Multi-factor USV term structure model

In recent works Collin-Dufresne and Goldstein [18], Heiddari and Wu [36], Jarrow, Li, and Zhao [45] and Li, Zhao [54] have extended the HJM approach to a framework, where either the volatility of forward rates, or the volatility of bond prices is driven by a subordinated stochastic process. One major implication of these new type of models is an additional source of uncertainty driving the volatility. This implies the existence of an additional market price of risk. Intuitively, this market price of risk cannot be hedged only by bonds. As a result of this, we have a new class of models causing incomplete bond markets.

The implications of this new model class are in contrast to most term structure models discussed in the literature, which assume that the bond markets are complete and fixed income derivatives are redundant securities. Collin-Dufresne and Goldstein [18] and Heiddari and Wu [36] show in an empirical work, using data of swap rates and caps/floors that there is evidence for one additional state variable that drives the volatility of the forward rates2. Following from that empirical findings, they conclude that the bond market do not span all risks driving the term structure. This framework is rather similar to the affine models of equity derivatives, where the volatility of the underlying asset price dynamics is driven by a subordinated stochastic volatility process (see e.g. Heston [38], Stein and Stein [71] and Schöbel and Zhu [69]).