Courses Stochastic Calculus for Finance
Course 3Coming soon
Stochastic Calculus for Finance
Measure-theoretic probability, martingales, Brownian motion, Itô calculus, SDEs, Black–Scholes and stochastic control, at the level of a graduate course.
- 2 parts
- 20 modules
- About 69 hours
- Before you start: Foundations Levels 2, 5 and 6, and Proof Gym stage PG6
Coming soon
Explore the syllabus below, and start with the courses it builds on.
How each module works
- 1Concept lessonsOne idea at a time, with runnable code
- 2Practice setQuick questions with instant feedback
- 3Case studiesThe ideas applied to real finance problems
1Coming soon
Part 1 · About 28 hours
Measure-theoretic probability and martingales
The rigorous framework that graduate stochastic calculus assumes, built on finite coin-toss spaces first.
1.1Why measure theoryComing soon
- You will learn:
- Infinite coin tossing, Events you cannot list, What breaks without care.
- Applied to:
- The binomial model with infinitely many steps
- In Python:
- Simulate long coin-toss paths
1.2σ-algebras as informationComing soon
- You will learn:
- Partitions, Generated σ-algebras, Borel sets.
- Applied to:
- What you know at time t, made precise
- In Python:
- Enumerate σ-algebras on small sample spaces
1.3Probability spaces and random variablesComing soon
- You will learn:
- Measures, Measurable functions, Distributions, Almost-sure events.
- Applied to:
- One random variable under two measures, P and Q
- In Python:
- Same payoff, two measures, two expectations
1.4Expectation as an integralComing soon
- You will learn:
- Simple functions, Lebesgue integral, Monotone and dominated convergence.
- Applied to:
- When you may swap a limit and an expectation in a pricing argument
- In Python:
- Approximate expectations with simple functions
1.5Modes of convergenceComing soon
- You will learn:
- Almost sure, In probability, In L², In distribution.
- Applied to:
- Monte Carlo and the strong law; the central limit theorem revisited
- In Python:
- Counterexamples that separate the modes
1.6Change of measureComing soon
- You will learn:
- Radon–Nikodym derivative, Equivalent measures.
- Applied to:
- The risk-neutral measure in one period; importance sampling
- In Python:
- Importance sampling for a tail probability
1.7Filtrations and conditional expectationComing soon
- You will learn:
- Filtrations, Adapted processes, E[X | G] and its properties, Conditional expectation as projection.
- Applied to:
- The tower property in multi-period pricing
- In Python:
- Compute E[X | Fₙ] on a binomial tree
1.8Discrete-time martingalesComing soon
- You will learn:
- Martingales, Sub- and supermartingales, Stopping times, Optional stopping, Markov property.
- Applied to:
- Discounted prices as Q-martingales; gambler's ruin
- In Python:
- Verify optional stopping by simulation
2Coming soon
Part 2 · About 35 hours
Stochastic calculus
From random walks to Brownian motion, the Itô integral, Itô's lemma, SDEs, Black–Scholes and optimal control, at the level of a graduate course.
2.1Random walk to Brownian motionComing soon
- You will learn:
- Symmetric and scaled random walks, Brownian motion as the limit, Definition.
- Applied to:
- Random-walk models of prices
- In Python:
- Watch scaled walks converge
2.2Brownian motion propertiesComing soon
- You will learn:
- Independent Gaussian increments, Continuity, Nowhere differentiable, Quadratic variation [W]ₜ = t, Reflection principle, First passage.
- Applied to:
- Barrier and first-passage problems
- In Python:
- Show Σ(ΔW)² → t while Σ|ΔW| blows up
2.3Continuous-time martingalesComing soon
- You will learn:
- W, W² − t and the exponential martingale, Markov property.
- Applied to:
- Filtrations and martingales in continuous time
- In Python:
- Test the martingale property by simulation
2.4The Itô integralComing soon
- You will learn:
- Simple integrands, Itô isometry, Properties, Why the left endpoint matters.
- Applied to:
- Trading gains as a stochastic integral
- In Python:
- Left and midpoint sums of ∫W dW give different limits
2.5Itô's lemmaComing soon
- You will learn:
- Derivation from Taylor plus (dW)² = dt, One-dimensional formula.
- Applied to:
- Dynamics of log-prices and of option values
- In Python:
- Verify ∫W dW = (W² − t)/2 numerically
2.6Multidimensional ItôComing soon
- You will learn:
- Itô processes, Multidimensional formula, Product rule, Correlated Brownian motions.
- Applied to:
- Two-asset dynamics
- In Python:
- Two-asset simulation with correlation
2.7Stochastic differential equationsComing soon
- You will learn:
- Existence and uniqueness (stated), Geometric Brownian motion, Ornstein–Uhlenbeck and Vasicek, CIR, Euler–Maruyama and Milstein, Strong and weak error.
- Applied to:
- Equity and short-rate models
- In Python:
- Convergence-order experiment
2.8Diffusions and Kolmogorov equationsComing soon
- You will learn:
- Generator, Backward and forward (Fokker–Planck) equations, Feynman–Kac.
- Applied to:
- Pricing by PDE and by expectation
- In Python:
- Price by PDE grid and by Monte Carlo and get the same answer
2.9Black–Scholes–Merton, derivedComing soon
- You will learn:
- Replication and delta hedging, The PDE, The formula, Greeks.
- Applied to:
- Black–Scholes and general diffusion models
- In Python:
- Delta-hedging simulation and hedging error
2.10Girsanov and risk-neutral pricingComing soon
- You will learn:
- Change of measure for Brownian motion, Market price of risk, Martingale representation.
- Applied to:
- Risk-neutral pricing
- In Python:
- Importance sampling for deep out-of-the-money options
2.11Stochastic optimal controlComing soon
- You will learn:
- Dynamic programming, The HJB equation, Verification.
- Applied to:
- Merton's portfolio problem
- In Python:
- Solve and simulate the optimal strategy
2.12Mixed problem setsComing soon
- You will learn:
- Mixed problems across Parts 1 and 2, Timed sets.
- Applied to:
- Every model from the course, in random order
- In Python:
- Exam-style practice with code checks
PGOpen the Proof Gym →
Proof Gym
About 6 hours, one session a week alongside the parts. The proofs this course leans on, in the same four formats as the Foundations Proof Gym. Probability inequalities first, then reading and writing proofs about conditional expectation and martingales.
PG7 · alongside Foundations Level 5
Probability inequalitiesExample: The weak law of large numbers from Chebyshev
PG8 · alongside Parts 1–2
Probability proofsExample: The tower property; W² − t is a martingale