Local Volatility vs Stochastic Volatility

The Fundamental Distinction

A local volatility model says: there is one true volatility function σ(S, t) that depends only on the current spot level S and time t. If the surface tomorrow has different shape, that's because spot moved to a different point on the same fixed function.

A stochastic volatility model says: variance itself is random, driven by its own Brownian motion (correlated with spot). The surface tomorrow has different shape because variance literally moved to a different state.

This difference doesn't matter for vanilla European options on the calibration surface (both produce the same prices by construction). It matters enormously for forward-vol-sensitive products and for understanding how the surface evolves through time.

Side-by-Side

When to Use Local Volatility

• Exotic options requiring exact surface match. Path-dependent options (American, Bermudan, barrier) where the price sensitivity to listed surface points matters. LV produces prices that are arbitrage-free relative to the listed surface by construction.
• Variance swap pricing. The variance swap rate from the LV model exactly equals the listed-surface variance swap rate (Carr-Madan replication). For dispersing or replicating variance swaps, LV is the most natural model.
• PDE-based numerical methods. American options, complex barriers, and other early-exercise products solved on PDE grids naturally accept σ(S, t) as the diffusion coefficient. The numerical infrastructure is well-developed.
• Historical / backward-looking analysis. If you want to price a hypothetical option against historical surfaces and stay consistent with then-listed market prices, LV gives the cleanest replay.
• Quick OTC quotes on European structures. Where the structure depends only on the terminal distribution at one expiration date, LV calibrated to that expiration's smile produces prices indistinguishable from SV.

When to Use Stochastic Volatility

• Forward-vol-sensitive products. Cliquets, forward-start options, ratchets, and reset structures depend on the smile at a future date. LV produces flat or near-flat forward smile (unrealistic); SV produces realistic forward smile that persists. For these products, LV systematically underprices.
• Long-dated options. Beyond ~1 year, mean reversion of variance dominates the IV term structure. LV's deterministic σ(S, t) cannot represent mean reversion in a structurally consistent way; Heston's κ and θ capture it directly.
• Vol-of-vol structures. VIX options, butterfly spreads on long-dated vol, and any vega-of-vega product requires explicit stochastic vol. LV implicitly prices vol-of-vol at zero.
• Smile dynamics modeling. When you need to forecast how the smile will evolve as spot moves (sticky-strike vs sticky-delta question), SV models produce the realistic answer (closer to sticky-delta in calm regimes). LV is structurally locked to sticky-strike.
• Risk management of vol risk. Vega and vanna under SV reflect actual vol-process dynamics. LV-derived vega is the sensitivity to a parallel shift in σ(S, t), which has a different (less practical) interpretation.

Where They Agree

By construction, on the calibration surface (vanilla European options at listed strikes and expirations), LV and SV produce identical prices. This is true after calibration: any model that fits the listed surface fits all listed European options.

For Greeks at ATM strikes in normal regimes, LV and SV produce delta and vega within a few percent of each other. The structural model differences emerge most strongly at OTM strikes and at long horizons.

Both are arbitrage-free under their respective assumptions. Both rely on the same risk-neutral pricing framework. Both are mature, production-grade models with decades of practitioner use.

Where They Diverge

• Forward smile. LV produces forward smile that flattens to ATM very quickly (often by ~50% within 30 days forward). SV produces forward smile that persists with realistic curvature. Cliquet pricing can differ by 30-50% between the two on 1-year structures.
• Sticky-strike vs sticky-delta. When spot moves $1, LV says the smile shape stays anchored to absolute strikes (sticky strike). SV says the smile re-anchors around the new spot (sticky delta). Empirically, equity index surfaces are closer to sticky delta in calm regimes and closer to sticky strike during crashes - a regime-dependent behavior that pure LV or pure SV cannot fully capture.
• Vol risk. Hedging vega exposure under LV means hedging against shifts in σ(S, t), which is a deterministic surface. Under SV, you're hedging against random innovations in v(t). The hedging instruments differ; the risk profile of vega-neutral books differs.
• Calibration stability. LV calibration is unique by construction (solve Dupire PDE). SV calibration is a nonlinear optimization with potential local minima. SV calibration can be unstable when the surface is sparse or noisy; LV is stable but can produce unrealistic σ(S, t) shapes if the listed surface itself is noisy.

The SLV Hybrid

Stochastic-local volatility (SLV) combines both: the model has a stochastic vol process v(t) but with a "leverage function" L(S, t) that is calibrated to match the listed surface exactly. SLV achieves both static fit (LV's strength) and realistic forward dynamics (SV's strength).

SLV is the practitioner standard for institutional exotic-option desks. The downside: calibration is more complex (requires both a parametric SV calibration and a particle-method calibration of L(S, t)), and the model has more parameters to manage. For most retail and quant-research use cases, pure LV or pure SV is sufficient.

Why They're Often Confused

Both produce the same vanilla prices after calibration, so beginners often see them as equivalent. They are not. The forward-vol dynamics differ qualitatively, with major implications for any path-dependent or vol-sensitive product.

The marketing terminology is also confusing. "Calibrated to the surface" applies to both: LV calibrates the function σ(S, t) to the surface exactly; SV calibrates parameters to the surface in a least-squares sense. The word "calibrated" hides this structural difference.

Further Reading

• Local Volatility Documentation
• Heston Model Documentation
• SABR Model Documentation
• Pricing Model Landscape
• Heston vs Black-Scholes
• SABR vs Heston

References

• Dupire, B. (1994). "Pricing with a Smile." Risk, 7(1), 18-20.
• Heston, S. L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility..." Review of Financial Studies, 6(2), 327-343.
• Gatheral, J. (2006). The Volatility Surface. Wiley. Chapter 4 covers Dupire/local vol; Chapter 3 covers Heston/SV.
• Tian, Y., Zhu, Z., Lee, G., Klebaner, F., and Hamza, K. (2015). "Calibrating and Pricing with a Stochastic-Local Volatility Model." Journal of Derivatives. The SLV calibration paper.

This is one of the model-vs-model comparison pages. For the full landscape of pricing models and their relationships, see the Pricing Model Landscape.