, Monte Carlo simulations and historical simulations, stress testing and worst-case scenarios.
- Volatility clustering: Concept and evidence.
- Properties of daily asset returns: Average values, standard deviations, departures from the normal distribution, squared returns.
- Properties of high-frequency returns: Five-minute returns contrasted with daily returns, intraday volatility patterns, impact of macroeconomic news, realized variance.
- Volatility models: The ARCH framework, why ARCH models are popular, the GARCH model, ARCH models, asymmetric
- ARCH models and econometric methods
|Equities & Currencies Derivatives
- The Black-Scholes model: A stochastic differential equation for an asset price, the delta-hedged portfolio and self-financing
- replication, no arbitrage, the pricing partial differential equation and simple solutions.
- Martingales: The probabilistic mathematics underlying derivatives theory, Girsanov, change of measure and Feynman-Kac.
- Risk-neutrality: Fair value of an option as an expectation with respect to a risk-neutral density function.
- Early exercise: American options, elimination of arbitrage, modifying the binomial method, gradient conditions, formulation as a free-boundary problem.
- The Greeks: delta, gamma, theta, vega and rho and their uses in hedging.
- Numerical analysis: Monte Carlo simulation and the explicit finite-difference method.
- Further numerical analysis: Crank-Nicolson, and Douglas multi-time level methods, convergence, accuracy and stability.
- Exotic options: OTC contracts and their mathematical analysis.
- Derivatives market practice: Examination of common practices and historical perspective of option pricing.
- Trading Simulator: Trading equity options, options strategies, Greeks, trading volatility
|Fixed Income & Commodities
- Fixed-income products: Fixed and floating rates, bonds, swaps, caps and floors, FRAs and other delta products.
- Yield, duration and convexity: Definitions, use and limitations, bootstrapping to build up the yield curve from bonds and swaps.
- Curve stripping: reference rates & basis spreads, OIS discounting and dual-curve stripping, cross-currency basis curve, cost of funds and the credit crisis.
- Interpolation methods: piece wise constant forwards, piece wise linear, cubic splines, smart quadratics, quartics, monotone convex splines.
- Current Market Practices: Money vs. scrip, holiday calendars, business day rules, and schedule generation, day count fractions.
- Stochastic interest rate models, one and two factors: Transferring ideas from the equity world, differences from the equity world, popular models, data analysis.
- Calibration: Fitting the yield curve in simple models, use and abuse.
- Data analysis: Examining interest rate and yield curve data to find the best model.
- Probabilistic methods for interest rates.
- Heath, Jarrow and Morton model: Modeling the yield curve. Determining risk factors of yield curve evolution and optimal volatility structure by PCA. Pricing interest rate derivatives by Monte Carlo.
- The Libor Market Model: (Also Brace, Gatarek and Musiela). Calibrating the reference volatility structure by fitting to caplet or swaption data.
- SABR Model: Managing volatility risks, smiles, local volatility models, using the SABR model and hedging stability.
- Arbitrage Free SABR model: Reduction to the effective forward equation, arbitrage free boundary conditions, comparison with historical data and hedging under SABR model.
|Credit Products and Risk
- Credit risk and credit derivatives: Products and uses, credit derivatives, qualitative description of instruments, applications.
- Structural and Intensity models used for credit risk.
- CDS pricing, market approach: Implied default probability, recovery rate, default time modeling, building a spreadsheet on CDS pricing.
- Synthetic CDO pricing: The default probability distribution, default correlation, tranche sensitivity, pricing spread.
- Implementation: CDO/copula modeling using spreadsheets.
- Correlation and state dependence: correlation, linear correlation, analyzing correlation, sensitivity and state dependence.
- Credit Valuation Adjustment (CVA): CVA a guided tour, exposure, modeling exposure, collateral, wrong way risk & right way risk, case study: CDO-Squared.
- Risk of default: The hazard rate, implied hazard rate, stochastic hazard rate and credit rating, capital structure arbitrage.
- Copulas: Pricing basket credit instruments by simulation.
- Statistical methods in estimating default probability: ratings migration and transition matrices and Markov processes
- Stochastic volatility and jump diffusion: Modeling and empirical evidence, pricing and hedging, mean-variance analysis, the Merton model, jump distributions, expectations and worst case analysis.
- Non-probabilistic models: Uncertainty in parameter values versus randomness in variables, nonlinear equations.
- Static hedging: Hedging exotic target contracts with exchange-traded vanilla contracts, optimal static hedging.
- Advanced Monte Carlo techniques: Low-discrepancy series for numerical quadrature. Use for option pricing, speculation and scenario analysis.
- Energy derivatives: Speculation using energy derivatives and risk management in energy derivatives
- Cointegration: Modeling long term relationships, statistical arbitrage using mean reversion.
- Dynamic Asset Allocation: Convexity management, stochastic control, multi-period projection, utility maximization and impact of transaction costs.
- Forecasting by using option prices: volatility forecasting using historical asset prices and current option prices) inserting option prices into ARCH models, Typical ARCH results.
- Density forecasting: Criteria for good forecast, estimating risk-neutral densities from option prices, risk-neutral to real-world densities.