What it means
An option's value depends partly on uncertainty about the underlying asset, and a constant-volatility model represents that uncertainty with one fixed parameter. Heston instead allows variance, the square of volatility, to evolve through time.
The model contains two linked random processes, one for asset prices and one for variance, and their correlation allows falling prices and rising uncertainty, for example, to occur together rather than behaving independently. Mean reversion applies to variance and represents a tendency to move toward a long-run variance level.
It does not mean that the asset price must return to its historical average or that tomorrow's volatility is known. Several parameters shape the process: initial variance describes the starting state, long-run variance the level toward which it tends, reversion speed the strength of that tendency, and volatility of variance how strongly variance itself fluctuates.
Correlation affects the asymmetry of returns and option prices across strikes, and with negative price-variance correlation, price declines can coincide with higher variance. This can help explain why downside options may imply different volatility from upside options.
The original research presents a characteristic-function method for valuing European options, but the phrase closed form does not mean a manager can calculate a price by inserting numbers into one short arithmetic expression, since practical implementations involve mathematical transforms and numerical integration. Calibration selects parameters that make model prices resemble observed option prices, and a useful fit depends on the selected strikes, expiries, weighting and market data quality.
Matching one day's quotes does not establish that the parameters will remain stable. The LSE-hosted calibration research discusses efficient parameter fitting and the shape of the calibration problem, which matters because different implementations or starting assumptions can affect results, so the price returned by software is only as useful as its model and calibration controls.
A model price is not necessarily an executable market price, because bid-ask spreads, transaction costs and available size still matter. A theoretical value should be compared with actual quotes before making a trading or hedging decision.
The basic model also does not include every source of market risk, since sudden jumps, liquidity changes and changing relationships can cause differences between modelled and observed prices, and more complicated extensions can address some limitations but introduce additional assumptions. Risk-neutral pricing parameters also differ from a straightforward forecast of future realised returns, because the model is designed to value claims consistently under pricing assumptions.
A manager should not interpret its drift parameter as the expected return available from buying the stock. For a non-finance manager using derivative valuations, ask which version was used, how it was calibrated and how sensitive the result is, and stress-test a model that fits quotes, since parameter uncertainty and market changes can matter as much as a small difference between quoted model prices.
In practice
Real-world examples.
Example
Two options on the same share have different strikes and implied volatilities. A Heston calibration can use the price-variance relationship to fit a skew rather than impose one fixed volatility.
Example
A risk team raises the variance-of-variance parameter in a sensitivity test. Option values change because future uncertainty about volatility itself has increased.
Example
A company receives two hedge valuations from different models. It compares market quotes, calibration inputs and sensitivities rather than choose the lower number without understanding the assumptions.
Formula
Calculation
In a basic risk-neutral version, dS = (r - q)S dt + sqrt(v)S dW1, and dv = kappa(theta - v)dt + sigma sqrt(v)dW2. S is price, v is variance, r the interest rate and q a dividend yield. Kappa controls reversion, theta long-run variance, sigma variance fluctuations, and the two random shocks have correlation rho.
These equations describe processes, not a standalone forecast or finished option-price calculation.Case study
Seen in the real world.
Fictional case study: Alder Treasury valued a long-dated equity-linked contract using one historical volatility estimate. The price differed substantially from a bank's quotation. Its adviser compared a constant-volatility valuation with a Heston calibration to observable options.
The team found that the strike pattern and uncertainty about future variance affected the contract's value. Alder documented calibration choices, tested alternative parameters and retained market-price comparisons. It used the model to understand valuation risk rather than treat the fitted output as a guaranteed transaction price.
Watch out
Common mistakes.
- Confusing variance with volatility. Volatility is the square root of variance in the model.
- Treating calibrated parameters as permanent facts. Refit and test sensitivity when market conditions change.
- Reading a theoretical price as a guaranteed fill. Market spreads and liquidity remain separate.
Questions
People also ask.
Does mean reversion make future variance certain?
No. Variance remains random even though the process tends toward a long-run level.
Is the basic model a stock-return forecast?
No. Its risk-neutral form supports derivative pricing rather than directly predicting investment returns.
Why use it instead of constant volatility?
It can represent stochastic variance and price-variance correlation, helping fit observed option-price patterns.
From the founder's library
Take it further with the book.
Build your financial confidence beyond this definition. Shihan's full-length guide, Accounting Fundamentals, takes the same plain-English approach and turns it into a complete, practical playbook for non-finance managers, business owners and students - with chapter-end quiz answers and presentation slides included.
25% off with code MMHQ25, applied at checkout. Priced in USD - checkout may show the equivalent in your local currency.
View the book and save 25%