Level I

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.

FokkerPlanck and Kolmogorov equations: similarity solutions.

Products and strategies: examination of different asset classes, derivatives products and common trading strategies.

Applied Itô calculus: Discretetime 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 selffinancing replication, no arbitrage, a pricing model and riskneutral 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 BlackLitterman Model.

Capital Asset Pricing Model: Singleindex model, beta, diversification, optimal portfolios, the multiindex 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 worstcase scenarios.

Volatility clustering: Concept and evidence.

Properties of daily asset returns: Average values, standard deviations, departures from the normal distribution, squared returns.

Properties of highfrequency returns: Fiveminute 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 BlackScholes model: A stochastic differential equation for an asset price, the deltahedged portfolio and selffinancing

replication, no arbitrage, the pricing partial differential equation and simple solutions.

Martingales: The probabilistic mathematics underlying derivatives theory, Girsanov, change of measure and FeynmanKac.

Riskneutrality: Fair value of an option as an expectation with respect to a riskneutral density function.

Early exercise: American options, elimination of arbitrage, modifying the binomial method, gradient conditions, formulation as a freeboundary problem.

The Greeks: delta, gamma, theta, vega and rho and their uses in hedging.

Numerical analysis: Monte Carlo simulation and the explicit finitedifference method.

Further numerical analysis: CrankNicolson, and Douglas multitime 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

Level II

Fixed Income & Commodities


Fixedincome 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 dualcurve stripping, crosscurrency 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

Advanced Topics


Stochastic volatility and jump diffusion: Modeling and empirical evidence, pricing and hedging, meanvariance analysis, the Merton model, jump distributions, expectations and worst case analysis.

Nonprobabilistic models: Uncertainty in parameter values versus randomness in variables, nonlinear equations.

Static hedging: Hedging exotic target contracts with exchangetraded vanilla contracts, optimal static hedging.

Advanced Monte Carlo techniques: Lowdiscrepancy 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, multiperiod 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 riskneutral densities from option prices, riskneutral to realworld densities.
