SOFR In-Advance Derivative Pricing: Convexity Adjustment in the Forward Market Model

As the USD Libor interest rate benchmark is in the process of being discontinued, there is an increasing occurrence of SOFR-based loans and derivatives. Whereas Libor was most commonly set in-advance, an important difference is that SOFR is commonly set in-arrears when it is used in instruments such as overnight index swaps. Loans and interest-rate cap options used as hedges may both choose to fix the SOFR rate in-advance, which is similar to Libor by allowing the rate to be known in-advance of the interest accrual period. Here we adopt the mild assumption of bivariate normal log-forward rates for adjacent rate periods in the forward market model to derive the convexity adjustment for contracts which reference SOFR in-advance. Due to martingale conditions, when adjacent forward rates are uncorrelated there is no convexity adjustment, but when there is non-zero correlation, there is a convexity adjustment for any derivative payoff including the SOFR in-advance forward rate and the caplet/floorlet payoff which references this rate. These results demonstrate that SOFR in-advance derivatives are correlation-sensitive in their pricing and risk management, which is notably more complex than the situation for Libor.