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Heston Model

Heston is the original stochastic vol model with a usable pricing formula. Vol follows a process that snaps back to a long-run level (it does not wander off to infinity), which is what we actually observe in markets -- vol spikes, then fades. The model produces skew and smile through the correlation between price moves and vol moves, generating a complete vol surface from a single set of parameters.

You do not need Heston for crypto. But every stochastic vol model since -- SABR, rough Bergomi, stochastic local vol -- is a descendant of this idea. Understanding Heston is understanding the DNA of modern implied volatility modeling.

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The conceptual ancestor

Heston is to stochastic vol what Black-Scholes is to option pricing: the foundational framework everything else extends or reacts against. You do not need to use it for crypto, but you need to understand it to make sense of the models you do use.

Parameter Intuition​

Adjust each parameter to see how the Heston smile changes.

Heston Smile Explorer

Typical equity smile. Strong put skew from negative rho, moderate vol-of-vol.
21%33%45%758595ATM105115125StrikeImplied Vol (%)
κ (mean reversion)2.0
How fast variance reverts to θ
θ (long-run var)0.040
Long-run variance level. Roughly equals long-dated ATM vol squared
σ (vol of vol)0.500
Controls smile curvature. 0 = flat (BS).
ρ (spot-vol corr)-0.700
Negative = put skew (usual)
v₀ (initial var)0.040
Current variance. Current vs long-run sets term structure slope
ATM IV
20.0%
Put wing slope
+0.28%/strike
Call wing slope
-0.13%/strike
Term structure
Current = long-run (flat term structure)

ρ controls skew (tilt), σ controls curvature (wing width), κ/θ/v₀ control the vol level and term structure.

The five parameters at a glance:

Parameter
What it controls
Effect on smile
kappa -- mean reversion speed
How fast vol snaps back to its long-run level
High kappa flattens the term structure quickly
theta -- long-run variance
The equilibrium vol level the process drifts toward
Sets the overall level of long-dated smiles
sigma -- vol of vol
How volatile the vol process itself is
Higher sigma lifts both wings (fatter tails)
rho -- spot-vol correlation
Link between price moves and vol moves
Negative rho steepens the left wing (put skew)
v0 -- initial variance
Where vol is right now
Gap between v0 and theta tilts the term structure

How each parameter feels​

  • kappa (mean reversion speed): How fast vol snaps back to normal. High kappa means vol shocks die quickly -- the term structure flattens. Low kappa means vol regimes stick around. In crypto, kappa tends to be low: vol regimes are sticky.

  • theta (long-run variance): The vol level the process gravitates toward over time. The square root of theta is roughly the long-dated ATM vol. In BTC, that is typically 50-70% annualized.

  • sigma (vol of vol): Controls smile width. When sigma = 0, there is no smile. As sigma goes up, both wings lift. Same idea as nu in SABR. High sigma = fat tails = expensive OTM wings.

  • rho (spot-vol correlation): Controls skew. Negative rho means vol goes up when the underlying drops. In crypto, rho is typically -0.5 to -0.8. More negative = steeper put skew. This directly drives delta hedging behavior.

  • v0 (initial variance): Where vol is right now. If v0 is above theta, the term structure slopes down (vol expected to fade). If v0 is below theta, it slopes up. After a vol spike, v0 >> theta and the term structure inverts.

ℹ️
Mean reversion separates Heston from SABR

Heston's vol process snaps back to a long-run level. SABR's does not -- it can drift forever. Heston's vol cannot explode to infinity. SABR's can, which is why SABR sometimes produces unrealistic long-dated smiles. For vega hedging, mean reversion means long-dated vega exposure decays predictably.

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Two parameters, two Greek exposures

rho maps to skew (and vanna exposure). sigma maps to smile curvature (and volga exposure). Those two links are the core of Heston.

Strengths and Limitations​

Strength
What it means for you
Fast pricing formula
Unlike most stochastic vol models, Heston options can be priced via a single integral. Thousands of prices per second.
Vol snaps back to normal
Realistic behavior -- vol spikes are followed by mean reversion. Produces a natural term structure.
Rich enough for skew and smile
rho controls skew, sigma controls curvature. Five parameters cover most liquid markets.
Massive tooling ecosystem
Studied since 1993. Libraries in every language. If you hit a problem, someone has solved it.
Limitation
What it means for you
5 parameters = unstable fits
Different parameter combos can produce similar smiles. Fits can jump around day to day.
Fitting is finicky
Multiple local minima. Needs good initial guesses and global search methods.
Cannot match crypto short-dated smiles
Crypto smiles are too steep and wide at short dates. Heston is too smooth for crypto vol dynamics.
Wings are too flat
Heston wings approach a constant slope. Real crypto smiles often have steeper wings at far OTM strikes.
⚠️
Do not use Heston for crypto smile fitting

If you are building a vol surface for crypto options, use SVI or SSVI. Heston's 5-parameter fitting is slower, less stable, and produces worse fits than purpose-built smile models. Heston is a pricing model, not a smile fitting tool. Its value is conceptual. You cannot avoid calendar arbitrage issues without additional constraints, whereas SSVI guarantees calendar-free surfaces by construction.

Heston vs. SABR​

Dimension
Heston
SABR
Vol dynamics
Snaps back to a long-run level
Random walk (no mean reversion)
Free parameters
5
3 (with beta fixed)
Pricing
Semi-closed-form (fast)
Approximation formula (faster)
Fitting
Global optimization, finicky
2-param fit, fast and stable
Term structure
Built in (mean reversion)
Per-slice only
Short-dated smiles
Too smooth
Better (but still limited)
Best for
Equity exotics, FX
Interest rates, FX vanillas
Crypto usage
Rare
Rare (SVI preferred)
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Heston vs. SABR tradeoff

Heston gives you built-in term structure consistency -- every strike is linked to the same variance process. The cost: harder fitting and more parameters. SABR is simpler and faster.

The Family Tree​

Every time you see a vol model with "stochastic variance" or "mean-reverting vol," you are looking at a Heston descendant.

Model
What it changed from Heston
SABR
Replaced mean-reverting variance with random-walk vol. Simpler fitting, clear parameter intuition.
Bates
Added jumps to Heston. Fatter wings from the jump component.
Rough Bergomi
Replaced smooth variance paths with rough, jagged paths. Matches observed vol roughness.
Stochastic local vol (SLV)
Combined Heston-style stochastic variance with local vol. Exact fit plus realistic dynamics.

Equation Explorer​

Convert between implied vol, total variance, log-moneyness, and option prices.

Equation Explorer

w = σ2 × Ttotal variance = IV2 × time
%
The implied volatility
days
Calendar days to expiration
Total Variance (w)
0.022225
Annualized Variance (σ²)
0.2704
Round-trip IV
52.00%
Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Building mathematical intuition​

Learn Heston from scratchInteractive lesson · no prerequisites

This lesson teaches Heston as a two-engine system: spot moves and variance moves. It walks through the five parameters, the two equations, and the exact reason negative rho creates skew.

See also:

Test your understanding before moving on.

Q: If kappa (mean reversion speed) is very high, what happens to the term structure of implied vol?
Q: Why is Heston a poor choice for fitting crypto vol smiles directly?
Q: What is the relationship between the Heston parameter rho and the Greek vanna?
Q: What does Heston give you that SABR does not?

💡 Tip: Try answering each question yourself before revealing the answer.

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