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.

For educational purposes only; not investment advice. Investing may result in loss.

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 rather than a forecast. Jump timing and magnitude are not known in advance, so continuous stock trading cannot perfectly replicate and hedge the option.

Model mechanics and pricing meaning

A common physical-measure 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, jump intensity, and jump distribution must instead be adjusted or calibrated consistently with traded prices and the chosen pricing measure.

Conditional on n jumps, the terminal log return is 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.

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%. If log jump size is Y~N(m,δ²), 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.

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.

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.