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Stochastic Local Vol (SLV)

SLV is what most production desks run. It blends local vol and stochastic vol. Neither alone is good enough for real trading. The goal: a vol surface that fits today's market and moves realistically.

💡
Local vol fits today; stochastic vol moves right; SLV does both

Local vol nails the smile today but gets the dynamics wrong (smile moves too much with spot). Stochastic vol gets the dynamics right but the smile wrong (not enough skew). SLV mixes them.

See the Mixing in Action​

Drag the slider between pure local vol and pure stochastic vol.

SLV Mixing Demo

Balanced mix. This is what most production desks actually run. Best of both worlds.
31%38%45%758595ATM105115125StrikeImplied Vol (%)SLV blendLocal volStochastic vol
Mixing ratio0.50
0 = Pure Local Vol50% Local / 50% Stochastic1 = Pure Stochastic Vol

The green SLV curve blends between the orange local vol smile and the blue stochastic vol smile. Most desks run near 50/50.

What you are looking at​

  • Orange dashed line (local vol): The Dupire local vol smile. Steep, realistic shape -- it matches today's market perfectly. But it implies the smile barely moves when spot moves, which is wrong.
  • Blue dashed line (stochastic vol): A Heston-style smile. Smoother, less skew. It predicts smile movement well, but it cannot match the current market shape on its own.
  • Green solid line (SLV blend): The production model. A weighted mix of both. At 50/50, you get a smile that fits today's market and moves realistically.

Why not just use one?​

Model
Fits today?
Dynamics right?
Production-ready?
Local vol (Dupire)
Perfect
Wrong
Rarely alone
Stochastic vol (Heston)
Approximate
Good
Sometimes
SLV (blended)
Perfect
Good
Yes

How the Mixing Works​

Take a stochastic vol model (like Heston) and multiply its implied volatility by a leverage function derived from local vol. The leverage function is the ratio that makes the blend match today's market exactly.

  • Mixing ratio near 0 (heavy local vol): The leverage function does most of the work. The smile fits perfectly but moves unrealistically.
  • Mixing ratio near 1 (heavy stochastic vol): The leverage function is nearly flat (close to 1 everywhere). The smile may not fit perfectly, but dynamics are realistic.
  • Mixing ratio around 0.5: The sweet spot most desks target. Good fit, good dynamics.
ℹ️
The leverage function does the calibration work

The leverage function L(S,t)L(S, t) absorbs whatever the stochastic vol component cannot explain. Flat leverage function = stochastic vol is doing all the work. Wildly varying = local vol is doing all the work. In production, you want it gently varying near ATM -- that means the mix is balanced.

When Does the Mixing Ratio Matter?​

For vanilla European options, it barely matters -- any mix that fits today's smile prices them the same. The mixing ratio matters for path-dependent products where smile dynamics affect the price. Delta and vega hedges can differ substantially between mixing ratios for exotic products.

Product
Mixing sensitivity
Why
Vanilla European
None
Price depends only on today's smile, which all mixes reproduce
Barrier option
High
Barrier hit probability depends on how the smile moves with spot
Cliquet / accumulator
Very high
Forward-starting options are pure bets on smile dynamics
American / Bermudan
Moderate
Early exercise decision depends on future smile shape

Strengths and Limitations​

Strength
What it means for you
Fits today's market exactly
The leverage function guarantees perfect calibration to all observed option prices.
Realistic dynamics
The stochastic vol component ensures the smile moves in a way that matches historical behavior.
Tunable
The mixing ratio gives you a dial between fit quality and dynamic realism.
Industry standard
This is what Goldman, JPM, and most tier-1 desks run for exotics pricing.
Limitation
What it means for you
Computationally expensive
Calibrating the leverage function requires Monte Carlo or PDE solvers. Not real-time.
Mixing ratio is a choice
There is no "correct" mixing ratio -- you pick it based on hedging performance or exotic P&L attribution.
Leverage function can be noisy
With sparse market data (like crypto), the leverage function can be unstable in the wings, especially far from ATM.
Hard to hedge the mixing
You cannot directly observe or trade the mixing ratio. It is a model assumption.
💡
Required for exotics, overkill for vanillas

Pricing or hedging barriers, cliquets, autocallables -- SLV is the minimum viable model. For vanilla options, use SVI or SABR instead. The term structure behavior comes from the stochastic vol component.

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.

Test your understanding before moving on.

Q: Why can't local vol alone produce realistic smile dynamics?
Q: What does the leverage function do in SLV?
Q: For vanilla European options, does the mixing ratio matter?

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

Building mathematical intuition​

Learn SLV from scratchInteractive lesson · no prerequisites

This lesson explains why local vol and stochastic vol each fail on their own, then shows how the leverage function mixes them into the production model many exotic desks actually use.

See also:

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