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As a market convention, interest rate volatility is measured by a swaption volatility surface. The volatility surface is a set of Black volatilities derived from quoted at-the-money swaption prices over a range of tenors and expiration dates. The volatility surface is similar to the swap curve in that it is stochastic, changing continually to reflect market expectations and the buy and sell order flows for interest rate volatilities. And this surface is an important practical input, often as important as the swap curve, to the valuation of many interest rate contingent claims.
The volatility surface is a subject of many previous studies. Heidari and Wu [2003] use principal component analysis to show that the volatility surface has three orthogonal movements, independent of the principal movements of the yield curve. They establish the stochastic nature of the volatility surface. Collin-Dufresne and Goldstein [2002] use interest rate straddles to provide empirical evidence of "unspanned stochastic volatility" showing that interest rate derivatives cannot be dynamically hedged or replicated by bonds alone (with no embedded options), because of the significant presence of volatility risk. Thus, there is a need to hedge the vega risks for any dynamic replication.
Another strand of empirical studies relates the volatility surface to the implied volatility functions of arbitrage-free interest rate models and has shown that the implied volatilities estimated are indeed stochastic (Amin and Morton [1994] and De Jong, Driessen, and Pelsser [2001]). Further, Amin and Ng [1997] have shown that implied volatilities have informational content in predicting future interest rate volatilities. These studies show that the vega measure should be used to manage volatility risk in investment, hedging, and risk reporting as an integral part of market risks on trading floors, and for portfolio management and enterprise risk management.
To manage the risk of an interest rate contingent claim, practitioners need both duration and vega measures which indicate an instrument's sensitivities to the shift in the swap curve and the volatility surface, respectively. To manage interest rate risk, practitioners use duration sensitivity along the yield curve which is called key rate duration (see Ho [1992]). However, to date, there are significant challenges in determining the vega buckets for interest rate derivatives. One challenge is the computational intensity required in determining the...