Variance Gamma from zero
1/5Time itself is random
Variance Gamma starts from a radical idea: instead of adding jumps to diffusion, make time itself stochastic. Brownian motion runs on a random clock.
Ordinary Brownian motion uses calendar time: one second per second, relentlessly uniform. VG says the market has its own internal clock — a gamma process G(t) — that sometimes races ahead and sometimes crawls. When the clock runs fast, Brownian motion gets more “effective time” and makes big moves. When the clock idles, the price barely moves.
The result: fat tails emerge naturally from the randomness of the clock, without explicitly specifying a jump size distribution. Fast clock periods create clusters of large moves. Slow periods create eerie calm. This matches what thin crypto order books actually look like — long stretches of nothing, then sudden bursts of activity.
G(t) — gamma process with mean rate 1 and variance rate ν. This is the random clock.
θ — drift inside the clock (creates skew).
σ — diffusion vol inside the clock.
Below, the top panel shows the gamma process G(t) — the random clock. The dashed line is calendar time (the straight diagonal). When G(t) jumps above the diagonal, time is running fast. The bottom panel shows the resulting VG process — Brownian motion evaluated at the random time G(t).
Crank up ν to make the clock more erratic. Watch how the VG process gets wilder — bigger moves, more clustering. That is the fat-tail mechanism.
Think of a movie with variable playback speed. Some scenes play in slow motion (quiet market). Some scenes fast-forward (panic selling, liquidation cascades). The underlying film is ordinary Brownian motion. The speed control is the gamma process. What the audience sees — the VG process — has all the drama of the speed changes baked in.
The three parameters
VG has the cleanest parameter interpretation of any smile model. Each parameter maps to exactly one statistical moment. No redundancy, no correlation headaches.
σ (sigma) — diffusion vol. The volatility of Brownian motion inside the random clock. Controls the overall level of the smile. Higher σ lifts everything. This is the analog of Black-Scholes vol.
θ (theta) — drift in subordinated BM. Controls skew. If θ < 0, the process drifts downward inside the random clock, and the smile tilts — put wing steeper than call wing. If θ = 0, the smile is symmetric.
ν (nu) — variance of gamma time. Controls excess kurtosis (tail fatness). Higher ν makes the clock more random, which creates fatter tails and steeper wings on both sides. This is the parameter that separates VG from Black-Scholes.
Three experiments:
1. Set θ = 0, ν = 0.01. Nearly flat smile — close to Black-Scholes. The clock is almost deterministic.
2. Set θ = −0.15, ν = 0.20. Negative skew with moderate kurtosis. Classic crypto smile shape.
3. Set θ = 0, ν = 0.50. Symmetric but extreme kurtosis. Both wings fly up. “Black swan regime.”
σ → variance (2nd moment). θ → skewness (3rd moment). ν → excess kurtosis (4th moment). This is the cleanest separation of smile shape in any jump or stochastic vol model. Heston has 5 parameters with correlations between them. VG has 3 orthogonal controls.
It's actually a pure jump process
Despite looking like time-changed Brownian motion (smooth + stretched), VG paths are technically pure jump. Every move is a jump. There is no continuous diffusion component in calendar time.
This is philosophically different from Merton. In Merton, the price moves smoothly most of the time (diffusion), with occasional big jumps. In VG, all movement is discontinuous. The process has infinite activity (infinitely many jumps in any interval) but finite variation (the total jump size is bounded).
Most of those jumps are tiny. A few are large. In the limit of many tiny jumps, the path looks almost continuous — it is approximated well by a smooth curve. But zoom in close enough and every move is technically a jump. No two adjacent prices are connected by a continuous path.
The left panel shows a VG path drawn as a step function — every time step is a distinct jump. The right panel shows a Merton path with smooth diffusion between rare big jumps (red bars). Hit Regenerate and compare:
VG: constant small jumps, occasionally large ones. No smooth sections. The path wiggles everywhere.
Merton: long smooth stretches interrupted by sudden vertical jumps. Two clearly distinct regimes (calm vs shock).
In a pure-jump world, delta-hedging is imperfect by construction — you cannot trade continuously because the price itself is discontinuous. This is actually more honest than Merton, which claims you can hedge the diffusion part perfectly and only the rare jumps are unhedgeable. In thin crypto order books, every fill is effectively a jump. VG acknowledges that reality.
The characteristic function
VG has a clean, closed-form characteristic function. This is what makes Fourier pricing practical — you can price European options fast and exactly without Monte Carlo.
σ enters via the u² term (variance contribution).
θ enters via the iu term (skew via imaginary part).
ν enters via the exponent −T/ν and in the base (kurtosis).
When ν → 0: the exponent → −∞, and the CF converges to the BS lognormal CF. VG nests BS as a limiting case.
The pricing workflow: take this CF, plug it into the Carr-Madan (1999) formula or the COS method, and apply a Fast Fourier Transform. You get option prices across all strikes in one shot — no per-strike computation, no simulation noise.
The exponent −T/ν is negative and gets more negative as T grows. This means the CF decays faster for longer maturities, which corresponds to the VG smile flattening with time. The clock randomness averages out over long horizons — a natural term structure effect.
VG in practice
VG is not the industry default — Bates (Heston + jumps) dominates equity and crypto desks. But VG's subordination idea shows up everywhere, and the model has specific niches.
Credit derivatives: VG was originally popular in credit modeling. Default is a jump event. VG's pure-jump nature handles discontinuous payoffs cleanly. Madan, Carr, and Chang (1998) introduced VG partly with credit in mind.
Equity exotics with simple smile requirements: When you need a 3-parameter smile fit with clear moment interpretation, VG is hard to beat. Calibration is fast because each parameter has an unambiguous effect.
Crypto on thin pairs: illiquid crypto pairs do not diffuse smoothly — they gap from one price to another as orders fill. VG's pure-jump character is a more honest description of that price action than any diffusion model.
The subordination idea: the concept of replacing calendar time with a random clock is foundational. It appears in stochastic clocks, business time models, activity-based models, and CGMY (a generalization of VG). Even if you never price a VG option, understanding time changes makes every other model clearer.
Black-Scholes: flat smile. Continuous paths. 1 parameter.
Merton: smile from rare big jumps. Smooth diffusion + Poisson jumps. 4 parameters.
Kou: smile from asymmetric jumps. Independent wing control. 5 parameters.
Variance Gamma: smile from a random clock. Pure jump, no diffusion. 3 parameters, one per moment.
Heston: smile from stochastic vol. Continuous paths. 5 parameters.
Bates: Heston + Merton jumps. The workhorse. 8 parameters.
Where to go next:
Merton Jump-Diffusion — diffusion + rare big jumps
Kou Jump-Diffusion — asymmetric jumps with independent wings
Heston Model — stochastic vol, the other approach to smiles
Bates Model — Heston + jumps: the industry workhorse