Analytic RFR Option Pricing with Smile and Skew

Paper: arXiv 2301.01260

Authors: Colin Turfus, Aurelio Romero-Bermúdez

Abstract

We extend the short rate model of Turfus and Romero-Bermúdez [2021] to facilitate accurate arbitrage-free analytic pricing of SOFR, SONIA or ESTR caplets, i.e. options on backward-looking compounded rates payments, in a manner consistent with the smile and skew levels observed in the market. These caplet pricing formulae and corresponding LIBOR or term-rate caplet results are translated into effective variance (implied volatility) formulae, which are seen to be of a particularly simple form. They show that the model is essentially equivalent to imposing on a Hull-White model an effective variance which is a quadratic function of the moneyness parameter (rather than a constant) for any given maturity. Results are also illustrated graphically.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents strong theoretical and mathematical derivations to extend short-rate models for analytic option pricing under new risk-free rates (SOFR/SONIA/ESTR). However, it relies primarily on graphical illustration rather than exhaustive empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Analytic RFR Caplet Pricing with Smile & Skew] --> B[Data/Inputs: SOFR, SONIA, €STR Market Smile & Skew]
    B --> C[Methodology: Extend Turfus & Romero-Bermúdez Short Rate Model]
    C --> D[Computational Process: Derive Caplet Pricing Formulae for Compounded Rates]
    D --> E[Computational Process: Translate Pricing Formulae to Effective Variance Formulae]
    E --> F[Key Outcome: Equivalent to Hull-White Model with Quadratic Effective Variance]
    F --> G[Key Outcome: Simple Closed-Form Formulae for Implied Volatility]