Why diffusion is not enough

In Black-Scholes the spot moves continuously: over a short interval it can only travel a distance proportional to \(\sigma\sqrt{\Delta t}\). A one-week option struck twenty percent away from the spot is therefore almost worthless. Markets disagree: such options trade at prices that imply very high volatility, because earnings surprises, central-bank decisions and crashes move prices in an instant. Stochastic volatility cannot fix this, since variance needs time to change. Jumps can.

The model

Merton (1976) adds a compound Poisson process to the diffusion:

$$ \frac{dS_t}{S_{t^-}} = (r - q - \lambda k)\, dt + \sigma\, dW_t + (e^{J} - 1)\, dN_t $$

• \(N_t\) is a Poisson process: jumps arrive at rate \(\lambda\) per year, so \(\lambda T\) are expected over the option's life.
• Each jump multiplies the spot by \(e^{J}\) with \(J \sim N(\mu_J, \sigma_J^2)\). Negative \(\mu_J\) means crashes are more likely than melt-ups.
• \(k = E[e^{J}] - 1 = e^{\mu_J + \sigma_J^2/2} - 1\) is the mean proportional jump. Subtracting \(\lambda k\) from the drift compensates for it, so the discounted spot is still a martingale.

Pricing

Conditional on exactly \(n\) jumps having happened, the spot at expiry is lognormal again, just with more variance and a shifted drift. So the price is an average of Black-Scholes prices weighted by the Poisson probabilities of \(n\) jumps:

$$ V = \sum_{n=0}^{\infty} \frac{e^{-\lambda' T} (\lambda' T)^n}{n!}\; V_{BS}\!\left(\sigma_n, r_n\right), \qquad \lambda' = \lambda (1 + k), \quad \sigma_n^2 = \sigma^2 + \frac{n \sigma_J^2}{T}, \quad r_n = r - \lambda k + \frac{n \ln(1+k)}{T} $$

The series converges fast; a few dozen terms are enough. This is why the calculator can price and compute Greeks instantly, with no integral to evaluate.

What jumps do to the smile

Jumps fatten both tails at once, so they lift both wings of the smile. With \(\mu_J \lt 0\) the left tail is fatter and the smile becomes a skew. The signature of jumps is in the term structure: because a jump can happen tomorrow, the effect is strongest for short maturities and washes out over long ones, where the diffusion dominates. Compare with Heston, where the smile builds up with maturity. Real surfaces show both patterns, which is why the Bates model combines the two.

Simulation

Between jumps the model is Black-Scholes, so each time step is an exact lognormal move followed by a Poisson-distributed number of jumps, each a lognormal factor. The calculator's path chart makes the jumps visible as vertical gaps, and the simulated price matches the series to within its standard error.