Gamma: accumulated activity
An increasing driver, infinitely many jumps, one simple likelihood ratio
The Poisson model counts isolated events. A gamma driver records a nonnegative amount of accumulated activity. It could represent an abstract operational clock or accumulated volume; the financial interpretation requires a model linking that quantity to the application.
The driver increases by jumps. Although its value at any fixed positive time has a continuous density, its paths are not continuous: a gamma process has infinitely many small jumps on every positive time interval.
Common shape, different rates
Specify one additive driver \(G(t)\) under two measures:
\[ \begin{aligned} G(t)&\sim\Gamma(\alpha t,\beta_X) &&\text{under }\mathbb P^X,\\ G(t)&\sim\Gamma(\alpha t,\beta_Y) &&\text{under }\mathbb P^Y. \end{aligned} \]
Here \(\alpha>0\) is the common shape rate. The parameters \(\beta_X,\beta_Y>0\) are rates, so the means and variances are
\[ E^j[G(t)]=\frac{\alpha t}{\beta_j}, \qquad \operatorname{Var}^j[G(t)]=\frac{\alpha t}{\beta_j^2}. \]
For \(z>0\) and \(j\in\{X,Y\}\), the density is
\[ f_j(z,t)=\frac{\beta_j^{\alpha t}}{\Gamma(\alpha t)} z^{\alpha t-1}e^{-\beta_jz}. \]
The common power of \(z\) and the gamma-function factor cancel. The resulting price is
\[ \begin{aligned} \frac{X(t)}{X(0)} &=\left(\frac{\beta_X}{\beta_Y}\right)^{\alpha t} e^{(\beta_Y-\beta_X)G(t)},\\ \log\frac{X(t)}{X(0)}&=aG(t)+bt, \end{aligned} \]
\[ a=\beta_Y-\beta_X, \qquad b=\alpha\log\frac{\beta_X}{\beta_Y}. \]
The common shape is essential for this affine expression. Different shape rates would leave an additional \(\log G(t)\) term.
Explore the model
The figure fixes \(\beta_Y=2\). Choose the common shape rate, forward information rate and jump direction; the corresponding \(\beta_X\) is then determined. The displayed horizon is at least \(0.5\) and \(\alpha\ge2\), keeping the horizon density finite at zero. This is an exploration range, not a restriction on the mathematical gamma family.
The interactive model requires JavaScript. The default common shape rate is \(\alpha=3\), the pricing rate is \(\beta_Y=2\) and the forward information rate is \(k=0.05\) nats per time unit.
Driver jumps become price jumps
For a driver jump of size \(\Delta G(t)>0\),
\[ \frac{X(t)}{X(t-)}=e^{a\Delta G(t)}. \]
If \(\beta_X>\beta_Y\), the asset measure favors smaller amounts of activity. An upward jump in observed activity then produces a downward price jump, while \(b>0\) supplies the compensating deterministic component. If \(\beta_X<\beta_Y\), the price jumps upward and \(b<0\).
There is no positive interval free of all gamma jumps. The decomposition \(aG(t)+bt\) still separates the jump contribution from the deterministic contribution, but it should not be described as a succession of isolated jumps with empty waiting periods.
The jump intensity density under measure \(\mathbb P^j\) is
\[ \nu_j(dz)=\alpha z^{-1}e^{-\beta_jz}\,dz, \qquad z>0. \]
Its integral diverges near zero, giving infinite activity. The integral of jump size against this density is finite, so total accumulated jump size is finite over bounded intervals.
Information accumulated per unit time
Taking the mean of the affine log likelihood gives the divergence between the two observation laws at horizon \(t\):
\[ \begin{aligned} D(\mathbb P^X\Vert\mathbb P^Y)&=k_Xt,\\ k_X&=\alpha\left[ \log\frac{\beta_X}{\beta_Y} -1+\frac{\beta_Y}{\beta_X}\right]. \end{aligned} \]
Reversing the two measures gives
\[ k_Y=\alpha\left[ \log\frac{\beta_Y}{\beta_X} -1+\frac{\beta_X}{\beta_Y}\right]. \]
The expected log-price slopes are \(+k_X\) under \(\mathbb P^X\) and \(-k_Y\) under \(\mathbb P^Y\). The price ratio itself remains a mean-one martingale under \(\mathbb P^Y\).
From discrete blocks to an additive clock
The negative-binomial model sums independent geometric blocks. Gamma laws supply a continuous-valued analogue: independent gamma variables with the same rate add their shape parameters,
\[ \begin{gathered} \Gamma(\alpha s,\beta)+\Gamma(\alpha t,\beta)\\ \overset{d}{=}\ \Gamma(\alpha(s+t),\beta). \end{gathered} \]
For integer shape, a gamma variable is a sum of exponential waiting times. Positive noninteger shapes extend that family consistently to every time increment. The simulation uses these exact increment distributions; the connecting segments do not reproduce the infinitely many individual jumps.
Continue: the inverse Gaussian model gives another additive positive clock, with a different density and a stronger small-jump singularity.
Jan Vecer, Principles of Portfolio Choice: An Information-Theoretic, Likelihood-Based Perspective, Chapter 5, “Gamma family.” Shape–rate convention: \(\Gamma(s,\beta)\) has mean \(s/\beta\). All prices are in units of \(Y\); \(\mathbb P^X\) and an investor’s \(\mathbb P^M\) are distinct unless explicitly identified.