Caplet Pricing with Backward-Looking Rates

Abstract

Colin Turfus considers the Hull-White short rate model and extends the known closed-form pricing kernel to include the integrated short rate as a separate independent variable, applying it to cap/floor pricing.

Analytic RFR Option Pricing with Smile and Skew

“…It will be observed that the leading-order contribution (n = 0) is in its functional form identical to the Hull-White kernel of Turfus [2022], with the caveat that the mean reversion function φ r (t, u) is adjusted up (reducing its impact) by inclusion of the smile factor ψ r (t, t 1 ) in (3.7), with no impact from skew. This means that solutions based on our expansion are likely to retain better accuracy even at longer maturities than those of Turfus and Romero-Bermúdez [2021].…”

“…We have successfully extended the short rate model of Turfus and Romero-Bermúdez [2021] to address the pricing of SOFR/SONIA/ESTR caplets based on compounded rates. This we achieved by expressing the result as a perturbation of the analytic kernel of Turfus [2022]. The model is seen to have the following properties:…”

“…This makes the situation similar to that captured by the model of Hull and White [1990]. An exact analytic pricing kernel for this model was derived by Van Steenkiste and Foresi [1999], Turfus [2019] and the extension of this kernel to address compounding rates by Turfus [2021aTurfus [ , 2022.…”

“…For a review of related work on options on compounded rates, the reader is referred to Turfus [2022], which follows the approach of Van Steenkiste and Foresi [1999] in introducing the integrated short rate as an additional state variable. Option prices are in this way derived analogous to the formulae of Henrard [2004], including taking account of delayed payment of the option premium.…”

“…We look to obtain analytic expressions for prices of SOFR options which take account of smile and skew by a method similar to that employed by Turfus and Romero-Bermúdez [2021] to price LIBOR caplets based on an extension of the Bachelier pricing kernel (for a Gaussian underlying). We do this by extending instead the Hull-White compounded rates kernel of Turfus [2022]. The challenge is that, as well as observing that the representations of the interest rate are similar (linear functions of a Gaussian variable) with and without smile or skew, we need to find a means of taking advantage of that observation systematically to obtain an explicit mathematical representation of the perturbation which must be applied to the known (Hull-White) analytic kernel.…”