Bates Model
Also known as: SVJ Model, Jump Diffusion
The Bates model (1996) combines stochastic volatility and jump diffusion to capture both the volatility smile and the implied volatility skew observed in equity and currency option markets. It extends the Heston model by adding a Poisson jump component to returns, making it suitable for pricing options when sudden price moves are expected.
Key highlights
- Captures both smile and skew: stochastic volatility + jumps explain more realistic option surfaces
- Jump-driven skew: naturally generates negative skew without ad-hoc modifications
- Flexibility: independent control over vol mean reversion, jump intensity, and jump size distribution
- Realistic price paths: incorporates sudden moves that match historical observations (e.g., 1987 crash, VIX spikes)
Intuition
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How it works
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When to use it
Use Bates when pricing equity or FX options where skew is pronounced and sudden moves are plausible (e.g., before earnings announcements or central bank decisions). The model is less suitable when jumps are extremely rare or when exotic path-dependent features (Asian, barrier) dominate pricing. It requires careful calibration and can be computationally expensive.
Strengths & limitations
- Captures both smile and skew: stochastic volatility + jumps explain more realistic option surfaces
- Jump-driven skew: naturally generates negative skew without ad-hoc modifications
- Flexibility: independent control over vol mean reversion, jump intensity, and jump size distribution
- Realistic price paths: incorporates sudden moves that match historical observations (e.g., 1987 crash, VIX spikes)
- Parameter proliferation: seven or more parameters to calibrate, leading to potential over-fitting and instability
- Jump probability specification: assuming Poisson jumps is simplistic; real jumps may cluster or have time-varying intensity
- Computational cost: Monte Carlo requires many simulations to resolve jump paths; PDE solvers are complex
- Jump size distribution: log-normal or normal assumption may not match tail behavior of real markets
Common pitfalls
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Applications
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Frequently asked
How does Bates differ from Heston?
Heston is pure stochastic volatility (Brownian motions for spot and vol). Bates adds a Poisson jump component to the spot process, introducing a third source of randomness. Bates generates skew in addition to smile; Heston alone generates primarily smile. Bates is more complex but fits market data better, especially for equity options.
What jump size distribution should I use?
Log-normal jumps (relative returns) are most common because they keep prices positive. Normal jumps (absolute) are simpler but can make prices negative. Empirically, jump sizes are often fat-tailed, so distributions like variance-gamma may fit better, but they increase computational complexity.
How do I calibrate Bates if I have only a few option prices?
Fix some parameters (e.g., jump intensity lambda, jump mean m) using historical realized jumps or market conventions, then optimize the remaining parameters (Heston parameters, jump variance) against available prices. Sensitivity analysis helps identify which parameters matter most for your problem.
Can Bates price American options accurately?
Yes, but it is computationally demanding. Jump-adapted PDE solvers and tree methods exist, but Monte Carlo with optimal stopping approximations (Longstaff-Schwartz) is more practical. Accuracy depends on proper jump discretization and convergence criteria.
Sources
- 1.Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies, 9(1), 69-107.
- 2.Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144.
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Cite this page
ScholarGate. (2026, June 3). Bates Model. ScholarGate. https://scholargate.app/quantitative-finance/bates-model