# Jump-Diffusion Option Model: Pricing Discontinuous Moves

Understand how Merton jump diffusion adds random gaps to continuous returns, how jump intensity and size affect tail options, and why jump risk cannot be continuously Delta-hedged away.

Canonical: https://wiki.fcontext.com/options/jump-diffusion-option-model/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

A **jump-diffusion model** supplements continuous price diffusion with randomly timed discontinuous moves. In Merton's model, a Poisson process determines jump arrivals and a separate distribution determines jump sizes. This gives short-dated and far-out-of-the-money options more tail probability than a constant-volatility geometric Brownian motion can produce. It can represent earnings gaps, takeover news, defaults, policy surprises, or crises more realistically, but it remains a model. Jump timing and magnitude are not known in advance, so continuous stock trading cannot perfectly replicate and hedge the option.

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## Model mechanics and pricing meaning

A common specification is `dS_t/S_{t-}=(μ-λκ)dt+σdW_t+(J-1)dN_t`. Here `σ` is diffusion volatility, `N_t` is a Poisson process with annual intensity `λ`, `J` is the multiplicative jump, and `κ=E[J-1]`. The compensator `-λκ` prevents the jump component from silently changing the intended expected drift. In risk-neutral pricing, the drift and jump distribution must be adjusted or calibrated consistently with traded prices.

Conditional on `n` jumps, the terminal log return is often normal under a lognormal-jump assumption. A European option can then be valued as a Poisson-weighted mixture of Black-Scholes-like conditional prices. More jump intensity, larger jump dispersion, or more negative downside jumps generally raises tail-option values and changes skew, but parameters can trade off against diffusion volatility and are not uniquely identified by a small option sample.

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## A 30-day jump calculation

Take `λ=4` jumps per year and `T=30/365` year. Expected jumps are `λT=0.329` and the Poisson probability of at least one jump is `1-e^(-λT)=28.0%`. This is the model probability under the selected measure—not a statement that four visible crashes occur every year.

Let diffusion volatility be `σ=20%`, mean log jump `m=-5%`, and jump-size standard deviation `δ=10%`. A simple annual log-return variance proxy is:

`σ²+λ(δ²+m²)=0.20²+4(0.10²+0.05²)=0.09`,

whose square root is `30%`. The jump compensator is `κ=e^(m+δ²/2)-1=e^(-0.045)-1≈-4.40%`. These calculations explain parameter scale; `30%` is not automatically an option's implied volatility, and the full price requires measure-consistent discounting and payoff integration.

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## Calibration and risk checklist

- State whether parameters are historical or risk-neutral. Physical jump frequency cannot be inserted into a pricing formula without a risk-premium assumption.
- Calibrate across strikes and expiries using synchronized executable quotes, forwards, rates, dividends, and contract terms.
- Test parameter identifiability: diffusion volatility, jump intensity, mean jump, and jump dispersion can produce similar smiles.
- Inspect fit outside the calibration set and across dates; an excellent in-sample fit can be unstable out of sample.
- Preserve no-arbitrage bounds and check numerical convergence of Poisson truncation, integration, simulation, or transform methods.
- Stress clustered jumps and state-dependent intensity. A constant independent Poisson process cannot capture every crisis or earnings calendar.
- Separate symmetric jump dispersion from negative jump asymmetry; they affect puts, calls, and skew differently.
- Delta hedging cannot trade during an instantaneous jump. Report gap P&L and residual jump Greeks rather than claiming a complete hedge.
- Include Bid/Ask, liquidity loss, margin, borrow, early exercise, dividends, and settlement when comparing model value with a trade.
- Use simpler Black-Scholes and observed intrinsic/no-arbitrage bounds as diagnostics before trusting a more flexible model.

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## Common misconceptions

- “Jump diffusion predicts the next gap.” It specifies a distribution, not timing or direction of a particular event.
- “Adding jumps always improves forecasts.” Extra parameters can overfit option noise and unstable regimes.
- “Lambda is the number of crashes per year.” It counts modeled jumps under a stated measure and threshold convention.
- “The 30% variance proxy is the option IV.” Smile, maturity, risk premium, and payoff integration remain.
- “More jumps only raise put prices.” Positive and negative jumps can raise both tails; asymmetry determines skew.
- “Delta hedging removes jump risk.” A discontinuous move occurs before a continuous hedge can rebalance.
- “Historical jump estimates equal pricing parameters.” Risk-neutral prices incorporate compensation for bearing jump risk.
- “Merton jump diffusion explains every smile.” Stochastic volatility, local volatility, liquidity, and multiple jump processes may also matter.

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## Related topics

- [Black-Scholes model](/options/black-scholes-model/)
- [Geometric Brownian motion](/options/geometric-brownian-motion/)
- [Heston model](/options/heston-model/)
- [Gap options](/options/gap-options/)

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## Primary and academic sources

- [Merton: Option Pricing When Underlying Stock Returns Are Discontinuous](https://doi.org/10.1016/0304-405X(76)90022-2)
- [Black and Scholes: The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062)
- [OCC: Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document)
- [Cboe: VIX Methodology](https://cdn.cboe.com/api/global/us_indices/governance/VIX_Methodology.pdf)