Quant Mechanics Lab

Stochastic Calculus

Categories: Stochastic Calculus
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About Course

The machinery every derivative price is built on — not a measure theory course, and not a formula sheet either.

Most treatments of this subject pick one of two failures: they bury you in abstraction before anything is computable, or they hand you Itô’s formula and hope you don’t ask where it came from. This one builds the apparatus in order, from conditioning and martingales to Brownian motion, the Itô integral, Itô’s formula, SDEs, Girsanov, and the bridge to PDEs — and every object is simulated and checked before it is used.

WHAT IT COVERS

Thirty-six lessons across eight submodules, each with its own executable notebook.

  • Filtrations, conditioning and martingales — what information actually means in a model
  • Brownian motion: construction, properties, and what makes it non-differentiable
  • Quadratic variation and the Itô integral, built from the ground up
  • Itô’s formula in one and several dimensions, with the product and quotient rules you’ll actually use
  • SDEs and the processes that matter: geometric Brownian motion, Ornstein-Uhlenbeck, Vasicek, CIR
  • Change of measure and Girsanov — the result that separates the real world from the pricing world
  • From SDEs to PDEs: Feynman-Kac, the fundamental theorem of asset pricing, and Black-Scholes derived three independent ways
  • Monte Carlo: discretisation bias, variance reduction, and error bars you can defend

WHO IT’S FOR

Quants, quant developers, risk and model-validation professionals, and students moving toward a pricing role. If you can follow a derivation and write Python, you’re ready. No measure theory assumed — what’s needed is built.

WHAT MAKES IT DIFFERENT

Nothing is asserted without being checked. Every figure in the course is reproduced by code you can run, every theorem is tested numerically against the object it describes, and where a result only holds under conditions, those conditions are stated and probed.

The canonical example runs through the whole module: the same option, priced by three independent routes, agreeing to the fourth decimal.

WHAT YOU KEEP

  • 36 executable, fully documented Jupyter notebooks — the exact code from the videos
  • Over 130 figures, each reproducible from source
  • A working mental model of the apparatus, not a collection of formulas
  • Lifetime access
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What Will You Learn?

  • Compute conditional expectations and solve stopping problems such as the gambler's ruin by optional stopping
  • Construct the Itô integral and compute the mean, variance and law of a stochastic integral
  • Apply Itô's formula in one and several dimensions, with the product rule, to derive the SDE of a function
  • Solve GBM, Ornstein–Uhlenbeck and linear SDEs and compute their moments, half-lives and exit times
  • Change measure with Girsanov, check that a density is a true martingale, and price under the T-forward measure
  • Derive the Black–Scholes price three ways: by hedging, by Feynman–Kac and by risk-neutral expectation
  • Simulate SDEs with Euler–Maruyama and Milstein and measure their strong and weak orders of convergence
  • Report Monte Carlo prices with standard errors and cut the variance with antithetics and control variates

Course Content

Start Here
One orientation lesson before the technique begins: what the module builds, what each of the eight submodules contributes, and how they chain into pricing and hedging. It sets out the order of the course and why each step is a prerequisite for the next.

  • Stochastic Calculus: The Map

Information, Conditioning and Martingales
The language of the rest of the module, in discrete time: σ-algebras and filtrations as information, conditional expectation and its rules, and martingales with optional stopping. By the end you can compute conditional expectations and solve stopping problems such as the gambler's ruin.

Brownian Motion
Brownian motion from a rescaled coin walk to its defining properties, the behaviour of its paths, and its first-passage laws through the reflection principle. The submodule ends with the tools to test whether a process is Brownian motion and to compute hitting and running-maximum probabilities.

Quadratic Variation and the Itô Integral
Quadratic variation as the sum that survives where total variation fails, and the Itô integral built on it, with its isometry and its martingale property. After it, means, variances and laws of Itô integrals are routine computations.

Itô’s Formula
Itô's formula in one and several dimensions, the product rule, and the formula used as an engine for building martingales and evaluating integrals. You finish able to differentiate any smooth function of one or several Itô processes and to explain the correction term.

Stochastic Differential Equations and Key Processes
Existence and uniqueness for SDEs, then geometric Brownian motion, Ornstein–Uhlenbeck and CIR solved, and the generator with Dynkin's formula and the Kolmogorov equations. It closes with linear SDEs solved and moments and exit times computed.

Change of Measure, Girsanov and Numéraires
Changing the probability measure: equivalent measures and densities, the stochastic exponential, Girsanov's theorem, and change of numéraire. By the end you can remove a drift, check that a density process is a true martingale, and price under the T-forward measure.

Feynman–Kac, the FTAP and Black–Scholes
From SDEs to PDEs and to prices: Feynman–Kac, the fundamental theorems of asset pricing, Black–Scholes derived three ways, and payoffs that are not smooth. The submodule leaves you able to price and hedge standard options and to say why a price is an expectation.

Monte Carlo and Numerical Stochastic Calculus
Monte Carlo treated as a measured tool: estimators and confidence intervals, discretisation of SDEs, variance reduction and Greeks, ending in an Asian option priced with every error accounted for. You leave able to simulate SDEs and to measure and reduce the error of an estimate.

Course materials — Python notebooks
All 36 companion notebooks for the module, one per lesson, in a single download. Each runs the code shown in its lesson on the same numbers, with the simulations, the pricing where the lesson prices, and the asserts that check every result. The notebooks ship already executed, so you can read them without running anything. Unzip and open them in Jupyter or Google Colab. The README lists the three dependencies.