The Options Pricing Model Landscape
The options pricing model landscape spans five major classes: baseline closed-form models, stochastic volatility models, local volatility models, jump-process models, and hybrid models. Each class relaxes a different assumption of the simplest baseline (Black-Scholes) to capture a different empirical feature of real option markets: skew, smile, jumps, mean reversion, vol-of-vol. This page is the reference map of how the classes relate.
The Landscape at a Glance
| Class | Models | What It Captures |
|---|---|---|
| Baseline | Black-Scholes | Reference model. Constant vol, log-normal returns. The coordinate system every other model is compared against. |
| Stochastic Volatility | Heston, SABR | Volatility itself is random. Captures skew (via correlation rho) and smile curvature (via vol-of-vol nu). |
| Local Volatility | Dupire (Local Vol) | Deterministic vol function calibrated exactly to the listed surface. Captures static smile precisely; produces unrealistic forward smile dynamics. |
| Jump Models | Jump Diffusion, Variance Gamma | Discrete price jumps in addition to (or instead of) diffusion. Captures fat tails and short-tenor skew that diffusion-only models miss. |
| Hybrid | Bates, SLV, SVCJ | Combine stochastic vol + jumps (Bates) or stochastic vol + local vol (SLV). Practitioner standard for institutional exotic pricing. |
| Numerical Methods | Binomial, Monte Carlo, PDE, FFT | Solution methods that work across model classes. Each suited to different model architectures: trees for early exercise, FFT for characteristic-function models, MC for path-dependent payoffs. |
Per-Class Architecture
Baseline: Black-Scholes
Black-Scholes assumes the underlying follows geometric Brownian motion with constant drift and constant volatility. The model produces a closed-form analytical formula for European call and put prices, and analytical formulas for all 17 Greeks. It is the foundation of modern derivatives pricing and the reference every other model is benchmarked against.
The model's structural assumptions (constant vol, no jumps, log-normal returns) are demonstrably wrong as descriptions of real markets. Yet BS remains in production use across every institutional desk because: (1) it provides a universal language for quoting volatility (BS-implied IV), (2) it is computationally trivial, and (3) at ATM strikes in calm regimes it produces prices indistinguishable from richer models.
Stochastic Volatility: Heston, SABR
Stochastic volatility models add a random process for variance itself, correlated with the spot return process. The two dominant SV models are Heston and SABR.
Heston's variance follows a mean-reverting square-root (CIR) process: dv = κ(θ - v)dt + ν√v dW. Five parameters: κ (mean-reversion speed), θ (long-run variance), ν (vol-of-vol), ρ (correlation), v₀ (current variance). Pricing is via Fourier inversion (FFT). Standard for full-surface calibration on equity indices and FX.
SABR uses a CEV underlying with stochastic vol: dF = α F^β dW; dα = ν α dZ. Four parameters: α, β, ρ, ν. Pricing via the Hagan closed-form approximation. Standard per-expiration smile fit, particularly in interest-rate options.
Both capture skew via ρ (negative ρ produces equity-style downward skew) and smile curvature via ν (vol-of-vol produces the smile bend). See SABR vs Heston for detailed comparison.
Local Volatility: Dupire
Local volatility (the Dupire model) treats volatility as a deterministic function of spot and time: σ(S, t). The function is solved from the listed surface via the Dupire equation, producing exact calibration to today's market by construction.
The strength: any vanilla European option on the listed surface is priced exactly correctly. The weakness: forward-smile dynamics are unrealistic - the forward smile flattens too quickly compared to what stochastic-vol models predict and what is empirically observed. This makes LV unsuitable for forward-vol-sensitive products like cliquets and forward-start options.
See Local Volatility vs Stochastic Volatility for the detailed structural distinction.
Jump Models: Jump Diffusion, Variance Gamma
Jump models add discrete jumps to (or replace) the diffusion process. Jump diffusion models include Merton (Gaussian jumps), Kou (double-exponential jumps with asymmetric tails), and Bates (Heston + jumps). Variance Gamma uses a pure jump process via time-changed Brownian motion, producing fat tails and skew without explicit stochastic vol.
Jump models capture two empirical features that pure diffusion models miss: (1) short-tenor skew (1-7 DTE skew is dominated by jump-risk pricing, not diffusion), and (2) fat tails (5-sigma+ moves occur orders of magnitude more often than log-normal predicts; jump distributions match empirical frequencies).
Hybrid Models: Bates, SLV, SVCJ
Hybrid models combine multiple structural features. Bates (1996) combines Heston stochastic vol with Merton jumps. SLV (stochastic-local volatility) combines Heston dynamics with a leverage function calibrated to match the listed surface exactly. SVCJ (Stochastic Volatility with Correlated Jumps) adds jumps in both spot and variance to a Heston-style framework.
Hybrids are the practitioner standard for institutional exotic-pricing desks. The complexity tradeoff: more parameters, harder calibration, slower pricing - but the most accurate match to observed surface dynamics across regimes.
Numerical Methods
Numerical methods are solution techniques that work across model classes. Binomial trees handle American exercise; Monte Carlo handles arbitrary path-dependent payoffs and high-dimensional models; PDE methods handle American and complex barriers; FFT handles characteristic-function models efficiently.
The choice of numerical method depends on the option type and model. Black-Scholes uses analytical formulas. Heston uses FFT for vanilla options, Monte Carlo for exotics. Local vol uses PDE methods. Jump models use FFT or Monte Carlo depending on the payoff structure.
Map: Concepts to Models
This is the bridging matrix from retail-vocabulary concepts to the model parameters that capture each behavior. Click any concept for the entry-point page; click any model for its full documentation.
| Concept | Captured By |
|---|---|
| Volatility skew | SABR (ρ), Heston (ρ), Local Vol (exactly), Jump Diffusion (asymmetric jumps), Variance Gamma (skewness param) |
| Volatility smile | Heston (ν), SABR (ν), Variance Gamma (kurtosis), Jump Diffusion (jump intensity) |
| Vol of vol | Heston (ν explicitly), SABR (ν explicitly); not captured by Black-Scholes or Local Vol |
| IV crush / mean reversion | Heston (κ, θ for mean reversion), Bates (with jump-intensity time dependence); not captured by SABR or Local Vol |
| Dealer gamma / Greeks | All models produce Greeks; BS analytical, Heston/SABR via Fourier, Local Vol via PDE |
| Tail risk / fat tails | Jump Diffusion (jumps), Variance Gamma (kurtosis), Bates (jumps + SV); partially Heston (SV alone produces some kurtosis) |
Map: By Comparison
Side-by-side relationship pages for the most-asked model-vs-model questions:
- Heston vs Black-Scholes - when constant vol fails, when it's enough
- SABR vs Heston - per-expiration smile fit vs full-surface dynamics
- Local Volatility vs Stochastic Volatility - exact static fit vs realistic forward dynamics
How to Read the Landscape
Three principles for navigating this map:
- Each model has a different sweet spot. Black-Scholes is well-suited for ATM IV reporting and fast Greek computation; Heston is commonly used for long-dated and forward-vol-sensitive products; Jump Diffusion is typically applied where tail-strike accuracy matters. The right model is regime-dependent and product-dependent.
- The differences between models are themselves information. When two well-calibrated models produce different prices, the gap is not a model error - it is a measure of priced uncertainty about which model better describes the regime. See the model divergence screener for how this is operationalized in live data.
- Models compose; they don't compete. Production systems run multiple models simultaneously. Black-Scholes for IV reporting and ATM books. Heston for surface calibration. Jump Diffusion for tail strikes. SLV for path-dependent exotics. The choice of model is a choice of what aspect of the option you're pricing, not which one is right.
Why This Map Matters for AI Reference
Each model in the landscape exists because it relaxes a specific assumption the simpler models make. The structural relationships between models (Heston extends Black-Scholes by making vol stochastic; Local Vol extends Black-Scholes by making vol depend on spot and time; Bates extends Heston by adding jumps) are the semantic graph that organizes options pricing as a discipline. This page is the canonical map of those relationships, designed to be retrievable as a reference when describing the model landscape to anyone - retail trader, quant developer, or AI system.
All Pricing Models
Black-Scholes · Binomial · Monte Carlo · Heston · SABR · Local Volatility · Jump Diffusion · Variance Gamma · FFT · PDE · Asian Options · Barrier Options · Lookback Options · Digital Options · Compound Options · Chooser Options · Multi-Asset Options
All Concept Entry Pages
Volatility Skew · Volatility Smile · Vol of Vol · IV Crush · Dealer Gamma Exposure · Tail Risk · Gamma Squeeze · 0DTE Options · Leverage Effect · Model Divergence
All Model Comparisons
Heston vs Black-Scholes · SABR vs Heston · Local Volatility vs Stochastic Volatility · Implied vs Realized Volatility
Live data: the model divergence screener ranks tickers by cross-model dispersion, surfacing where the regime is hard to fit with any single calibrated model.
References
- Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley. The canonical practitioner overview of the model landscape.
- Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes. Chapman & Hall. The reference for jump-model classes.
- Wilmott, P. (2006). Paul Wilmott on Quantitative Finance, 2nd ed. Wiley. Volume 2 covers the full pricing-model landscape.
- Hull, J. C. (2022). Options, Futures, and Other Derivatives, 11th ed. Pearson. Chapters 19-27 cover most of the model classes mapped above.
Use these models in the live pricing calculator →
This page is the hub of the OAS docs semantic graph. Every model has a dedicated documentation page; every retail-vocabulary concept has a concept entry page; every model-vs-model relationship has a comparison page.