Kou Jump-Diffusion from zero
1/5Merton's jumps are too symmetric
Merton uses lognormal jumps. The jump size distribution is a single bell curve, centered somewhere. Up-jumps and down-jumps are drawn from the same family. That is a problem.
Real crashes are sharper than rallies. A stablecoin depeg does not look like a mirror image of a short squeeze. The -20% gap happens in a single block. The +20% rally takes a week. You need a model where the left tail and the right tail are controlled separately.
Kou (2002) fixes this by replacing the lognormal jump distribution with a double exponential. Up-jumps decay at one rate. Down-jumps decay at a different rate. Two separate knobs for two separate tails.
In Merton: ln(J) ~ Normal(μJ, σJ²).
In Kou: the jump size Y = ln(J) follows a double exponential with separate decay rates for positive and negative values.
The practical consequence: in Merton, when you steepen the left wing of the smile (by making μJ more negative), you also drag the right wing around. The normal distribution is symmetric about its mean. Kou decouples the wings entirely.
The double exponential
The jump size Y has a density made of two exponential halves spliced together at zero. Each half decays at its own rate. This is the core innovation.
f(y) = (1−p)·η₂·eη₂y for y < 0 (down-jumps)
η₂ controls down-jump decay. Small η₂ means down-jumps can be large (fat left tail). Mean down-jump = 1/η₂.
p is the probability that a given jump is upward.
Slide the parameters below and watch the density change. The key experiment: set η₁ much larger than η₂. The right tail (up-jumps) becomes thin and concentrated near zero, while the left tail (down-jumps) extends far out. That is the shape of crash risk.
Three experiments to try:
1. Set η₁ = η₂ = 5, p = 0.5. The density is symmetric. Both tails are identical. This is equivalent in spirit to Merton with zero mean jump.
2. Set η₁ = 10, η₂ = 2, p = 0.3. Fat left tail, thin right tail, most jumps go down. Classic crash regime.
3. Crank p toward 0.9. Most jumps go up, but the down-jumps that do happen are still governed by η₂ independently.
Why asymmetric jumps matter
The ratio of η₁ to η₂ and the parameter p together control the skew of the implied vol smile. Crucially, they control each wing independently.
Consider a crypto token. A depeg crash is sharp and deep — that means a small η₂ (fat left tail). Normal upward price action is incremental — that means a large η₁ (thin right tail). The resulting smile has a steep put wing and a gentle call wing. Exactly what you see in the market.
In the explorer below, watch how changing η₂ alone steepens the left wing without moving the right wing. Then try changing η₁ — it steepens the right wing independently. This is Kou's practical edge: you fit each wing to the market separately.
Why p matters for skew: if p = 0.3 (most jumps are downward), the left wing inflates because OTM puts are seeing a steady stream of downward jump risk. The right wing is quieter — fewer jumps land there.
Why η ratio matters for skew: even with p = 0.5 (equal jump probability), if η₂ is much smaller than η₁, the down-jumps are on average much larger. That lifts the put wing because the same number of down-jumps covers more ground per jump.
Closed-form advantage
The exponential distribution has a special property: it is memoryless. If you know a jump exceeds some barrier x, the overshoot (jump − x) has the exact same distribution as a fresh jump. This is what gives Kou closed-form barrier prices.
Think about what a barrier option needs: you need to know the distribution of where the price lands after it crosses the barrier. With Gaussian jumps (Merton), the overshoot distribution is a mess — it depends on how far past the barrier you went. With exponential jumps, the overshoot is memoryless: the conditional distribution given you crossed the barrier is the same as the unconditional distribution. This makes the math tractable.
The result: Kou (2004) derived closed-form solutions for knock-in/knock-out barriers, lookback options, and perpetual Americans. Merton has no such formulas. If you price exotics and need analytical Greeks, Kou wins.
The left panel shows the full exponential density with a threshold x marked. The shaded region is the probability of exceeding x. The right panel shows the conditional density of the excess (Y − x), given Y > x. Slide the threshold around: the conditional density is always the same shape as the original. That is the memoryless property.
Move η and notice both panels rescale identically. The shape of the excess never depends on where you set the threshold. For barrier pricing, this means the overshoot distribution at the barrier is known analytically — no simulation needed.
Kou vs Merton vs Heston
Each model has a role. Understanding where Kou fits relative to Merton and Heston is the final piece.
Kou: asymmetric jumps, independent wing control, closed-form exotics. Best for markets with pronounced crash asymmetry (crypto, single-name equity) and when you need analytical barrier or lookback prices.
Merton: simpler, symmetric jumps. Fewer parameters. Good enough when the smile is roughly symmetric or when you only price vanillas. The industry starting point for jump models.
Heston: stochastic vol, no jumps. Generates skew via vol-spot correlation (ρ). Dominates at long maturities where vol-of-vol drives the term structure. Cannot produce the short-dated wing steepness that jumps create.
The chart above overlays Kou and Merton smiles with the same total jump variance. Both models add the same amount of jump risk in aggregate, but Kou allocates more of it to the left tail. Notice how Kou's left wing is fatter (steeper put wing) while its right wing is thinner. Merton splits the risk more evenly.
Black-Scholes: flat smile. No skew, no wings.
Merton: smile with wings. Symmetric jump distribution means both wings move together. Good for short-dated vanillas.
Kou: smile with independent wings. Asymmetric jump distribution. Closed-form barriers and lookbacks. Better crypto fit.
Heston: smile from stochastic vol. Persists at long maturities. No jumps, so short-dated wings are too flat.
Bates: Heston + Merton jumps. The workhorse. For the most demanding applications, replace the Merton jump component with Kou-style double exponential jumps.
Where to go next:
Merton Jump-Diffusion — the symmetric-jump predecessor
Variance Gamma — a pure-jump model with no diffusion at all
Heston Model — stochastic vol, no jumps
Bates Model — Heston + jumps: the industry workhorse