Researcher course in quantitative foundations
Stochastic Differential Equations for Finance
How stochastic differential equations are solved, simulated and estimated, with the mean-reverting, short-rate and volatility models that use them.
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 modelled 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.
Where are SDEs used in finance?
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?
Researcher-path students and practitioners who build or validate pricing and risk models. It is part of the Fellowship in Computational Finance.
What you will be able to do
- State existence and uniqueness conditions for an SDE and check them for financial models.
- Solve linear SDEs in closed form, including Ornstein-Uhlenbeck and Vasicek.
- Simulate SDEs with Euler-Maruyama and Milstein schemes and measure their error.
- Estimate SDE parameters from discretely observed data.
- Explain how the choice of SDE shapes the prices a model produces.
Syllabus, module by module
SDEs and their solutions
Strong and weak solutions, existence and uniqueness, and when the conditions fail.
Linear SDEs
Closed-form solutions for geometric Brownian motion and Ornstein-Uhlenbeck processes.
Mean reversion
Ornstein-Uhlenbeck dynamics for spreads and commodities, half-life and stationary distributions.
Short-rate models
Vasicek and Cox-Ingersoll-Ross: dynamics, bond prices and the positivity question.
Discretisation
Euler-Maruyama and Milstein schemes, strong and weak convergence.
Estimation
Exact and approximate maximum likelihood for discretely sampled diffusions.
Multidimensional systems
Correlated diffusions and the stochastic volatility setting.
Research lab
Supervised work on the written assessment.
How the certificate is earned
A written, paper-style assessment: solve and simulate an assigned model, estimate it on market data and discuss the fit and its limits. On a pass you receive the Course certificate in Stochastic Differential Equations for Finance, issued by the Institute with a verification code. Read how certification works.
Before you start
Stochastic calculus at the level of Stochastic Calculus for Finance.
This course counts toward
Questions applicants ask
How does this course differ from Stochastic Calculus for Finance?
Stochastic calculus builds the integral and the pricing theory. This course works with the equations themselves: solving, simulating and estimating them.
Is there programming?
Yes. Simulation and estimation are done in Python.