1. The financial market
(a) The financial market in discrete time
i. One period market: model with n assets and s states of the world
ii. The asset portfolio: present and future financial wealth
iii. The arbitrage on the market: strong and weak arbitrage, Farkas-Stiemke lemma
iv. Redundant assets (replication)
v. Riskless assets: uniqueness and interest rate
vi. Asset pricing: the fundamental theorem of asset pricing
vii. Compounding and discounting
viii. A new probability: «risk neutral» or «martingale equivalent»
ix. Uniqueness of non-arbitrage price
x. Market completeness
xi. Future price estimation: historical simulation
(b) A multi-period model
i. The stochastic riskless asset
ii. Multi-period compounding
iii. Backward pricing for a cash flow paying asset
iv. The fundamental theorem of asset pricing
v. Infra-period analysis (bond pricing)
vi. A stock price model
(c) Continuous time and space market
i. The riskless asset and the continuous compounding
ii. A stochastic process representing a risky asset
iii. Derivatives and Itô's lemma
iv. Stochastic processes simulation (Euler discretization)
v. Geometric Brownian motion: theoretical properties and parameter estimation
vi. A multi-asset market: variances, covariances, and correlations
vii. Pricing of derivatives written on many underlying assets (multi-dimensional Itô's lemma)
viii. Portfolio dynamics in continuous time
ix. Arbitrage
x. Market price of risk
xi. The risk neutral (or martingale equivalent) probability
xii. The fundamental theorem of asset pricing
xiii. Market completeness
xiv. Switching between probabilities (Girsanov's theorem)
xv. The numéraire and the change in probability
(d) Assets with jumps
i. Poisson jumps
ii. Jump-diffusion stochastic processes for asset prices
iii. Simulation of jump-diffusion processes
iv. Itô's lemma for Poisson jumps
(e) Financial markets and utility
i. Utility and preferences (HARA, CARA, CRRA...)
ii. Critiques to the expected utility
iii. Risk aversion: the Arrow-Pratt index
iv. Inter-temporal utility maximization and asset pricing
2. Interest rate risk
(a) The interest rate
i. Spot and forward interest rates
ii. Relationship between spot and forward rates
iii. Instantaneous interest rates
iv. Interest rate dynamics
v. Historical interest rate
(b) Bond pricing
i. Zero-coupon bonds
ii. Fixed coupon bonds
iii. Floating coupon bonds
iv. Yield to maturity
v. Par yield
(c) Stochastic models for interest rates and bonds
i. Instantaneous interest rate and bond pricing
ii. Forward probability
iii. Duration
iv. Immunization
v. Affine processes for interest rate
vi. Affine jump-diffusion processes
vii. Merton model: theoretical properties and parameter estimation
viii. Vasiček model: theoretical properties and parameter estimation
ix. Cox-Ingersoll-Ross model: theoretical properties and parameter estimation
(d) Yield curve interpolation
i. Nelson-Siegel and Svensson models
3. Measuring risk
(a) The risk
i. Properties of a coherent risk measure (ADEH)
ii. The variance is not a coherent risk measure
iii. (ADEH) Representation theorem
iv. Coherent risk measures
v. Expected Shortfall: theoretical properties and historical simulation
vi. Spectral risk measures
vii. The portfolio optimizing spectral risk measures
(b) Value at Risk
i. The case of the Gaussian distribution
ii. What's wrong with VaR
iii. A comparison between Expected Shortfall and VaR about diversification
iv. CVaR
v. Relationship between Expected Shortfall and VaR
(c) Back-testing
i. Basel proposals
ii. Bask-test thresholds