The negative-binomial model

The same binary observations, a new clock, and fewer trading opportunities

Start with the binomial example: the stock rises by \(u=3/2\) or falls by \(d=1/2\), with up probabilities \(1/2\) under the market law, \(3/4\) under the stock law, and \(2/3\) under investor beliefs.

One completed shock consists of \(G\) down moves followed by the first up move. For example, \(G=0\) means an immediate up move; \(G=2\) means down, down, up. The duration is \(\tau=G+1\) binary trials. This page changes the observation clock; the portfolio comparison also restricts trading to the boundaries between shocks.

Three laws for the same geometric shock

Let \(a^Y=1/2\), \(a^X=1/4\), and \(a^M=1/3\) denote the down probabilities. For \(g=0,1,2,\ldots\),

\[ \begin{gathered} \mathbb P^i(G=g)=(1-a^i)(a^i)^g,\\ i\in\{Y,X,M\}. \end{gathered} \]

Consequently, the stock return and desired likelihood payoff are

\[ R=\frac{\mathbb P^X(G)}{\mathbb P^Y(G)} =\frac32\left(\frac12\right)^G. \]

\[ L_M=\frac{\mathbb P^M(G)}{\mathbb P^Y(G)} =\frac43\left(\frac23\right)^G. \]

Down moves \(G\)	Duration \(\tau\)	Stock return \(R\)	Desired payoff \(L_M\)
\(0\)	\(1\)	\(3/2\)	\(4/3\)
\(1\)	\(2\)	\(3/4\)	\(8/9\)
\(2\)	\(3\)	\(3/8\)	\(16/27\)
\(g\)	\(g+1\)	\((3/2)(1/2)^g\)	\((4/3)(2/3)^g\)

All values are normalized by initial wealth and expressed in reference-asset units. The terminal stock return is positive in every state but can approach zero as \(G\) grows.

From geometric shocks to a negative-binomial count

Independent shocks \(G(1),\ldots,G(n)\) give the total number of down moves

\[ Z(n)=G(1)+\cdots+G(n). \]

Under each law,

\[ \begin{aligned} &\mathbb P^i(Z(n)=k)\\ &\quad=\binom{n+k-1}{k}(1-a^i)^n(a^i)^k,\\ &\qquad k=0,1,\ldots. \end{aligned} \]

This is the negative-binomial convention used here: count down moves before the \(n\)th up move. Other texts may exchange the names “success” and “failure”; the displayed mass function fixes the convention without ambiguity.

The combinatorial factor cancels in the likelihood ratio:

\[ \begin{aligned} \frac{X(n)}{X(0)} &=\left(\frac{a^X}{a^Y}\right)^{Z(n)} \left(\frac{1-a^X}{1-a^Y}\right)^n\\ &=u^n d^{Z(n)}. \end{aligned} \]

Thus the log price is affine in accumulated count and completed shocks:

\[ \log\frac{X(n)}{X(0)}=A Z(n)+Bn, \]

where

\[ \begin{aligned} A&=\log\frac{a^X}{a^Y},\\ B&=\log\frac{1-a^X}{1-a^Y}. \end{aligned} \]

The one-shock likelihood ratio has expectation one under \(\mathbb P^Y\). Products across independent shocks therefore form a positive \(\mathbb P^Y\)-martingale; their reciprocals form a \(\mathbb P^X\)-martingale. This follows directly from the geometric laws, without requiring an unrestricted optional-stopping argument.

The interactive figure requires JavaScript. At the Columbia settings \(a^X=1/4\), \(a^Y=1/2\), and \(a^M=1/3\), ideal information is \(0.0849495\) nats per completed shock, while the stock’s own-law information is \(0.1744160\) nats per shock.

The first plot compares the market, stock, and investor laws on the same count axis. The price and log-price simulations show the same six paths under each of \(\mathbb P^X\) and \(\mathbb P^Y\), sampled at completed shocks. Trials inside each shock are not displayed. The infinite distributions are truncated only for plotting; neither their displayed probabilities nor the simulated shocks are renormalized or truncated.

The dashed blue line on the price plot is \(\mathbb E^Y[X(n)/X(0)]=1\). Dashed lines on the log plot are expected log prices. Exponentiating an expected log price does not give an expected price. Extremely large prices are marked as off scale; their log paths remain visible in full.

What the sliders count

The counted trial is the down move in the Columbia preset, where \(a^X<a^Y\). The laboratory also permits \(a^X>a^Y\); then that trial raises the stock price and the terminating trial lowers it. The geometric count and likelihood formulas remain valid. The neutral label “counted trials” in the charts accommodates both directions.

Linear information in the shock clock

For geometric parameters \(a,b\in(0,1)\), define

\[ \begin{aligned} I(a,b)&=\log\frac{1-a}{1-b}\\ &\quad+\frac{a}{1-a}\log\frac ab. \end{aligned} \]

Relative entropy between the corresponding negative-binomial laws is \(nI(a,b)\). Applied to investor beliefs and the market,

\[ \begin{aligned} &D_{\mathrm{KL}}(\mathbb P^M_n\Vert\mathbb P^Y_n)\\ &\quad=nI(a^M,a^Y)\\ &\quad=n\times0.0849495. \end{aligned} \]

This is ideal expected log growth, attained if the payoff \(L_M(n)=\mathbb P^M_n/\mathbb P^Y_n\) can be implemented. The stock has its own separate information rate:

\[ \mathbb E^X\!\left[\log\frac{X(n)}{X(0)}\right] =nI(a^X,a^Y). \]

\[ \mathbb E^Y\!\left[\log\frac{X(n)}{X(0)}\right] =-nI(a^Y,a^X). \]

Under investor beliefs the stock’s expected log return instead equals

\[ \begin{aligned} &\mathbb E^M\!\left[\log\frac{X(n)}{X(0)}\right]\\ &\quad=n\{I(a^M,a^Y)-I(a^M,a^X)\}. \end{aligned} \]

This keeps ideal information, the stock’s induced law, and the investor’s actual assessment distinct.

Wald’s identity: information per trial versus per shock

The total number of binary trials is \(\tau(n)=Z(n)+n\). Under beliefs,

\[ \mathbb E^M[\tau(n)]=\frac{n}{1-a^M}=\frac32n. \]

The one-trial log-likelihood increments are bounded and this stopping time has finite expectation. With \(d(a\Vert b)\) denoting Bernoulli relative entropy, as on the binomial page, Wald’s identity gives

\[ I(a^M,a^Y)=\frac{d(a^M\Vert a^Y)}{1-a^M}. \]

Numerically, information per shock equals expected trials per shock times information per trial:

\[ 0.0849495=\frac32\times0.0566330. \]

The new clock creates no extra information. It packages a random number of observations into each unit of time.

If trading is available after every binary trial, rebalancing to the stock weight \(w=2/3\) produces an up multiplier \(4/3\) and a down multiplier \(2/3\). Its wealth at the end of a shock is exactly \(L_M=(4/3)(2/3)^G\), so it attains the full \(0.0849495\) nats.

What changes when trading is restricted?

Now permit trading only between completed shocks. A stock/cash portfolio held through one shock has payoff

\[ V_w=1+w(R-1). \]

\[ \mathbb P^{V_w}=w\mathbb P^X+(1-w)\mathbb P^Y. \]

There are infinitely many possible \(G\) values but only two traded securities. No single \(w\) matches the desired payoff in every state. The old binary weight \(2/3\) matches the mass of \(G=0\) and the combined tail \(G\geq1\); it does not match the individual tail outcomes.

The best fixed shock-level weight is \(w_{\mathrm{KL}}\approx0.745331\). Its expected log growth is \(0.0814623\), leaving a KL gap of \(0.0034872\) nats per shock. That gap is the gain available from the ideal within-shock rebalancing strategy in this frictionless example.

Compare exact KL, KL2, and quadratic hedging for the derivation, the full comparison table, and the distinction between ordinary squared error and a local approximation to information loss.

Optional detail: what binary coarsening discards

Keep only the partition \(C=\{G=0,G\geq1\}\). Its market probabilities are \((1/2,1/2)\) and its belief probabilities are \((2/3,1/3)\). The coarse KL divergence is the same \(0.0566330\) as one binary trial.

The KL chain rule separates coarse information from conditional detail. Write \(M_c=\mathbb P^M(G\mid c)\) and \(Y_c=\mathbb P^Y(G\mid c)\) for the two conditional laws:

\[ \begin{aligned} &D_{\mathrm{KL}}(\mathbb P^M_G\Vert\mathbb P^Y_G)\\ &\quad=D_{\mathrm{KL}}(\mathbb P^M_C\Vert\mathbb P^Y_C)\\ &\qquad+\sum_{c\in C}\mathbb P^M(c)D_{\mathrm{KL}}(M_c\Vert Y_c). \end{aligned} \]

The event \(G=0\) contains no remaining uncertainty. Conditional on \(G\geq1\), the shifted count \(G-1\) has the original geometric law under each measure. Hence

\[ D_{\mathrm{fine}}=D_{\mathrm{coarse}}+\frac13D_{\mathrm{fine}}, \]

\[ 0.0849495=0.0566330+0.0283165. \]

The coarsening loss \(0.0283165\) and the portfolio implementation gap \(0.0034872\) are different quantities. The first discards outcome detail; the second retains the full outcome space but restricts the available payoff.

Optional detail: the likelihood principle

Suppose the observed data contain \(k\) counted down-type trials and \(n\) terminating up-type trials. For an unknown counted-trial probability \(a\), both a fixed-trial design and a design stopped at the \(n\)th terminating trial have likelihood proportional to \(a^k(1-a)^n\). Their combinatorial factors depend on the observed counts and design but not on \(a\).

Thus the realized likelihood ratio is the same in either design. This does not make their sampling distributions identical: procedures depending on unobserved outcomes, including fixed-design p-values, can differ. Evaluating the original binary wealth process at the \(n\)th terminating trial produces the negative-binomial wealth shown here.

References

Jan Vecer, Principles of Portfolio Choice: An Information-Theoretic, Likelihood-Based Perspective, Chapter 5, “Negative binomial family.”
Jan Vecer, Portfolio Optimization through Information Theory: Discrete Models (Prediction Markets), Columbia University, 9 October 2026; available in talk resources.
A. Wald (1947). Sequential Analysis.
T. M. Cover and J. A. Thomas (2006). Elements of Information Theory, Chapter 2.
D. V. Lindley and L. D. Phillips (1976). “Inference for a Bernoulli Process (A Bayesian View).” The American Statistician 30(3), 112–119.

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