Maths

Advanced Heston Model Pricing Calculator

Analyze stochastic volatility pricing with flexible market parameters. Test sensitivities, compare payoffs, and inspect convergence. Built for precise option valuation under changing volatility regimes.

Calculator Inputs

Enter annualized inputs for the Heston stochastic volatility model. The page uses a single-column flow, while the input fields use a responsive 3-column, 2-column, and 1-column grid.

Spot Price (S)

Current underlying asset price.

Strike Price (K)

Exercise price of the option.

Time to Maturity (T, years)

Use years, such as 0.5 or 1.25.

Risk-Free Rate (r)

Annual continuously compounded rate as decimal.

Dividend Yield (q)

Annual dividend yield as decimal.

Option Type

Select the target price to highlight.

Initial Variance (v0)

Starting variance level, usually σ₀².

Long-Run Variance (θ)

Mean level that variance reverts toward.

Mean Reversion Speed (κ)

Higher values pull variance back faster.

Volatility of Variance (σ)

Controls the variance process volatility.

Correlation (ρ)

Correlation between spot and variance shocks.

Integration Limit

Larger values can improve tail capture.

Integration Steps

Use even values for Simpson integration.

Chart Points

Recommended range is 7 to 15 points.

Displayed Decimals

Controls numeric precision in the result panel.

Reset Calculator

Formula Used

The calculator prices European options under the Heston stochastic volatility framework. The underlying and variance processes are:

dSₜ = (r − q)Sₜdt + √vₜ Sₜ dW₁

dvₜ = κ(θ − vₜ)dt + σ√vₜ dW₂

with correlation corr(dW₁, dW₂) = ρ.

European call price: C = S·e^-qT·P₁ − K·e^-rT·P₂

European put price uses put-call parity: P = C − S·e^-qT + K·e^-rT

The probabilities P₁ and P₂ are obtained from the Heston characteristic function and evaluated numerically with Simpson integration.

Greeks in this page are finite-difference approximations, not closed-form symbolic derivatives.

How to Use This Calculator

Enter the spot price, strike price, and maturity in years.
Provide the risk-free rate and dividend yield as decimals.
Set Heston parameters: initial variance, long-run variance, mean reversion, variance volatility, and correlation.
Choose call or put as the highlighted output.
Adjust integration settings for a balance between speed and numerical smoothness.
Submit the form to display the results above the form, along with Greeks, diagnostics, and the Plotly chart.
Use the CSV or PDF buttons to export the output summary.

Example Data Table

These sample rows illustrate common Heston input sets. They are useful for testing the page layout, validating inputs, and comparing parameter effects.

Scenario	Spot	Strike	T	r	q	v0	θ	κ	σ	ρ	Expected Effect
Base ATM	100	100	1.00	0.05	0.02	0.04	0.04	2.00	0.50	-0.70	Balanced baseline for call and put testing.
High Variance Shock	100	105	0.75	0.04	0.01	0.09	0.05	1.60	0.85	-0.55	Raises convexity and skew sensitivity.
Long-Dated Mean Reversion	95	100	2.50	0.03	0.00	0.03	0.06	3.20	0.40	-0.30	Shows long-horizon pull toward θ.

Frequently Asked Questions

1) What makes the Heston model different from constant-volatility pricing?

It allows variance to move over time instead of staying fixed. That helps the model reflect volatility clustering, skew, and smile behavior that simpler constant-volatility models often miss.

2) Why are v0 and θ entered separately?

Initial variance v0 describes today’s starting variance. Long-run variance θ is the level the process tends to revert toward. They can differ when current market stress is above or below equilibrium conditions.

3) What does a negative ρ usually imply?

A negative correlation means spot declines often coincide with rising variance. This commonly creates downside skew, which is one reason Heston is widely used for equity-style option surfaces.

4) Why can integration settings change the answer slightly?

The model uses numerical integration of the characteristic function. More steps or a wider upper limit can improve stability, but they also increase computation time. Small output changes are normal.

5) Does the Feller condition have to hold?

Not always. It is a useful diagnostic for keeping variance strictly positive in the continuous-time process. Real calibrations may violate it, but the flag still helps you judge parameter quality.

6) Are the Greeks exact?

No. This page estimates Greeks with finite-difference bumps around the chosen inputs. They are practical approximations for analysis, not closed-form symbolic values from a separate derivation.

7) Can this calculator be used for put options too?

Yes. Select Put in the form. The page still computes both call and put values, then highlights the selected side while also showing probabilities, diagnostics, and exportable summary results.

8) How should I interpret long-run volatility?

Long-run volatility is the square root of θ. It is not an immediate forecast for tomorrow. Instead, it is the long-horizon variance anchor implied by the model’s mean-reverting variance process.