**By Gifford Fong, Oldrich Vasicek, and Daihyun Yoo**

*Risk*, 5 (2) (1992), 62–65.

Investors have long understood the need to measure how changes in interest rates will affect the value of fixed-income portfolios. Duration and convexity, used to measure these effects, belong in every portfolio manager's tool kit. But these alone do not give a complete picture of the risk in a portfolio. Changes in interest rates are not the only source of risk in fixed-income investment. What about changes in interest rate volatility?

Nearly all fixed-income instruments contain embedded options. The price of a callable bond, for example, depends on the value of the call option; this, in turn, depends on the volatility of interest rates. Measuring an instrument's sensitivity to interest rate volatility is thus central to valuing the instrument as a whole.

The Black-Scholes formula shows that options' sensitivity to volatility, and the value of callable bonds, pass-throughs, futures, and other instruments with option-like features also depends on market volatility. Even noncallable bonds are volatility-dependent. The published results from Vasicek (1977) (Chapter 6 of this volume), Cox, Ingersoll, and Ross (1985), and others on the behavior of the term structure of interest rates show the presence of the volatility parameter in the bond pricing formula.

Just as the fixed-income investor needs to know how changing interest rates affect portfolio ...

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