Researcher course in quantitative foundations
Numerical Methods in Finance
The numerical methods behind every pricing and risk system: Monte Carlo and its variance reduction, finite differences for pricing equations, lattices, Fourier methods, root finding and optimisation for calibration, with attention to error and stability.
Numerical methods in finance turn pricing models into prices. Almost no product beyond a plain European option has a closed-form price under a realistic model, so desks rely on simulation, finite differences, lattices and Fourier transforms, and on the calibration routines that fit models to the market. This course teaches those methods and how to know when they are wrong.
Why does error analysis matter so much?
Because a pricer that converges slowly, or not at all, still prints a number. A Monte Carlo price without a confidence interval, or an explicit finite-difference scheme run outside its stability limit, can be badly wrong without any warning. Each unit pairs a method with the checks that show whether it can be trusted.
How is the course taught?
Audio lessons give the reasoning, explainer films show the methods converging and failing, and the workbooks carry derivations, notebooks and cheat sheets. Assignments and the project receive written feedback from the faculty.
Who is this course for?
Quants heading for pricing, model validation or risk roles. It counts toward the Certificate in Derivatives Pricing and so toward the Fellowship in Computational Finance.
What you will be able to do
- Price options by Monte Carlo and reduce the variance with antithetic, control-variate and quasi-random methods.
- Solve the Black-Scholes equation with explicit, implicit and Crank-Nicolson schemes and judge their stability.
- Price under Heston and other models with Fourier methods.
- Calibrate a model to market prices and diagnose a poor fit.
Syllabus, module by module
Error, stability and cost
Floating point, truncation error, convergence order and how to measure each.
Monte Carlo
Path simulation, discretisation of SDEs, confidence intervals and Greeks by simulation.
Variance reduction
Antithetic variates, control variates, importance sampling and quasi-random sequences.
Finite differences
Explicit, implicit and Crank-Nicolson schemes, boundary conditions and American exercise.
Trees and Fourier methods
Binomial and trinomial lattices; characteristic functions and the FFT and COS methods.
Calibration
Root finding, least squares fitting to an implied volatility surface, and stability of the fit.
How the certificate is earned
Graded computational assignments and a final project: a pricer for one exotic option by two methods, with a written error analysis. On a pass you receive the Course certificate in Numerical Methods in Finance, issued by the Institute with a verification code. Read how certification works.
Before you start
Stochastic calculus and the Black-Scholes model at the Institute's level, and confident Python.
This course counts toward
- Certificate in Derivatives PricingDerivative pricing in one semester of ten courses and a capstone, from stochastic calculus to stochastic volatility and interest rate models: how options are priced, hedged, traded and calibrated to the market.
- Fellowship in Computational FinanceThe top credential on the researcher path. Four semesters, 38 courses, and a dissertation supervised in writing and examined by two faculty members.
Questions applicants ask
Which numerical methods are used most in finance?
Monte Carlo simulation for path-dependent and multi-asset products and for risk, finite differences for products with early exercise, and Fourier methods for fast calibration of models such as Heston.
Is the course taught in Python or C++?
In Python with NumPy, so the methods stay readable. Students who have taken C++ for Quantitative Finance may submit the project in C++.