The trinomial model
One stock. Three states. Two different questions.
The binomial model has just enough traded payoffs to distinguish its two states. A third state exposes two separate issues: which pricing measure fits the traded returns, and which investor beliefs can be implemented by trading?
Work in units of \(Y\) and take the book’s three-state stock return:
\[ R^X=\left(\frac32,1,\frac12\right). \]
Only \(X\) and \(Y\) are traded. Every state is possible; borrowing and short selling are available unless a portfolio constraint is explicitly imposed.
First, find all consistent pricing measures
The pricing weights must satisfy
\[ q_1+q_2+q_3=1, \]
\[ \frac32q_1+q_2+\frac12q_3=1. \]
Subtracting these equations gives \(q_1=q_3\). Thus the positive pricing measures form a line:
\[ \mathbb P^Y(t)=(t,1-2t,t), \qquad 0<t<\frac12. \]
Each choice gives an asset measure by multiplying the pricing weights by the stock return:
\[ \mathbb P^X(t) =\left(\frac{3t}{2},1-2t,\frac t2\right). \]
Both martingale checks now hold:
\[ \mathbb E^Y[R^X]=1, \qquad \mathbb E^X[1/R^X]=1. \]
The first prices \(X\) in units of \(Y\). The second prices \(Y\) in units of \(X\). Choosing another \(t\) changes both measures while preserving the same stock returns.
Then choose an attainable portfolio
Put a fraction \(w\) of initial wealth in \(X\), and \(1-w\) in \(Y\). Its return is
\[ R^{V,w} =\left(1+\frac w2,1,1-\frac w2\right). \]
For a fixed pricing measure, its implied density is
\[ \mathbb P^{V,w} =(1-w)\mathbb P^Y+w\mathbb P^X. \]
With no short sales, \(0\le w\le1\): the investor can move along a line segment in the probability triangle. Allowing borrowing and short selling extends the line; nonnegative terminal wealth imposes \(-2\le w\le2\). Neither case fills the triangle.
The interactive figure requires JavaScript. At the book’s default \(t=1/3\), the pricing measure is \((1/3,1/3,1/3)\) and the stock measure is \((1/2,1/3,1/6)\). Every stock-and-cash portfolio retains middle-state probability \(1/3\).
The left triangle varies the pricing rule, holding the stock’s returns fixed. The right triangle varies the portfolio, holding the selected pricing rule fixed. Each corner represents all probability concentrated on one state; interior points assign weight to all three.
Why a belief can be unattainable
At the book’s choice \(t=1/3\),
\[ \mathbb P^{V,w} =\left(\frac{2+w}{6},\frac13,\frac{2-w}{6}\right). \]
Now consider the investor’s beliefs
\[ \mathbb P^M=\left(\frac12,\frac16,\frac13\right). \]
Every attainable portfolio has middle-state weight \(1/3\), while the investor assigns \(1/6\). No choice of \(w\) can make the two distributions equal. The unrestricted likelihood payoff \(\mathbb P^M/\mathbb P^Y\) exists as a mathematical claim, but it cannot be replicated using only this stock and cash.
The best attainable portfolio instead minimizes
\[ D_{\mathrm{KL}}(\mathbb P^M\Vert\mathbb P^{V,w}). \]
Equivalently, it maximizes
\[ \begin{aligned} g(w)&=\frac12\log\left(1+\frac w2\right)\\ &\quad+\frac13\log\left(1-\frac w2\right). \end{aligned} \]
Here the maximizing weight is \(w=2/5\), under either of the displayed constraints. The figure shows its remaining information mismatch. This weight depends on the beliefs and the traded returns, so it stays the same as \(t\) changes. The two divergences in the growth decomposition change together.
At \(t=5/12\), this particular belief lies on the attainable line and the mismatch vanishes at \(w=2/5\). The market remains incomplete: implementing one selected belief does not mean that every payoff is spanned.
The exact decision is pricing-law invariant
For any strictly positive pricing law \(q(t)\) consistent with the stock and cash, and any positive unit-cost portfolio \(V_w\),
\[ \begin{aligned} &D_{\mathrm{KL}}(\mathbb P^M\Vert q(t)V_w)\\ &\quad=D_{\mathrm{KL}}(\mathbb P^M\Vert q(t)) -\mathbb E^M[\log V_w]. \end{aligned} \]
Hold beliefs, traded returns, and the feasible portfolio set fixed. The first term depends on the pricing-law choice, but not on \(w\). Thus exact KL selects the same portfolio under every \(t\). Quadratic hedging of the corresponding ideal target \(L_M=\mathbb P^M/q(t)\) instead gives
\[ \begin{aligned} w_Q(t)&=\frac{\mathbb E^M[R^X]-1}{\operatorname{Var}^{q(t)}(R^X)}\\ &=\frac{1/12}{t/2}=\frac1{6t}. \end{aligned} \]
| Pricing parameter \(t\) | Exact-KL fraction | Quadratic fraction |
|---|---|---|
| \(1/4\) | 0.400000 | 0.666667 |
| \(1/3\) | 0.400000 | 0.500000 |
| \(5/12\) | 0.400000 | 0.400000 |
These positions satisfy the displayed long-only bounds. Exact attained growth and portfolio rankings stay the same. Absolute KL gaps, ideal likelihood payoffs, and quadratic criteria can change. The geometric-shock comparison repeats this contrast in a countably infinite state space, with an interactive pricing-law control.
The closest pricing law describes attainable growth
There is also a dual question: among all laws that price the traded assets, which is closest to the investor’s beliefs? In this finite-state example, the exact optimum is interior, all belief probabilities are positive, and wealth is strictly positive. Write
\[ \begin{aligned} V^*&=(6/5,1,4/5),\\ Q^*_i&=\frac{\mathbb P^M_i}{V^*_i} =\left(\frac5{12},\frac16,\frac5{12}\right)_i. \end{aligned} \]
This is precisely the pricing law \(t=5/12\) identified above. The first-order condition gives
\[ \mathbb E^M\left[\frac{R^X-1}{V^*}\right]=0. \]
Using \(V^*=1+w^*(R^X-1)\) then gives \(\mathbb E^M[1/V^*]=1\). Therefore \(Q^*\) sums to one and \(\mathbb E^{Q^*}[R^X]=1\): it really is a pricing law.
For every other positive pricing law \(Q\) and every positive attainable unit-cost portfolio \(V\),
\[ \begin{aligned} &D_{\mathrm{KL}}(\mathbb P^M\Vert Q)\\ &\quad=\mathbb E^M[\log V] +D_{\mathrm{KL}}(\mathbb P^M\Vert QV)\\ &\quad\geq\mathbb E^M[\log V]. \end{aligned} \]
At \(Q^*,V^*\), the last divergence vanishes. Hence
\[ \max_V\mathbb E^M[\log V] =\min_{Q\text{ pricing law}}D_{\mathrm{KL}}(\mathbb P^M\Vert Q). \]
The proof uses an interior optimum with all traded risky directions locally available in both signs. A binding portfolio constraint generally replaces the first-order equality by inequalities. In that case \(\mathbb P^M/V^*\) should not automatically be called a probability or an equivalent pricing law; the appropriate dual object must respect the constraints. Infinite-state extensions additionally need integrability and existence conditions. The finite example above meets the stated requirements.
Incompleteness also leaves a price range
A claim paying one in the middle state and zero otherwise would have price
\[ C(0)=\mathbb P^Y(\omega_2)=1-2t. \]
The original stock and cash prices permit every value in \((0,1)\). This is an open no-arbitrage interval; its endpoints are limiting bounds, not admissible prices when all three states remain possible. Trading that additional claim at a price inside the interval would identify \(t\) and complete this one-period market.
This is the distinction behind the fundamental theorem of asset pricing: no arbitrage gives existence of a pricing measure; completeness gives uniqueness. A likelihood description respects the same distinction.
Jan Vecer, Principles of Portfolio Choice: An Information-Theoretic, Likelihood-Based Perspective, Chapter 1, “Trinomial model” and “Numerical trinomial mixture: rigid asset densities”; Chapter 3, incomplete markets and relative-entropy projection.
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