dorsal/arxiv
View SchemaUnifying the BGM and SABR Models: A short Ride in Hyperbolic Geometry
| Authors | Pierre Henry-Labordere |
|---|---|
| Categories | |
| ArXiv ID | physics/0602102 |
| URL | https://arxiv.org/abs/physics/0602102 |
Abstract
In this short note, using our geometric method introduced in a previous paper \cite{phl} and initiated by \cite{ave}, we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matrix. We apply this formula to a specific model where the forward rates are assumed to follow a multi-dimensional CEV process correlated to a SABR process. For a caplet, this model degenerates to the classical SABR model and our asymptotic swaption implied volatility reduces naturally to the Hagan-al formula \cite{sab}. The geometry underlying this model is the hyperbolic manifold $\HH^{n+1}$ with $n$ the number of Libor forward rates.
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"date_created": "2026-03-02T18:01:07.015000Z",
"date_modified": "2026-03-02T18:01:07.015000Z",
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"abstract": "In this short note, using our geometric method introduced in a previous paper\n\\cite{phl} and initiated by \\cite{ave}, we derive an asymptotic swaption\nimplied volatility at the first-order for a general stochastic volatility Libor\nMarket Model. This formula is useful to quickly calibrate a model to a full\nswaption matrix. We apply this formula to a specific model where the forward\nrates are assumed to follow a multi-dimensional CEV process correlated to a\nSABR process. For a caplet, this model degenerates to the classical SABR model\nand our asymptotic swaption implied volatility reduces naturally to the\nHagan-al formula \\cite{sab}. The geometry underlying this model is the\nhyperbolic manifold $\\HH^{n+1}$ with $n$ the number of Libor forward rates.",
"arxiv_id": "physics/0602102",
"authors": [
"Pierre Henry-Labordere"
],
"categories": [
"physics.soc-ph",
"cond-mat.other",
"q-fin.CP"
],
"title": "Unifying the BGM and SABR Models: A short Ride in Hyperbolic Geometry",
"url": "https://arxiv.org/abs/physics/0602102"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c192909f-1f35-4e8f-8286-adda1f86077d",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
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