Poisson: events become price jumps
Disagreement about arrival rates determines both jump size and drift
In the normal model, the observation is a continuous return driver. Here it is an event count. The pricing measure and the asset measure attach different probabilities to the same sequence of arrivals; their likelihood ratio determines how the price responds to each arrival.
Two arrival rates
Let \(N(t)\) count events. Under measure \(\mathbb P^j\), for \(j\in\{X,Y\}\), use a Poisson process with intensity \(\lambda_j>0\):
\[ N(t)\sim\operatorname{Poisson}(\lambda_jt). \]
The mass assigned to \(n\) events is
\[ f_j(n,t)=e^{-\lambda_jt}\frac{(\lambda_jt)^n}{n!}. \]
The factorial and the common power of \(t\) cancel when the two masses are divided:
\[ \begin{aligned} \frac{X(t)}{X(0)} &=\left(\frac{\lambda_X}{\lambda_Y}\right)^{N(t)} e^{(\lambda_Y-\lambda_X)t},\\ \log\frac{X(t)}{X(0)}&=aN(t)+bt, \end{aligned} \]
\[ a=\log\frac{\lambda_X}{\lambda_Y}, \qquad b=\lambda_Y-\lambda_X. \]
This is again linear information flow. The count is sufficient for the likelihood ratio: with constant intensities, the precise arrival times carry no further likelihood information once the horizon and total count are known.
Explore arrival frequency and price response
Choose a pricing intensity, an information rate and a jump direction. The figure solves for \(\lambda_X\) from those choices. The intensity ratio, and hence the size of a price jump, is derived rather than chosen independently.
The interactive model requires JavaScript. Its default pricing intensity is \(\lambda_Y=3\) and the information rate is \(k=0.05\) nats per time unit; the asset intensity is chosen on the downward-jump branch.
Read the path from the two rates
At an arrival, \(N\) increases by one and the price is multiplied by
\[ \frac{X(t)}{X(t-)}=\frac{\lambda_X}{\lambda_Y}. \]
Between arrivals, log price changes at the deterministic rate \(b=\lambda_Y-\lambda_X\).
| Rate comparison | At an event | Between events |
|---|---|---|
| \(\lambda_X>\lambda_Y\) | Price jumps up. | Price drifts down. |
| \(\lambda_X<\lambda_Y\) | Price jumps down. | Price drifts up. |
| \(\lambda_X=\lambda_Y\) | No price response. | Price is constant. |
An event that the asset measure regards as more likely than the pricing measure favors the asset measure, so its likelihood-ratio price rises. The drift compensates for this jump response and preserves the pricing-measure martingale identity.
If a local dynamics description is useful, the same process satisfies
\[ \begin{aligned} \frac{dX(t)}{X(t-)} &=(r-1)\{dN(t)-\lambda_Ydt\},\\ r&=\frac{\lambda_X}{\lambda_Y}. \end{aligned} \]
Under \(\mathbb P^Y\), the expression in braces is a compensated Poisson increment. This equation is a consequence of the likelihood specification.
The two information rates need not match
Define \(k_X\) as the forward rate and \(k_Y\) as the reverse rate:
\[ \begin{aligned} k_X&=\lambda_X\log\frac{\lambda_X}{\lambda_Y} +\lambda_Y-\lambda_X,\\ k_Y&=\lambda_Y\log\frac{\lambda_Y}{\lambda_X} +\lambda_X-\lambda_Y. \end{aligned} \]
Then
\[ \begin{aligned} E^X\!\left[\log\frac{X(t)}{X(0)}\right]&=k_Xt,\\ E^Y\!\left[\log\frac{X(t)}{X(0)}\right]&=-k_Yt. \end{aligned} \]
Both rates are nonnegative, and both vanish only when the intensities coincide. Unlike the normal example, the divergences are generally asymmetric. The widget controls \(k_X\) and displays the resulting \(k_Y\).
Exact at every horizon
Independent counts add: Poisson laws with means \(\lambda_js\) and \(\lambda_jt\) sum to a Poisson law with mean \(\lambda_j(s+t)\). The same model therefore works at daily, hourly or finer horizons. In the figure, exponential waiting times produce actual arrival times and the finite sequence of jumps, with the exact drift between them.
Continue: the gamma model replaces a count of isolated events with accumulated positive activity, generated by infinitely many small jumps.
Jan Vecer, Principles of Portfolio Choice: An Information-Theoretic, Likelihood-Based Perspective, Chapter 5, “Poisson family (counting driver).” The price is measured in units of \(Y\). The asset measure \(\mathbb P^X\) is a subjective investor model only when the investor chooses \(\mathbb P^M=\mathbb P^X\).