Researcher course in quantitative foundations
Stochastic Calculus for Finance
Brownian motion, the Ito integral, Ito's lemma, change of measure and martingale pricing, derived carefully and applied to the models used on derivatives desks.
Stochastic calculus is the language of continuous-time finance. Without it, the Black-Scholes formula is a formula to memorise; with it, Black-Scholes, the Greeks, stochastic volatility and interest rate models all follow from a few ideas: a random process with independent increments, a way to integrate against it and a rule for changing probabilities.
What is stochastic calculus used for in finance?
Pricing and hedging derivatives, modelling interest rates and volatility, and solving optimal control problems in portfolio choice and execution. Every researcher-level course in derivatives and continuous-time models assumes it.
Who is this course for?
Students preparing for quant researcher roles, practitioners who use pricing models and want to derive them, and anyone on the researcher path. It counts toward the Certificate in Derivatives Pricing and the Fellowship in Computational Finance.
What you will be able to do
- Construct Brownian motion and prove its basic properties.
- Define the Ito integral and compute with it.
- Apply Ito's lemma to derive the dynamics of functions of diffusions.
- Use Girsanov's theorem to change measure and price derivatives as expectations.
- Connect expectations to partial differential equations through the Feynman-Kac formula.
Syllabus, module by module
Information and martingales
Filtrations, conditional expectation and martingales in discrete time.
Binomial pricing
Replication and risk-neutral pricing in the binomial model, the template for everything that follows.
Brownian motion
Construction, quadratic variation and why Brownian paths are nowhere differentiable.
The Ito integral
Definition for simple and general integrands, isometry and martingale property.
Ito's lemma
The one-dimensional and multi-dimensional formulas, with geometric Brownian motion and the Ornstein-Uhlenbeck process.
Change of measure
Radon-Nikodym derivatives, Girsanov's theorem and the market price of risk.
Risk-neutral pricing
The fundamental theorems of asset pricing in continuous time and Black-Scholes as a worked case.
Feynman-Kac
From expectations to partial differential equations and back.
Jumps
Poisson processes and a first look at jump-diffusion models.
Numerical methods
Euler-Maruyama simulation, convergence and checking a derivation numerically.
How the certificate is earned
A written, paper-style assessment: a set of derivations, one pricing problem solved both by expectation and by PDE, and a numerical check by simulation. On a pass you receive the Course certificate in Stochastic Calculus for Finance, issued by the Institute with a verification code. Read how certification works.
Before you start
University calculus, linear algebra and probability, including conditional expectation.
This course counts toward
- Certificate in Derivatives PricingDerivative pricing in five courses and a capstone, from stochastic calculus to stochastic volatility: how options are priced, hedged and calibrated to the market.
- Fellowship in Computational FinanceThe top credential on the researcher path. Nine courses, eight of them at researcher level, and a supervised dissertation, examined by two faculty members.
Questions applicants ask
Is measure theory required?
No. The course states the measure-theoretic results it needs and proves what can be proved without them. Readers who know measure theory get the full statements in the notes.
Which textbook does it follow?
It does not follow one text. Steven Shreve’s Stochastic Calculus for Finance II is the closest companion and is recommended for further reading.