Researcher course in derivatives
Stochastic Volatility and the Heston Model
Stochastic volatility in depth: why volatility needs its own process, the Heston model and its characteristic function, Fourier pricing methods, calibration to a volatility surface, simulation schemes, and rough volatility in outline.
In the Black-Scholes model volatility is a constant. In markets it moves, it tends to rise when prices fall, and option prices reflect both facts. Stochastic volatility models give volatility its own random process. The Heston model, published in 1993, is the most widely used of them because it fits the smile reasonably well and can still be priced quickly.
How are options priced in the Heston model?
Heston showed that the model’s characteristic function, the Fourier transform of the log-price distribution, has a closed form. Option prices then follow from a numerical integral. Fourier methods such as Carr-Madan and the COS method make this fast enough for calibration. You will derive the characteristic function, implement two pricing methods and compare them.
What are the limits of the Heston model?
It struggles to fit the steep short-dated skew seen in equity markets, and its parameters often change from day to day when recalibrated. Those limits motivated jump models, local-stochastic volatility and, more recently, rough volatility, which the course introduces in outline.
Who is this course for?
Quant researchers and derivatives quants who build or validate pricing models.
What you will be able to do
- Explain the features of the volatility surface that stochastic volatility models capture.
- Derive the Heston pricing PDE and its characteristic function solution.
- Price options with Fourier methods such as Carr-Madan or COS.
- Calibrate the Heston model to a market surface and judge the fit.
- Simulate Heston paths with schemes that respect the variance process.
Syllabus, module by module
Why stochastic volatility
Volatility clustering, the leverage effect and the dynamics of the smile.
The Heston model
The square-root variance process, correlation, the Feller condition and parameter roles.
The pricing PDE
Hedging with two instruments, the market price of volatility risk and the PDE.
Characteristic functions
Heston's 1993 solution, the affine structure and numerical pitfalls.
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Transcript. A full transcript of every film and every audio recording, searchable and quotable.
- 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.
- 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.
- 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 with a computational component: you calibrate the Heston model to market data, compare pricing methods, and write up the derivation, calibration and results. On a pass you receive the Course certificate in Stochastic Volatility and the Heston Model, 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
This course assumes the material of the courses below, or the same knowledge gained elsewhere.
This course counts toward
- Certificate in Derivatives PricingDerivative pricing in seven 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. Twenty-one courses and a dissertation supervised in writing and examined by two faculty members.
Questions applicants ask
What is the Heston model?
A model, published by Steven Heston in 1993, in which the variance of returns follows its own mean-reverting random process, correlated with the price. It produces a volatility smile and still has a near closed-form solution for European options.
Why is calibration of the Heston model difficult?
The fit is often flat in some directions, so different parameter sets match the market almost equally well, and the optimizer can stop at local minima. The course covers methods that make calibration more stable and how to judge the result.