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The resulting integral representation is written in terms of normalized implied volatilities. Just as Fukasawa's work provides rigourous ground for Chriss and Morokoff's (1999) model-free formula for the log-contract (related to the Variance swap implied variance), we prove an expression for the moment generating function $\\mathbb E[e^{p X_T}]$ on its analyticity domain, that encompasses (and extends) Matytsin's formula "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1703.00957","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-fin.PR","submitted_at":"2017-03-02T21:31:56Z","cross_cats_sorted":[],"title_canon_sha256":"3a9b9305b4caa12dbd634221c9c5fbdb31eb8526376d1d285bc83e4edbaf8702","abstract_canon_sha256":"37521cfd224992443d88fddc7e074c575e67a40eecc616235ca9de3bf1f34105"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:06.103859Z","signature_b64":"uEX1eVYAEFX8ITzb1e1mpEDxRH32nhpCKvh2NXM3ci8trYUDzoKZqVucD6uct71D4JPZfgoYjGxAsK0z2mPkBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdd3a71ae3a685c9859f914cd0d1bb47c7536f6f43217d25dd314004ff524425","last_reissued_at":"2026-05-18T00:45:06.103372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:06.103372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Moment generating functions and Normalized implied volatilities: unification and extension via Fukasawa's pricing formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"q-fin.PR","authors_text":"Claude Martini, Stefano De Marco","submitted_at":"2017-03-02T21:31:56Z","abstract_excerpt":"We extend the model-free formula of [Fukasawa 2012] for $\\mathbb E[\\Psi(X_T)]$, where $X_T=\\log S_T/F$ is the log-price of an asset, to functions $\\Psi$ of exponential growth. 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