Guide
Lessons meant to be read in order. Each one only uses what came before it.
Stochastic models
Brownian motion, geometric Brownian motion, Itô's lemma and Monte Carlo.
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- Random walks and Brownian motionThe coin-flip walk, variance that grows linearly in time, and its limit: Brownian motion.
- Probability spaces, σ-algebras and random variablesSample space, σ-algebra, probability measure, measurable maps and the σ-algebra generated by a variable.
- Filtrations, adapted processes and stopping timesInformation that grows over time, adapted and predictable processes, and when a random time is a stopping time.
- Conditional expectation and martingalesConditional expectation given the past, fair games, why no betting strategy beats a martingale, and discounted prices under Q.
- Properties of Brownian motionDefinition, the Gaussian characterisation, scaling, time inversion, the Markov property, quadratic variation and rough paths.
- The reflection principle and first passage timesThe strong Markov property, the law of the running maximum and the density of the first passage time.
- The Brownian bridgeBrownian motion with a fixed endpoint: construction, covariance, independence and applications.
- The Itô integralSimple integrands, the Itô isometry, the martingale property and the derivation of ∫W dW = (W² − t)/2.
- Itô's lemmaQuadratic variation, the chain rule for Brownian motion, and the −½σ² drift of the log price.
- Kolmogorov equations and the Feynman–Kac formulaThe generator of a diffusion, the backward equation, and how an expectation becomes the solution of a PDE.
- Change of measure and Girsanov's theoremReweighting probabilities with a density, equivalent measures, Girsanov's theorem and the market price of risk.
- Monte Carlo option pricingEstimating option prices by simulation: standard error, confidence intervals and convergence to Black–Scholes.
- Simulating SDEs: the Euler–Maruyama schemeDiscretising a stochastic differential equation, strong and weak error, step size, and checking against the exact GBM solution.
- Mean reversion: Ornstein–Uhlenbeck and VasicekA process pulled toward a long-run level, its exact solution and stationary distribution, and the short rate in the Vasicek model.
- Jumps and the Merton jump-diffusion modelThe Poisson process, compound jumps, the Merton model, and how jumps create fat tails and a smile.
Check yourself
Questions from the lessons of this series. Open a lesson's group whenever you want to test yourself.