One mismatch, three criteria
How trading restrictions separate exact information loss from quadratic hedging
The desired payoff can be clear even when trading cannot produce it. The investor supplies a target distribution; stock and cash span a smaller set of distributions. The choice of loss determines which attainable portfolio comes closest.
This page continues the binomial example and the negative-binomial trading clock. All payoffs are expressed in units of the reference asset \(Y\) and normalized to initial cost one.
All returns are measured in units of a strictly positive reference asset \(Y\). Its own return is one, so interest and reinvestment are included in the choice of unit.
| Symbol | Role |
|---|---|
| \(\mathbb P^M\) | Investor’s beliefs about outcomes |
| \(\mathbb P^Y\) | Pricing law in reference-asset units |
| \(\mathbb P^X\) | Law induced by the traded stock |
| \(\mathbb P^V\) | Law induced by the implemented portfolio |
| \(\mathbb P^U\) | Law induced by a utility-selected payoff |
| \(R=d\mathbb P^X/d\mathbb P^Y\) | Stock’s gross return |
| \(L_M=d\mathbb P^M/d\mathbb P^Y\) | Ideal payoff for the belief |
| \(V_w=1+w(R-1)\) | Unit-cost stock-and-cash payoff |
For \(0\leq w\leq1\), \(\mathbb P^{V_w}=w\mathbb P^X+(1-w)\mathbb P^Y\) is a convex mixture. Outside that range, admissibility still requires nonnegative wealth in every relevant state. In incomplete markets, the chosen pricing law need not be unique.
The same binary tree, a different trading rule
In the binary model, stock returns are \(u=3/2\) and \(d=1/2\). The up probabilities are \(1/2\) under the pricing law and \(2/3\) under the investor’s beliefs. Rebalancing a stock fraction \(w=2/3\) after every step produces the desired likelihood payoff.
Now let one trading period contain \(G\) down moves followed by the first up move. You may trade before and after the completed shock, but not during it. For \(g=0,1,\ldots\),
\[ \begin{aligned} q_g&=\mathbb P^Y(G=g)=2^{-(g+1)},\\ m_g&=\mathbb P^M(G=g)=\frac23\left(\frac13\right)^g. \end{aligned} \]
The stock return and desired payoff are
\[ \begin{aligned} R_g&=\frac32\left(\frac12\right)^g,\\ L_{M,g}&=\frac{m_g}{q_g}=\frac43\left(\frac23\right)^g. \end{aligned} \]
The stock’s law is \(p^X_g=q_gR_g=\tfrac34(\tfrac14)^g\). Holding a stock fraction \(w\) unchanged throughout the shock gives
\[ \begin{aligned} V_w&=1+w(R-1),\\ v_g(w)&=q_gV_w(g). \end{aligned} \]
This is an affine payoff in \(R\). The desired \(L_M\) is nonlinear in \(R\), so no single stock-and-cash holding matches it in every state. Strictly positive wealth in every state requires \(-2<w\leq1\). The explorer focuses on \(0\leq w\leq1\), which contains all three optima in the examples below.
At \(w=2/3\), the portfolio matches the probability of \(G=0\) and the total probability of \(G\geq1\). It does not match the individual outcomes inside that tail. This is why the binary solution need not solve the completed-shock problem.
Three ways to measure the remaining mismatch
Write \(e_g=v_g(w)-m_g\). Exact information loss is
\[ \begin{aligned} D(w)&=D_{\mathrm{KL}}(m\Vert v(w))\\ &=\sum_gm_g\log\frac{m_g}{v_g(w)}. \end{aligned} \]
It is exactly equivalent to maximizing expected log growth:
\[ \mathbb E^M[\log V_w]=D_{\mathrm{KL}}(m\Vert q)-D(w). \]
Two quadratic alternatives are
\[ \begin{aligned} \operatorname{KL}_2(w) &=\frac12\sum_g\frac{e_g^2}{m_g}\\ &=\frac12\mathbb E^Y\left[\frac{(V_w-L_M)^2}{L_M}\right], \end{aligned} \]
\[ Q(w)=\sum_g\frac{e_g^2}{q_g} =\mathbb E^Y[(V_w-L_M)^2]. \]
The target-centered \(\operatorname{KL}_2\) is the second-order expansion of exact KL when the relative mismatch \((V_w-L_M)/L_M\) is small with suitable integrability. Its factor of \(1/2\) comes from that expansion. Ordinary quadratic hedging weights payoff errors using the pricing law; we define its loss \(Q\) without a factor of \(1/2\) throughout. Rescaling a loss by a positive constant changes its value but not its minimizing portfolio. The two quadratic criteria have different denominators; exact KL also has a different, logarithmic loss.
The target approaches zero in the tail, while a portfolio with \(w<1\) retains a positive cash intercept. Relative errors can therefore be large even when absolute errors are small. There is no guarantee that minimizing \(\operatorname{KL}_2\) gives a better exact-KL fit than minimizing ordinary squared error.
The resulting portfolios
The quadratic hedge has the regression coefficient
\[ \begin{aligned} w_Q&=\frac{\mathbb E^M[R]-1}{\operatorname{Var}^Y(R)}\\ &=\frac{1/5}{2/7}=\frac7{10}. \end{aligned} \]
For this target, the other quadratic criterion is explicitly
\[ \begin{aligned} \operatorname{KL}_2(w)&=\frac{24}{65}w^2-\frac35w+\frac14,\\ w_{\mathrm{KL}_2}&=\frac{13}{16}. \end{aligned} \]
Exact log growth is strictly concave. Its interior maximizer solves
\[ \mathbb E^M\left[\frac{R-1}{1+w(R-1)}\right]=0. \]
The solution is \(w_{\mathrm{KL}}\approx0.745331\).
| Portfolio | Stock fraction | Expected log growth | Exact KL gap |
|---|---|---|---|
| Binary-model fraction, held throughout | 0.666667 | 0.0802308 | 0.0047187 |
| Target-centered quadratic (\(\operatorname{KL}_2\)) | 0.812500 | 0.0804213 | 0.0045282 |
| Ordinary quadratic hedge | 0.700000 | 0.0810414 | 0.0039081 |
| Exact KL optimum | 0.745331 | 0.0814623 | 0.0034872 |
Growth and exact KL are in nats per completed shock. In each row their sum is \(D_{\mathrm{KL}}(m\Vert q)=0.0849495\). Ordinary quadratic hedging beats \(\operatorname{KL}_2\) on exact growth in this example; each quadratic criterion still minimizes its own loss.
Explore the losses
Move the stock fraction or select an optimizer. The graph shows exact KL, \(\operatorname{KL}_2\), and the ordinary squared error \(Q\), with the normalizations defined above. Only the exact-KL curve is an information loss in nats. Compare each curve’s minimizing stock fraction; the displayed heights use different loss scales. A local view makes the distinct minima easier to see.
The interactive explorer requires JavaScript. The table above contains its default results. With the alternative pricing law below, the exact-KL optimum remains 0.745331, while the quadratic optimum becomes 0.640858 and the KL2 optimum becomes 0.827898.
The explorer evaluates the geometric sums over \(g=0,\ldots,159\) and solves the exact first-order condition by bisection. This truncation is far below displayed precision on the selected range, including the slower \((3/4)^g\) tail in the target-centered quadratic. The original-law quadratic solutions above provide independent analytical checks. The slider restricts only the displayed allocation range; the stated optima are also the optima on the full positive-wealth domain.
What changes when the pricing law changes?
With only terminal stock and cash traded, many probability laws price those same payoffs. Keep the original probabilities for \(g\geq3\), and set
\[ q'_0=\frac{17}{32},\qquad q'_1=\frac5{32},\qquad q'_2=\frac3{16}. \]
Both \(q\) and \(q'\) sum to one and satisfy \(\mathbb E^Q[R]=1\). Hold beliefs \(m\), stock returns \(R\), and the portfolio menu \(V_w\) fixed. Relative to any such law,
\[ \begin{aligned} &D_{\mathrm{KL}}(m\Vert QV_w)\\ &\quad=D_{\mathrm{KL}}(m\Vert Q)-\mathbb E^M[\log V_w]. \end{aligned} \]
| Quantity | Original \(q\) | Alternative \(q'\) |
|---|---|---|
| Exact-KL stock fraction | 0.745331 | 0.745331 |
| Attained optimal log growth | 0.0814623 | 0.0814623 |
| Quadratic-hedge stock fraction | 0.700000 | 0.640858 |
| KL2 stock fraction | 0.812500 | 0.827898 |
| Ideal information \(D_{\mathrm{KL}}(m\Vert Q)\) | 0.0849495 | 0.1189439 |
| Optimal exact KL gap | 0.0034872 | 0.0374816 |
The exact decision and attained growth are invariant. The ideal payoff \(m/Q\), the absolute information benchmark, and the two quadratic criteria change. This comparison holds beliefs fixed; it does not hold a separately specified hedge target fixed while changing its valuation.
The gap has a trading interpretation
On the original binary tree, rebalancing to \(w=2/3\) at every binary step produces
\[ \frac43\left(\frac23\right)^G=L_M. \]
Its expected log gain per completed shock is \(0.0849495\). The best portfolio held unchanged during the shock earns \(0.0814623\). Thus
\[ 0.0849495-0.0814623=0.0034872 \]
is the expected log growth lost because intermediate rebalancing is unavailable. This difference between two attainable growth values does not depend on a chosen pricing law. Its equality to the absolute KL gap above uses the original law \(q\), which is fixed by pricing every step of the binary tree. The alternative \(q'\) prices the terminal stock and cash, but not those intermediate trading opportunities.
Continue: Mean–variance and the local quadratic limits · Hedging a specified option payoff · Pricing-law geometry and the dual problem
The worked example follows Jan Vecer’s Columbia presentation, Portfolio Optimization through Information Theory: Discrete Models (Prediction Markets), 9 October 2026, especially slides 22–37 and 43–48. The presentation is available from Research and resources. All portfolio comparisons here concern expected log growth under the stated beliefs. A full-market utility-selected target can also be approximated by KL, but that procedure does not generally equal constrained expected-utility maximization for the original utility.