Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing

Abstract

We propose to use the linearity-generating framework to accommodate the evidence of unspanned stochastic volatility: Variations in implied volatilities on interest-rate options such as caps and swaptions are independent of the variations on the interest rate term structure. Under this framework, bond valuation depends only on the transition dynamics of interest-rate factors, but not on their volatilities. Thus, interest-rate volatility is truly unspanned. Furthermore, this framework allows tractable pricing of options on any bond portfolios, including both caps and swaptions. This feat is not possible under existing exponential-affine or quadratic frameworks. Finally, the framework allows sequential estimation of the interest-rate term structure and the interest-rate option implied volatility surface, thus facilitating joint empirical analysis. Within this framework, we perform specification analysis on interest-rate factor transition dynamics and its relation to the interest-rate term structure; we also analyze the interest-rate volatility dynamics and its impact on interest-rate option pricing. We estimate several specifications for the transition dynamics to ten years worth of U.S. dollar LIBOR and swap rates across 15 maturities. We also estimate several interest-rate volatility dynamics specifications using ten years of swaption implied volatilities across a matrix of ten option maturities and seven swap tenors. The estimation results show that the volatility dynamics dictate the option implied volatility variation along the option maturity dimension, whereas the interest-rate transition dynamics dictate the implied volatility variation along the underlying swap maturity dimension.