Both, Actually: Bates Combines Jumps and Stochastic Volatility

By the mid-1990s there were two competing explanations for why option prices refuse to match Black-Scholes.

Explanation one: volatility is random. Hull and White proposed it, Heston made it tractable. Volatility wanders, mean-reverts, and correlates negatively with returns, which produces fat tails and a skew.

Explanation two: prices jump. Merton proposed it. Prices gap on news, which also produces fat tails and, with asymmetric jumps, a skew.

Both stories fit the broad facts. So which is it? David Bates set out to answer that question empirically, using the deutsche mark currency options market, and his answer was: neither one alone is sufficient. You need both. The model he built to prove it, now universally called the Bates model or the stochastic volatility jump diffusion (SVJD), became a standard.

The problem: a diagnostic test that separates the two stories

The two explanations look interchangeable if you only look at one maturity. Bates's contribution was to notice that they make different predictions about how the smile behaves across time to expiry, and that this gives you a way to tell them apart.

Here is the argument, and it is the intellectual heart of the paper.

Stochastic volatility is a slow effect. Volatility drifts around, but it drifts continuously. Over a very short horizon, a few days, volatility barely has time to move at all. So over short horizons a stochastic volatility model behaves almost exactly like Black-Scholes with a fixed volatility: nearly normal returns, thin tails, and therefore a nearly flat smile. It takes months for the randomness in volatility to accumulate into meaningfully fat tails. Stochastic volatility generates a smile that is shallow at short maturities and deepens with time.

Jumps are a fast effect. A jump can happen tomorrow. It does not need time to accumulate. So a jump model produces a very steep, sharply curved smile at short maturities, where a single gap would completely dominate the modest diffusive wiggle. But as maturity lengthens, the central limit theorem starts to bite: over a long horizon, many jumps and lots of diffusion average out toward normality, so the jump-induced smile flattens with time.

The two effects therefore have opposite maturity signatures. And that means you can look at the real market and see which is present.

The key idea via analogy: two kinds of surprise

Think of the exchange rate as being buffeted by two kinds of shock.

Weather: conditions gradually get stormier or calmer. This is stochastic volatility. It changes the character of the market for weeks at a time. It is why a currency can be quiet for months and then enter a period of sustained agitation.

Earthquakes: sudden, unpredictable, discrete. A central bank intervenes. A peg breaks. This is the jump component. It does not build up gradually and it does not announce itself.

Bates simply put both into one model: Heston's mean-reverting stochastic variance for the weather, and Merton's Poisson jumps for the earthquakes, with the jumps allowed to be asymmetric (a bias toward gapping in one direction).

Crucially, he kept the model within the class where a characteristic function can be written down explicitly, which meant it stayed fast enough to calibrate. He also developed methods to price American options under it, which mattered because the deutsche mark options he studied were American-style.

What the data said

Bates estimated the model and its restricted sub-models (jumps only, stochastic volatility only) on deutsche mark options over 1984 to 1991, and tested them against futures prices and the observed path of implied volatility.

The key finding was a rejection. Stochastic volatility alone could not explain the smile evidence, except under parameter values that were implausible given how implied volatilities actually behave over time. To make a pure stochastic volatility model produce enough excess kurtosis, the fat tails the option market is clearly pricing, you have to crank the volatility-of-volatility to a level that would make implied volatility itself far more erratic than it is observed to be. The model can fit the option prices or it can match the volatility time series, but not both.

That is a genuinely powerful form of evidence. It is not "the fit is a bit worse." It is: the parameters required to fit the cross-section of option prices are inconsistent with the time series of the very quantity the model is modelling. The model is being asked to be two different things at once, and it cannot.

Jumps supply the missing ingredient, because they generate excess kurtosis immediately, without requiring volatility itself to be wild.

Why it mattered

• It settled a real debate with evidence, not preference. The jumps-versus-stochastic-volatility argument could have run forever on aesthetic grounds. Bates gave it a decisive empirical test with a clear diagnostic, the maturity structure of the smile, and an answer.
• The Bates model became a workhorse. Heston plus Merton jumps is now a standard production model, particularly where short-dated options matter. It fits the short end (jumps) and the long end (stochastic volatility) simultaneously, which neither parent can do.
• It fed directly into the affine framework. Duffie, Pan and Singleton generalised the Bates structure into a full affine jump-diffusion theory in 2000, which is the umbrella under which Black-Scholes, Heston, Merton, Bates and many others all sit as special cases, all sharing one computational engine.
• It explained the steep short-dated smile. The single most stubborn failure of pure Heston is that it cannot produce a steep enough smile at short maturities. Bates diagnosed exactly why, and supplied the fix. That fix is why practitioners routinely add jumps rather than simply cranking the vol-of-vol.
• It applies far beyond currencies. The same diagnostic and the same conclusion hold in equity index options, where the short-dated skew is even more dramatic and even less explicable without jumps.

The honest limitations

• The model has a lot of parameters, and they fight each other. Heston's five, plus jump intensity, mean jump size and jump volatility. Eight or more parameters fitted to a noisy surface. The result is that different parameter combinations fit almost equally well, and the fitted values are unstable from day to day. That instability is not a cosmetic problem: your hedges depend on those numbers.
• Fitting well is not the same as being right. With enough parameters you can fit almost any surface, which weakens the evidence that the mechanism is correct. A good fit is necessary, not sufficient.
• Jump risk cannot be hedged, so its price is an assumption. As in Merton, the market is incomplete. Bates must assume something about how the market compensates jump risk, and the estimated jump risk premium is large and not well identified. That means part of the model's fit is being bought with a parameter nobody can independently verify.
• Jumps do not arrive at a constant rate. The Poisson assumption says crashes are equally likely on any given day. In reality they cluster: turbulence breeds turbulence. Self-exciting (Hawkes) processes were developed to address exactly this.
• Volatility can jump too. Bates lets the price jump, but volatility moves continuously. In a real crisis, implied volatility itself gaps. Later models add jumps to the variance process.
• The results are from one market and one era. Deutsche mark options in the 1980s are a specific market with specific institutional features, notably central bank intervention, which is a jump generator. The conclusions have held up broadly, but the original evidence is narrower than its influence.

The one-line takeaway

Bates showed that jumps and stochastic volatility leave different fingerprints on how the smile changes with maturity, that the real market shows both fingerprints, and that a pure stochastic volatility model can only fit option prices by adopting a volatility dynamic contradicted by the volatility data itself, which is why the standard production model became Heston plus jumps rather than either one alone.