Stochastic Calculus and Stochastic Differential Equations

Stochastic calculus is the language of continuous-time finance. Without it, the Black-Scholes formula is a formula to memorize; 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, modeling 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.

Where are SDEs used in finance?

A stochastic differential equation describes how a quantity evolves when part of the change is random. Asset prices, interest rates, volatility and trading spreads are all modeled with stochastic differential equations. Writing down an SDE is easy; knowing whether it has a solution, simulating it without bias and estimating it from data are where the work lies.

In derivatives pricing, in interest rate and credit models, in stochastic volatility, and in trading, where mean-reverting SDEs model spreads and optimal execution problems are posed as controlled SDEs.

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.

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.
• State existence and uniqueness conditions for an SDE and check them for financial models.

Syllabus, module by module

1. Information and martingales

   Filtrations, conditional expectation and martingales in discrete time.

2. Binomial pricing

   Replication and risk-neutral pricing in the binomial model, the template for everything that follows.

3. Brownian motion

   Construction, quadratic variation and why Brownian paths are nowhere differentiable.

4. The Ito integral

   Definition for simple and general integrands, isometry and martingale property.

5. Ito's lemma

   The one-dimensional and multi-dimensional formulas, with geometric Brownian motion and the Ornstein-Uhlenbeck process.

6. SDEs and their solutions

   Strong and weak solutions, existence and uniqueness, and when the conditions fail.

7. Linear SDEs

   Closed-form solutions for geometric Brownian motion and Ornstein-Uhlenbeck processes.

What every course contains

Every paid course comes with the first seven and the certificate, whichever level it sits at. The proctored exam and the credential come when the course is taken as part of a Certificate, Diploma or Fellowship. Everything is self-paced: the material is yours to study in any order, at any hour, for the whole of the license.

1. Workbooks. Written to train the thinking patterns and the important concepts of the course, with worked problems and the reasoning behind each answer, so that you learn to think the way the subject is thought.
2. Audio edition. Focused, well-presented recordings with explainers and examples of the material, made to be interesting, relevant and entertaining, so that the general subject can be learned away from the desk.
3. Notes. Pre-produced notes with every fact the course relies on, written out in full, so that nothing has to be transcribed from the films or the audio.
4. Cheat sheet. One sheet with every important learning concept of the course on it, to keep beside you during the work and to revise from before the exam.
5. Named sources. Every fact in the notes names its source, the paper, the book or the dataset, for anyone who wants to dig in further.
6. Explainer films. Focused films, in the style of the sample films on the catalog page, that walk through the most difficult concepts of the course with visual explanations of a high standard.
7. Transcript. A full transcript of every film and every audio recording, searchable and quotable.
8. Proctored exam. Taken as part of a Certificate, Diploma or Fellowship, the course ends in a proctored multiple-choice questionnaire, with a pass mark of 60% and three retakes included. A course taken on its own is assessed by its coursework and has no exam.
9. Verifiable certificate. A certificate of completion that a third party can verify: an employer sends the code on it to the credentials office and receives written confirmation of the course, the date and the result.
10. Credential of the program. The Certificate, Diploma or Fellowship of the program, track or path the course belongs to, awarded when every course in it is passed. Which one depends on the program you choose.

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 and Stochastic Differential Equations, issued by the Institute with a verification code. A course taken as part of a Certificate, Diploma or Fellowship ends in a proctored online exam with a pass mark of 60%. The fee includes three retakes; each attempt after that carries a reduced proctored exam fee. A course taken on its own is assessed by its coursework and has no exam. Read how certification works.

Before you start

University calculus, linear algebra and probability, including conditional expectation.

• Mathematics for Quantitative Finance: Calculus and Linear Algebra
• Probability for Quantitative Finance, with Statistics