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. It counts toward the Certificate in Derivatives Pricing and the Fellowship in Computational Finance.
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.
Fourier pricing
Carr-Madan, the COS method and their accuracy and speed.
Calibration
Objective functions, weighting, local minima and parameter stability.
Simulation
Euler schemes, truncation, the Andersen QE scheme and exact simulation in outline.
Beyond Heston
SABR, local-stochastic volatility and rough volatility in outline.
Assessment preparation
Supervised work on the written assessment.
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, issued by the Institute with a verification code. Read how certification works.
Before you start
Stochastic calculus including multidimensional Ito's lemma, complex numbers and Fourier transforms, and the material of Implied Volatility and the Volatility Smile.
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
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 optimiser can stop at local minima. The course covers methods that make calibration more stable and how to judge the result.