Nonparametric Estimation of State-Price Densities Implicit in Interest Rate Cap Prices

Abstract

Based on a multivariate extension of the constrained locally polynomial estimator of At-Sahalia and Duarte (2003), we provide nonparametric estimates of the probability densities of LIBOR rates under forward martingale measures and the state-price densities (SPDs) implicit in interest rate cap prices conditional on the slope and volatility factors of LIBOR rates. Both the forward densities and the SPDs depend signicantly on the volatility of LIBOR rates, and there is a signicant impact of mortgage prepayment activities on the forward densities. The SPDs exhibit a pronounced U-shape as a function of future LIBOR rates, suggesting that the state prices are high at both extremely low and high interest rates, which tend to be associated with periods of economic recessions and high in ations, respectively. Our results provide nonparametric evidence of unspanned stochastic volatility and suggest that the unspanned factors could be partly driven by renancing activities in the mortgage markets. Over-the-counter interest rate derivatives, such as caps and swaptions, are among the most widely traded interest rate derivatives in the world. According to the Bank for International Settlements, in recent years, the notional value of caps and swaptions exceeds $ 10 trillion, which is many time