Why do different options pricing models give different prices for the same option?

Each model makes different assumptions about how the underlying moves. Black-Scholes assumes constant volatility and continuous paths, producing a single fair value per option. Heston adds stochastic volatility, capturing the smile that Black-Scholes can't reproduce. Jump-diffusion models (Merton, Kou, Bates) add discontinuous price jumps for event risk. Variance Gamma uses pure-jump Lévy dynamics with explicit fat tails. Local Volatility fits the listed surface exactly by construction. The 'correct' price depends on which assumptions best match the underlying's actual dynamics, and the gap between two models is itself priced information about which assumptions the market is paying for.

Which model best handles options around earnings?

Around earnings the front-month tenor concentrates jump risk that continuous models (Black-Scholes, Heston, Local Vol) systematically under-price. The right tools are jump-diffusion variants: Merton for symmetric jumps, Kou for asymmetric (downside-skewed) jumps, Bates for jumps plus stochastic volatility. The model divergence between Black-Scholes and Bates on a front-month ATM straddle is roughly the jump premium the market is paying for the event. After the event, the front-month IV typically collapses (IV crush); the post-event jump premium effectively goes to zero, and Black-Scholes regains accuracy.

Which pricing model should I use for which situation?

Black-Scholes for fast pre-trade pricing, IV computation, and as the reference for divergence checks. Heston for surface-aware pricing where the smile and term structure matter jointly. SABR for fitting per-expiration smiles cleanly. Local Volatility when you need calibration-perfect repricing of the listed surface. Jump-diffusion (Merton, Kou, Bates) for options around earnings or known event jumps. Variance Gamma for fat-tailed Lévy returns. Binomial trees or PDE for American-exercise pricing. Monte Carlo for path-dependent exotics and risk-neutral probability distributions. The platform exposes all of these so you can compare.

How does model calibration to the listed surface work?

Calibration finds the model parameters that minimize the squared error between model-computed prices and listed market prices across a chosen strike-tenor grid. For Heston, the five parameters (κ, θ, ξ, ρ, v₀) are jointly fit; for SABR, the three parameters (α, ρ, ν, with β fixed) are fit per-expiration. The optimization is typically Levenberg-Marquardt on the log-IV space (more stable than fitting prices directly). The platform calibrates daily after market close and exposes the resulting parameter sets, the residual error per option, and the implied IV surface from the calibrated model; this lets users see where each model is fitting and where it's straining.

Jump-Diffusion Models - Merton, Kou & Bates

Last reviewed: May 17, 2026 by Options Analysis Suite Research.

What Is Jump Diffusion?

Jump diffusion models extend Black-Scholes by adding a jump component on top of the continuous Brownian-motion price process. Where Black-Scholes assumes the underlying evolves smoothly with normally-distributed log-returns, jump diffusion allows for sudden discontinuous moves, captured by a Poisson process that fires at random times with a random jump size. This matches the empirical reality of equity markets, where earnings, FDA decisions, FOMC announcements, and tail events produce sharp gaps that no continuous model can replicate.

The original jump-diffusion formulation is Merton (1976), with the canonical extension to asymmetric jumps coming from Kou (2002). Bates (1996) combined Heston's stochastic volatility with Merton's jumps to give the SVJD (stochastic volatility jump diffusion) model used widely in quant equity desks. The platform exposes Merton, Kou, and Bates as three distinct jump-aware models.

The Merton Specification

The underlying follows dS/S = (μ − λκ)·dt + σ·dW + (Y − 1)·dN where dW is the Brownian shock, dN is a Poisson process with intensity λ, and Y is the random jump multiplier. Merton models ln(Y) as normal with mean μ_J and standard deviation σ_J, giving the model five parameters: σ (continuous vol), λ (jump frequency), μ_J (mean log-jump), σ_J (jump volatility), and the implied κ = E[Y − 1] (jump-compensator drift).

Kou's Asymmetric Jumps

Kou's variant replaces normal log-jumps with a double-exponential distribution: jumps are positive or negative with separate decay rates and a probability mix. This better captures the equity-market asymmetry where downside jumps tend to be larger and more frequent than upside jumps, and where a fat-tailed downside is the dominant feature of left-tail risk.

Bates SVJD

Bates (1996) layers Merton-style jumps on top of Heston's stochastic-volatility dynamics. Bates is the most parameter-rich model in the jump family: eight parameters joint-calibrate to capture term structure, smile curvature, and tail risk simultaneously. It's also the most computationally expensive; calibration is typically Fourier-based.

What Do Jump Models Capture?

Earnings and event jumps: discontinuous price moves Black-Scholes assigns zero probability.
Fat tails: both left and right, with controlled asymmetry under Kou.
Steep short-dated smiles: the smile shape produced by jumps at short tenors is closer to market-observed than what continuous models can produce.
Volatility-of-volatility correlation with jumps under Bates: the regime-change behaviour around tail events.

When Should You Use Jump Diffusion?

Pricing options around known jump events (earnings, FDA, scheduled macro releases).
Fitting steep short-dated smiles where Heston alone leaves residual error in the wings.
Modeling tail risk for risk management: VaR and ES estimates that take left-tail jumps seriously.
Research on regime detection: λ and σ_J shifts often precede observable volatility regime changes.

When Should You Not Use Jump Diffusion?

For fast pre-trade pricing where the Bates calibration overhead isn't justified; Heston or Black-Scholes are cheaper and often "close enough" mid-tenor.
For path-dependent exotics where the jump-process simulation needs careful tuning to converge; Monte Carlo with jumps requires more paths than a continuous model.
For instruments without a meaningful jump risk (very-low-vol commodity baskets, idiosyncratically smooth underlyings).

Calibration Considerations for Jump Models

Calibrating jump models presents distinct challenges relative to continuous-vol models. The jump intensity λ and the jump-size distribution parameters trade off against each other in ways that produce non-unique calibrations: a high-intensity small-jump model can produce similar prices to a low-intensity large-jump model. Bound-constrained optimization with economically-sensible parameter ranges is essential. Calibration is typically done via Fourier-based pricing (Carr-Madan FFT or Fourier-COS) because jump models have closed-form characteristic functions even when they lack closed-form prices. Recalibration cadence matters: jump parameters can shift dramatically around event windows and during regime transitions, so daily nightly recalibration is the production-grade baseline.

The Jump Premium and What It Encodes

The price difference between a jump-aware model (Merton, Kou, Bates) and Black-Scholes on the same option is approximately the dollar value of the jump premium the market is paying. For OTM puts on a name with upcoming earnings, this premium can be substantial: the jump model values the tail risk that Black-Scholes ignores. Tracking the jump premium across tenors and strikes provides a direct readout of how the market is pricing event risk and tail-event probability. The platform's model-divergence view exposes this gap as a first-class output.

How OAS Uses Jump Models

The platform calibrates Merton, Kou, and Bates jump-diffusion variants and exposes them in the model selector alongside Black-Scholes, Heston, and the local-volatility / FFT / PDE engines. Around earnings, the model divergence view often shows the jump models pricing the front month materially differently from Black-Scholes; that gap is the jump premium the market is paying for tail protection. The platform's pre-earnings IV expansion screener pairs naturally with this view: stocks where jump premium is loading in the days before an event are exactly the names where the jump models pull furthest from the continuous baseline. Bates calibrations are exposed for users who want the full stochastic-vol-plus-jumps surface but at higher computational cost than Heston alone.

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Related Concepts

Heston (vs) · Variance Gamma (vs) · Black-Scholes · SABR · Local Volatility · Tail Risk · IV Crush · 0DTE Options · Volatility Smile · Volatility Skew · Risk-Neutral Density · Model Divergence · Implied Volatility · Leverage Effect · Variance Risk Premium · Vol of Vol · Realized Volatility · Monte Carlo · Jump Diffusion vs Variance Gamma · Calibration · Model Landscape

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As of the latest snapshot, AAPL has an ATM implied volatility of 23.4%, IV rank 37% (percentile 25%); 20-day realized vol 22.2%. 25-delta skew is +2.8%, meaning OTM puts trade richer than OTM calls. The IV here is the input that pricing-model walkthroughs (Black-Scholes, Heston, SABR, local-vol) take as their starting point: each model decomposes the same observed quote into different latent dynamics (constant vol, stochastic vol, surface-fitted vol, etc.) which is why two models can agree on price but disagree on Greeks and on how vol will evolve.