Black-Scholes vs Local Volatility
Black-Scholes uses a single scalar volatility parameter and produces a flat IV surface that ignores skew and smile. Local volatility (Dupire) constructs a deterministic spot-and-time-dependent vol function sigma(S, t) that exactly reproduces every observed option price by solving the Dupire forward equation. Side-by-side comparison of fit quality, dynamics, calibration, and operational use.
Side-by-Side
| Property | Black-Scholes | Local Volatility (Dupire) |
|---|---|---|
| Vol structure | Single constant scalar sigma | Deterministic function sigma(S, t) |
| Surface fit | Cannot fit skew or smile - flat IV | Exact fit to every traded option price (by construction) |
| Stochasticity of vol | None - vol is constant | None - vol is deterministic in spot and time |
| Parameter count | 1 (sigma) | Implicit - sigma(S, t) is a function |
| Pricing form | Closed-form analytical | PDE solver (finite-difference grid) |
| Pricing speed | Microseconds per option | Milliseconds per option (PDE-based) |
| Calibration | Trivial (1D root-find for sigma) | Solve Dupire PDE on listed surface |
| Forward smile dynamics | Flat at all forward dates | Flattens too quickly relative to observed |
| Primary use cases | IV reporting, ATM Greeks, real-time aggregation | Vanilla pricing on calibrated surfaces |
| Use with stochastic vol | Separate model class | Combined as SLV (stochastic-local volatility) |
| Variance swap consistency | Approximate via single scalar | Exact match to listed surface |
When to Use Black-Scholes
- Implied volatility extraction. Every market quote is reported in BS-implied IV space. To compute IV from a market price, you invert BS analytically. Even when subsequently pricing under local-vol, you start from BS-implied IV and end with a BS-implied IV-equivalent.
- ATM Greeks aggregation. For real-time portfolio-level delta, gamma, and vega aggregation across thousands of positions, BS analytical Greeks are 100-1000x faster than PDE-based local-vol Greeks. The precision difference is small for ATM books.
- Quick mental math. The Brenner-Subrahmanyam ATM straddle approximately 0.8 · S · sigma · sqrt(T) identity is BS-derived and useful for on-the-fly expected-move calculations.
- Liquid ATM strikes. For SPY 50-delta options 30 days out in a normal vol regime, BS pricing matches local-vol within a few cents. The complexity premium of local-vol is wasted on these strikes.
- Baseline comparison. Every smile-aware model is most cleanly evaluated by computing the BS-implied vol it produces and comparing to the BS-implied vol of the listed market price. The deviation tells you the model's smile-fitting quality.
When to Use Local Volatility
- Exact surface fitting. When you need a model that prices every listed option exactly, local volatility is the only choice in this comparison. By construction, local-vol matches every traded price. This is essential for vanilla-option market-making and for situations where any pricing residual is unacceptable.
- Vanilla pricing on calibrated surfaces. If the goal is to price a non-listed strike on the same surface (e.g., interpolating between listed expirations or strikes), local-vol is the standard model because it inherits the no-arbitrage structure of the listed surface.
- Path-dependent options on listed surfaces. Asian options, lookbacks, and barriers on equity indexes are routinely priced under local-vol because the calibrated function sigma(S, t) feeds the relevant PDEs and Monte Carlo simulations cleanly.
- SLV (stochastic-local volatility) as production standard. The hybrid SLV model combines local-vol's exact surface fit with stochastic-vol's realistic forward-vol dynamics. SLV is the practitioner standard for exotic-option pricing because it inherits the strengths of both classes.
- Variance swap pricing. Local-vol matches variance swap fair values exactly because its construction guarantees consistency with the calibrated surface. BS variance-swap pricing requires additional adjustments.
Where They Agree
For a single ATM strike at intermediate maturity, BS and local-vol produce nearly identical prices in a calibrated surface. The local-vol function at that (S, t) point is essentially the BS-implied IV at that strike (with a small correction term that vanishes for ATM). The difference appears at OTM strikes, where the local-vol function takes different values than the BS-implied IV.
Both models use the same underlying spot dynamics (geometric Brownian motion). Both require the same input data (spot, strike, time, rate, dividends). Both produce arbitrage-free vanilla prices when properly applied.
Both produce identical Greeks structure: delta, gamma, theta, vega all exist and behave consistently. Local-vol Greeks differ in numerical value at OTM strikes but the conceptual interpretation is identical.
Where They Diverge
- Surface fit at OTM strikes. A 5-delta SPX put 30 DTE: BS prices ~$0.40 (using ATM IV), local-vol prices ~$0.85 (matching market). The market price typically matches local-vol within 1%; BS underprices by 50%+.
- Forward smile dynamics. BS produces a flat forward smile (smile at every future date is flat). Local-vol produces a forward smile that flattens too quickly with forward time, which is a known limitation. Stochastic-vol models capture the persistent forward smile that local-vol misses.
- Cliquet and forward-start pricing. Cliquets and forward-start options are sensitive to forward-vol dynamics. BS produces zero forward smile and underprices these structures. Local-vol produces flat-too-quickly forward smile and underprices them differently. Stochastic-vol or SLV is required for accurate cliquet pricing.
- Sticky behavior. Local-vol implies sticky-strike behavior (smile translates with spot). BS has no smile to be sticky. Empirically, market behavior is closer to sticky-delta in calm regimes (closer to stochastic-vol prediction) and closer to sticky-strike during crashes (closer to local-vol prediction).
Why They're Often Confused
Both produce IV surfaces, and both feed into option pricing engines. The casual observer may not distinguish the model class behind a "calibrated IV surface" - it could be parametric (eSSVI), local-vol (Dupire), or stochastic-vol (Heston/SABR). The class matters operationally because it determines forward-vol dynamics and the price of path-dependent and forward-start contracts.
Local volatility is sometimes described as "Black-Scholes with a strike-dependent vol." This is technically wrong: local vol is sigma(S, t), not sigma(K). The distinction matters because BS-implied IV is a strike-and-tenor lookup; local vol is a deterministic function of the underlying spot and clock time, evaluated along the path.
The practitioner choice is rarely BS-or-local-vol. BS handles fast Greek computation and IV reporting; local-vol handles surface-consistent vanilla and path-dependent pricing; stochastic-vol or SLV handles forward-vol dynamics. All three coexist in production systems.
Further Reading
- Black-Scholes Model Documentation
- Local Volatility Model Documentation
- Heston Stochastic Volatility
- Local Volatility vs Stochastic Volatility
- What Is Volatility Skew?
- Pricing Model Landscape
References
- Black, F. and Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, 81(3), 637-654. The original BS paper.
- Dupire, B. (1994). "Pricing with a Smile." Risk, 7(1), 18-20. The local volatility paper.
- Derman, E. and Kani, I. (1994). "Riding on a Smile." Risk, 7(2), 32-39. Independent derivation of local volatility for binomial trees.
- Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley. Practitioner reference for stochastic-volatility and local-volatility modeling.
- Ren, Y., Madan, D., and Qian, M. Q. (2007). "Calibrating and pricing with embedded local volatility models." Risk, 20(9), 138-143. Local-vol calibration in practice.
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.