Pricing and hedging
Represent the claim, identify what trades, then choose the approximation
A hedge starts with a target claim and the assets available to reproduce it. Information theory supplies a probability representation of both the target and the implemented position.
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.
Fix a pricing law \(\mathbb P^Y\) and a nonnegative claim \(H\) with finite, positive price \(H(0)=\mathbb E^Y[H]\). Its induced law is
\[ \mathbb P^H(\omega) =\mathbb P^Y(\omega)\frac{H(\omega)}{H(0)}. \]
The formula is written for finite or countable outcomes; densities give the corresponding continuous-state construction. A claim that vanishes on some outcomes induces a law with zero mass there. All payoffs and prices here use the reference asset as numeraire.
Exact replication matches the distribution
In two states, cash and a nonconstant stock span every payoff. The usual delta hedge and the mixture of induced probability laws express the same replication. The binomial tutorial connects shares, cost weights, and probability matching in one numerical example.
Incomplete trading leaves a choice of criterion
With more outcomes than independent traded payoffs, exact replication may fail. Squared payoff error, target-relative quadratic error, and exact relative entropy can then choose different hedges.
The three-criteria comparison uses a single negative-binomial market throughout. It explains the weighting and loss function of each criterion, compares the optimal portfolios, and identifies which conclusions survive a change of pricing law.
The information interpretation must retain the target measure. Minimizing \(D_{\mathrm{KL}}(\mathbb P^H\Vert\mathbb P^V)\) maximizes expected log payoff under the claim-induced law \(\mathbb P^H\) for equal-cost positive candidates. This criterion is distinct from minimizing payoff error under a pricing or physical law.
Options are distributions too
The options page displays the probability law induced by a call, put, digital, or straddle. It connects familiar option-price formulas with weighted tail probabilities and shows how a cash reserve changes the induced distribution.
Rebalance over a finite interval
The discrete hedging laboratory projects a target onto stock and cash over one conditional trading interval in a normal model. It separates approximation error from positivity constraints and explains when the short-interval hedge approaches the classical delta.
For the portfolio-selection interpretation of the same geometry, continue with beliefs and attainable portfolios.