Numeraire Invariance of Entropy-Projected Martingale Measures

Battle Count: 68

The paper is primarily theoretical and foundational rather than directly implementable for trading strategies. However, it has significant indirect relevance: (1) It establishes that the forward entropy projection provides a numeraire-invariant pricing functional, which is critical for consistent derivative pricing across currencies and asset classes in multi-asset trading. (2) The log-optimal portfolio connection (Theorem 10) links the forward projection to growth-optimal portfolio construction. (3) The pricing consistency result (Proposition 4) has direct implications for how quantitative traders should select martingale measures when pricing nonreplicable derivatives. (4) The MEMM failure example shows a concrete risk of pricing disagreement in cross-currency derivatives. The paper is more relevant to quantitative finance theory, derivative pricing, and portfolio construction than to algorithmic trading or market microstructure.

The theoretical framework is mathematically sophisticated (measure theory, convex analysis, utility duality, f-divergence theory), but the finite-state computations are straightforward. The forward entropy projection in a finite-state market reduces to minimizing a sum of p_i log(p_i/q_i) subject to linear martingale constraints, which is a standard convex optimization problem solvable via Lagrange multipliers or interior-point methods. The trinomial example is trivially computable. However, extending to continuous-time semimartingale models would require significant additional machinery (deflator domains, supermartingale theory). The paper itself does not provide algorithms or code.

The paper is purely theoretical with self-contained proofs. All theorems, lemmas, and corollaries include complete proofs. The trinomial counterexample (Theorem 12) is fully explicit with numerical values. No computational experiments or code are needed. The finite-state constructions are transparent and verifiable. The only limitation is that some results (existence, first-order conditions) are stated for finite-state markets, and the continuous-time extension is discussed qualitatively in Section 7 without full proofs.