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Pricing and Hedging Volatility Risk in Fixed Income Markets

Though dynamic models with a small number of risk factors (e.g., two or three) have had considerable success at pricing bonds across a broad spectrum of maturities, they typically generate large errors when pricing options on these bonds. Mean-squared relative pricing errors for options on the order of 30% are reported in Buhler et al. (1999), Dreissen et al. (2003), and Jagannathan et al. (2003). Moreover, model-free principal components analyses, e.g., Heidari and Wu (2003), show that the level, slope, and curvature term-structure factors explain only about 60% of the cross sectional variation in option-implied volatilities.

This paper develops a dynamic term structure model with four risk factors that resolves both of these empirical puzzles: the model’s mean-squared pricing errors on options are less than the bid-ask spreads, and one of the risk factors drives option volatilities while being only weakly correlated with bond yields, leading to the ability of the model to price both bonds and bond options.

There are two critical features of my model that underlie its relative success in simultaneously pricing bonds and bond options. First, I focus on members of the affine family of term structure models (Duffie and Kan (1996)) that are known to be successful in pricing bonds and allow flexibility in the conditional covariances of the risk factors. In particular, I use the affine process specification given in Joslin (2006) which allows for a richer covariance structure among risk factors than the specification of Dai and Singleton (2000).

In contrast, Jagannathan et al. (2003) examine multi factor Cox-Ingersoll-Ross models which fails to capture both the first and second moment properties of bond yields (e.g., Dai and Singleton (2000,2002)), and so would not be expected to accurately price options on bonds. Similarly, Buhler et al. (1999) consider only special cases of the general specification of Dai and Singleton (2000), requiring independent risk factors with restricted conditional first-moments. The covariance structure of the risk factors is critical both in the pricing of derivatives and in capturing risks that drive option implied volatilities but do not affect the level, slope, and curvature.

Pricing and Hedging Volatility Risk in Fixed Income Markets