INTRODUCTION
Simple interest rate options may be valued using variants of the Black-Scholes approach. For example, LIBOR rates, swap rates or bond prices may be taken as analogous to equity prices in the Black-Scholes model. The Black-Scholes formula then gives the value of the option. It is easy to describe the local behavior of the option value with respect to the parameters of the formula by taking partial derivatives, giving delta, vega, rho, theta, and so forth. These so-called "Greeks" are widely used for hedging and risk management purposes. Of course, since the underlying quantity that the Black-Scholes formula is applied to varies according to the particular instrument being priced, these Greeks cannot be simply added together to describe the exposure of the entire position.
More complex interest rate derivatives may depend in an essential way on the entire term structure of interest rates rather than a single scalar underlying. In addition, volatility often will be described by a term structure or surface as well. For such instruments, it is more natural to look at their dependence on the entire term structure and its volatility. To do so, it is necessary to extend the usual notion of partial derivative with respect to scalar-valued quantities to curve- or surface-valued quantities. The term "generalized Greeks" describes the resulting partial derivatives.
The use of generalized Greeks provides a solution to the problem of not being able to add the Greeks from different assest classes together. The idea is, instead of using multiple instrument-specific valuation models with a few scalar parameters, to use a single comprehensive valuation model with functional parameters that can be applied to all instruments. The resulting generalized Greeks can be meaningfully added across assest classes to provide a unified description of portfolio exposure to interest rate and interest rate volatility risk.
FUNCTIONAL DERIVATIVES
A real-valued function of a curve or surface is termed a functional. For example, consider an interest rate instrument whose value v is a function of the current term structure. The current term structure is described by an instantaneous forward-rate curve for maturities in the interval [0, T]. This curve is represented as a function, f : [0, T] *R, where f(T) denotes the instantaneous forward rate for maturity T. Denote the set of all continuous real-valued functions on [0, T]by F. The value of our instrument is given as a functionalv : F *R.
The sensitivity of our instrument to various perturbations of the forward curve is described by taking directional derivatives. We define the directional derivative of v at f0 in the direction f by
Note that Dv(f0) .f is just an approximation of the change in v induced by a perturbation f in f at f0, the approximation being increasingly good for small perturbations f. We thus define the derivative Dv(f0) of v at f0 to be the linear function that maps the perturbation f * F to the approximate change Dv(f0) . f in v under that perturbation.
Example 1
Consider a bond paying amount ci at time Ti for i = 1,..., n. The value at time 0 is given by
Define the zero-coupon discount factor for maturity T by P(T). A straightforward calculation shows that
where 1{[0,Ti]}denotes the indicator function for the interval [0,Ti].
In example 1, the derivativeDv(f0) was represented by the integral over [0,T] of the product of f with another function, namely -(sum)ni=1ciP0(Ti)1{[0,Ti]}.Quite generally, the derivative of any valution function can be represented in the form
where *v/*f*f=f0 is a real-valued function on [0,T] termed the functional derivative of v with respect to f at f0. Intuitively, *v/*f(T) describes the sensitivity of v to a localized change in f at T. As *v/*f is simply a function in F, it provides a convenient graphical representation of Dv(f0).
GENERALIZED DELTA
Generalized delta is defined as the functional derivative of v with respect to f at f0.
Example 2
In Figure 1, generalized delta for a bond paying annual USD6 coupons and returning principal of USD100 in five years. It is assumed that interest rates are flat at 5%. From the graph it can be seen that, e.g., a 0.1% increase in forward rates over the two-three year maturity interval would decrease the value of the bond by approximately USD0.09.
GENERALIZED GAMMA
Generalized gamma is the second derivative of the function v with respect to f at f0 .
GENERALIZED VEGA
Generalized vega is the partial derivative of the valuation functional v with respect to the volatility surface, *, used to value the instrument. For example, in the Heath-Jarrow-Morton model, volatility of instantaneous forward rates is described by a volatility surface that is a function of maturity and time.
As an example, consider an at-the-money European call that expires in two years on a zero-coupon bond that pays USD100 in three years. The option is valued using a single-factor HJM model with a volatility surface of the form *(t, T) = 0.01e-t/10, i.e., it is flat with respect to maturity, has an absolute standard deviation of 1% at time 0, and is exponentially decaying in time with a time constant of 10 years. It is assumed that the current term structure is flat with instantaneous rates at 5%. Figure 2 displays generalized vega. Note that generalized vega for this option is non zero only on the maturity interval between two and three years and the time interval between zero and two years.
This week's Learning Curve was written by Alvin Kuruc, a senior v.p. of Infinity, a Sungard Company.
