Stochastic volatility for factor Heath–Jarrow–Morton framework

We introduce an extension of Factor Heath–Jarrow–Morton (FHJM) framework augmented with a stochastic volatility (SV) driver correlated with factors dynamics. Our model fits the initial yield curve by construction and it is able to produce positive implied volatility skews as observed in market prices of interest rate derivatives. We develop a general framework for analytical valuation of swaptions, interest rate (SOFR) futures, and options on rate futures using moment generating function (MGF) corresponding to the FHJM model with the SV driver. We implement our results to the FHJM model with, first, the CIR SV driver, for which we find the exact solution for model MGF, and, second, for the log-normal SV driver, for which we derive semi-closed-form solution for the MGF based on drift-freezing technique. As a base case, we apply the widespread Nelson–Siegel term structure model for the basis in the FHJM model. We show that Nelson–Siegel model with log-normal SV driver is able to fit accurately market implied volatilities of swaptions across different tenors and expiries and implied volatilities of options on rates futures. Our framework allows for valuation, risk-management and scenario generation for fixed income derivatives including SOFR futures, swaptions, options on rate futures across several tenors and expiries in an arbitrage-free and consistent way.