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Admissible projections are constrained by sign restrictions on risk premia, which yield bounds on physical moments. 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Predictive regressions report coefficients with standard errors in square brackets. Out-of-sample R2s are defined relative to a sample-mean forecast, following Campbell-Thompson. 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Theory sections build intuition in a Black-Scholes setting and then generalize to HJ projections and moment bounds. A long empirical section on S&P 500 options covers recovered moments, predictability, risk premia, kernel shape, trading strategies, and tests of the assumptions. Proofs, implementation details, and affine-model illustrations are placed in appendices.",253 "writing_profile_keywords": [254 "option-implied recovery",255 "pricing kernel projection",256 "minimum variance",257 "model-free bounds",258 "figure-heavy empirics",259 "theory-then-application"260 ],261 "confidence": 0.72,262 "notes": "Several section entries (idx 3-7) are equation fragments created by parsing errors and belong to Section II. Standard error type and significance conventions are not visible in the excerpts. 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