Volatility Models — Heston & SABR
Generate the implied-volatility smile parametrically and compare Heston against SABR on the same axes.
Real option markets don't have a single volatility — implied vol varies by strike and maturity, forming the volatility smile and skew. This Heston and SABR calculator generates that surface parametrically so you can see how each parameter reshapes it.
Heston vs SABR
The Heston (1993)model treats variance itself as a mean-reverting stochastic process and is priced here by Fourier integration — it's the workhorse for equity and index smiles. SABR (Hagan 2002)is a stochastic-alpha-beta-rho model with a famous closed-form implied-vol approximation, ubiquitous on rates and swaption desks. Plotting them side by side makes the trade-off concrete: Heston's richer dynamics versus SABR's speed and closed form.
Reading the parameters
Correlation ρ tilts the smile into a skew — negative ρ (the usual equity case) lifts the downside puts. Vol-of-vol(ν in SABR, ξ in Heston) inflates the wings, steepening the smile away from the money. Mean reversion and the long-run variance set the term structure. It's the fastest way to build intuition for skew before an interview — pair it with the volatility vs direction concept and the Coach questions on skew and vega risk.
Frequently asked questions
What is the volatility smile?
Implied volatility is not one number — it varies by strike and maturity. Plotted against strike it forms the smile or skew, which flat Black-Scholes cannot reproduce. This calculator generates that curve parametrically.
Heston or SABR — what is the difference?
Heston (1993) models variance itself as a mean-reverting stochastic process and is priced here by Fourier integration — the workhorse for equity and index smiles. SABR (Hagan 2002) has a famous closed-form implied-vol approximation and rules rates and swaption desks. The tool plots both on the same axes.
What do correlation and vol-of-vol do to the smile?
Negative correlation ρ tilts the smile into a skew, lifting downside puts (the usual equity case). Vol-of-vol (ν in SABR, ξ in Heston) inflates the wings, steepening the smile away from the money.
Is the Heston/SABR calculator free?
Yes, fully free with no account. Move the parameters and the smile regenerates live against a flat Black-Scholes baseline.