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Stochastic Vol Models

Volatility moves. It spikes during crashes, compresses during quiet periods, and mean-reverts over time. Stochastic vol models make vol itself a random process that evolves alongside the price.

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Random vol produces the smile

Black-Scholes assumes vol is constant — no smile. The moment vol is random, the smile appears. Every model in this section does that, with different assumptions about how vol moves.

The Family Tree​

Model
Year
Key idea
Used in production?
1993
Vol mean-reverts. Has a pricing formula (fast).
Rarely alone
1996
Heston + jumps. Handles both smooth dynamics and gap risk.
Exotic desks
2002
Vol is random + backbone skew. Predicts smile movement.
Rates, FX
2011
SABR with a flexible backbone. Fits unusual shapes.
Niche rates
2016
Vol paths are rough (jagged). Explains steep short-dated skew.
Research only

What they share​

All stochastic vol models have the same basic structure: the price process has a random vol component, and that vol component follows its own stochastic process. The differences are in what that vol process looks like:

Model
Vol dynamics
Mean-reverts?
Jumps?
Closed-form pricing?
CIR process (square-root diffusion)
Yes
No
Yes (Fourier)
CIR + compound Poisson jumps
Yes
Yes
Yes (Fourier)
Geometric Brownian motion on vol
No
No
Approx (Hagan)
GBM on vol + custom backbone
No
No
No (PDE)
Fractional Brownian motion
No
No
No (Monte Carlo)

How to choose​

  • Just need to fit the current smile? → You probably want SVI, not a stochastic vol model.
  • Need to predict how the smile moves? → SABR is the standard.
  • Pricing exotics across the full term structure? → Bates or SLV.
  • Building a production exotic pricer? → Stochastic Local Vol (a hybrid) is what most desks run.
  • Want to understand why short-dated skew is steep? → Rough Bergomi.
  • Learning the foundations? → Start with Heston. Everything else builds on it.

How they relate to each other​

Heston is the foundation. Bates adds jumps to Heston. SABR takes a different path — no mean reversion, but a backbone that links vol to the price level. ZABR generalizes SABR's backbone. Rough Bergomi replaces the whole vol process with something rougher and more empirically grounded, but too slow for production.

In crypto, SABR matters most for understanding smile dynamics and initializing SVI fits. Bates matters for exotic desks that need jumps. Heston and Rough Bergomi are conceptual — they explain why smiles look the way they do.

Models in this section:

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