The Kernelyze technology is able to approximate *any strictly sign-regular* kernel, as defined by Karlin (1968), *Total Positivity Volume 1*. A strictly sign-regular kernel can be characterized as one with all order *n* minors having the same sign. Integrating a function with *m* sign changes against a strictly sign regular kernel produces, subject to weak regularity conditions, a function with no more than *m* sign changes. (This is a “variation-diminishing” property.)

Examples include the discounting kernel *e ^{-rt}*, the normal distribution’s probability density function (along with any other probability density function of a one-parameter exponential family of distributions, such as the gamma distribution), the Black-Scholes option pricing formula, the non-central

*t-*distribution’s density, Green’s function of an even-order differential operator (subject to regularity conditions and appropriate boundary conditions), the generalized convolution of any two strictly sign-regular kernels, and the solution of the heat equation (via the Gaussian density).

## Financial

The Kernelyze technology offers a superior solution to hedging bond and option portfolios.

### Bonds

The Kernelyze technology allows one to hedge against moves in the interest rate curve, both parallel shifts and steepening/flattening of the curve.

Understanding the movement of a bond’s price if all interest rates go up or down by the same amount (a “parallel shift” of rates) boils down to an analysis of the kernel *e ^{-st}*, where s is the amount of the shift and t is time. Since this is a sign-regular kernel, KernelyzeBase can determine the (numerically best possible) functions

*f1, g1, . . . , fn, gn*to use in an approximation of the form

*f1(s) g1(t) + f2(s) g2(t) + . . . + fn(s) gn(t)*.

A bond’s value in shift scenarios is an integral (more specifically, a weighted sum) of the kernel e-st, where the sum is over the t variable and the weights are present values of the payments using the base (unshifted) interest rate curve. Taking the same weighted sum over *t* of the numerically-optimal approximation of *e ^{-st}* then provides an approximation for the corresponding bond. The numerically-optimal approximation is the best starting point for this exercise in that it minimizes (numerically) the worst-case error over shifts s in the user’s given range for the hardest bond to approximate whose payments occur in the range of

*t*given by the user. Importantly, the approximate integrals (approximate bonds) are all linear combinations of the same n functions, which leads naturally via the solution of a linear system to hedging and immunization portfolios: using a rank-n approximation, any bond can be approximated by at most

*n – 1*other bonds. Because the approximation works well over a range of rates, this hedging portfolio also works well over a range of rates (rather than deteriorating in quality as duration and convexity hedging portfolios will in larger rates moves).

For sufficiently small shifts, nothing does better than a truncated Taylor series (which leads to duration and convexity approximations and their higher-order cousins). However, the new approximation here is much better than the traditional duration or convexity approaches for even moderate movements in rates, and it adapts easily to problems in which rates curves move in different ways (steepening or flattening, yield curve twists, etc.).

The new approximation here can also handle a change in curve slope, a shift in the curvature of the term structure, or combinations of such types of shifts.

### Options

The Kernelyze technology allows one to hedge against moves in both the forward price and implied vol for a portfolio of options.

Popular option pricing formulae (such as the Black (1976) formula and the normal option pricing formula discovered much earlier by Bachelier (1900)) are strictly sign-regular as functions of the forward and strike, so are subject to the Kernelyze numerically-optimal approximation.

Simultaneous changes in forwards and implied vols can also be addressed by the Kernelyze numerically-optimal approximation with excellent results as shown here.

## Thermodynamics

The Kernelyze technology allows one to approximate heat conduction on a wire, a flat surface, or a three-dimensional body.

## Statistics

The Gaussian (or normal) probability density function is a strictly sign-regular function of two variables, where the two variables are the mean and the usual argument of the density. In fact, all single-parameter exponential-family densities are strictly sign-regular as functions of their canonical parameters and sufficient statistics. The non-central t-distribution also qualifies. Consequently, all of these density functions can be approximated using the Kernelyze technology.