Pricing and modelling of averaged Risk-Free Rate derivatives

Abstract

The transition from the London InterBank Offered Rate (LIBOR) to the on transaction-based Risk-Free Rates (RFRs) such as SOFR and ESTR, has fundamental consequences for the valuation and modelling of interest rate products. Where old interest rates were known upfront and based on estimates from banks, are RFRs determined on a daily basis based on real transactions in the market, and only known afterwards. This new interest rate leads to more transparency, but also more uncertainty in future cash flows. Because of this, the valuation of interest rate products has become more challenging, especially contracts based on the average of daily rates, instead of daily compounded rates used in the market.

In this study, we analyse how we can value products based on the arithmetic average of daily RFRs efficiently. We first investigate how we can use the data available in the market to approximate the dynamics of the average RFR. This allows us to value contracts based on the average RFR, and we will specifically focus on options on this average. We introduce two analytic approximations: the Asymmetric Quadratic Option method, the Quadratic Option method and propose a new method, the Distributional method, based on the replication formula of Carr and Madan (2002).

The proposed methods are evaluated in a comparative study, where we will consider various market conditions and volatility scenarios, and compare based on accuracy, evaluation speed, and stability. The results show that there is no uniformly superior method. Depending on skew, convexity, and maturity, another method can be more optimal to use. To conclude, we present a case study where averaging swaps and caplets are valued under a realistic yield-curve calibration. In this setting, we will compare the associated sensitivities with the sensitivities of the vanilla counterparts.