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ProbabilityHilbert SpacesConditional Expectation
Conditional Expectation as an L² Projection
Level: Medium12 min
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Module Library — 29 modules
Probability Theory1 module
Stochastic Calculus & SDEs4 modules
Brownian Motion and Quadratic Variation
MediumStochastic CalculusBrownian MotionQuadratic Variation
Itô's Lemma: Derivation and Applications
HardStochastic CalculusItô's LemmaStochastic Differential Equations
Girsanov's Theorem and Equivalent Martingale Measures
HardStochastic CalculusGirsanov's TheoremRisk-Neutral Pricing
Derivatives Pricing4 modules
Black-Scholes: Derivation, Greeks, Limitations
MediumDerivatives PricingBlack-ScholesGreeks
Implied Vol Surfaces and Smile Dynamics
HardDerivatives PricingImplied VolatilityLocal Volatility
Heston Model: Calibration and Simulation
HardDerivatives PricingStochastic VolatilityHeston Model
Numerical Methods4 modules
Monte Carlo: Antithetic Variates, Control Variates, Quasi-MC
MediumNumerical MethodsMonte CarloVariance Reduction
Finite Difference Schemes and Convergence Analysis
HardNumerical MethodsFinite DifferencesBlack-Scholes PDE
Greeks & Risk4 modules
Numerical Differentiation and Bump-Reval
MediumRisk & GreeksNumerical MethodsBump-Reval
Calibration4 modules
Newton-Raphson and Brent for Implied Volatility
HardCalibrationImplied VolatilityRoot-Finding
Levenberg-Marquardt for Model Calibration
HardCalibrationNumerical OptimisationHeston Model
Statistical & ML Methods4 modules
Factor Models: CAPM, APT, and Fama-French
MediumStatistical / ML for QuantsFactor ModelsCAPM
Time Series for Quants: ARIMA and GARCH
MediumStatistical / ML for QuantsTime SeriesARIMA
ML in Signal Generation
HardStatistical / ML for QuantsMachine LearningAlpha Signals
Market Microstructure4 modules
Adverse Selection and Inventory Models
HardMarket MicrostructureMarket MakingAdverse Selection