When should I use a jump-diffusion model like Bates instead of pure stochastic vol?

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AcadiFi · Accepted Answer

Heston can't generate enough short-dated skew. Implied vol smiles for 1-week options are dramatically steeper than any diffusion-only model can produce. Jumps are the cure...

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Bates extends Heston by adding Poisson jumps: dS/S = (r - lambda*m) dt + sqrt(V) dW1 + J dN, where N is a Poisson process with intensity lambda and jumps J ~ N(mu_J, sigma_J^2). Six parameters now: Heston's five plus lambda, mu_J, sigma_J.

Why it matters: Dasha Pietroniewicz at Prague Structured Products prices weekly SPX options. The 1-week 90% put IV is 34% while ATM is 14% — a 20-vol skew. Heston alone needs unrealistic parameters to fit this. Bates with lambda=0.8 jumps/year and mu_J=-0.05 fits naturally.

Practical considerations:
1. Jumps are non-hedgeable — you can't replicate them with continuous trading
2. This creates genuine incompleteness; Bates is a jump-diffusion, not arbitrage-free in the hedging sense
3. Calibration adds parameters that are often poorly identified; short-dated data is crucial
4. For risk management, tail scenarios need explicit jump simulation

Beyond Bates, practitioners use Merton jump-diffusion (no stochastic vol, just jumps), CGMY / variance gamma (pure jump Levy processes), and rough volatility models (Bayer-Friz-Gatheral) which use fractional Brownian motion with Hurst exponent ~0.1 to generate steep short-dated skew without jumps. The right choice depends on what you're pricing — weekly options need jumps, long-dated need stochastic vol, barriers need LSV.

When should I use a jump-diffusion model like Bates instead of pure stochastic vol?

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