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Handling Discontinuities

Many functions have jumps or discontinuities at which points no mathematical derivatives exist. These are typically written as conditionals in the source code or by using the math functions abs, max, or min.

XAD generally defines the derivatives of standard math functions as the average of the left and right derivatives at the discontinuity points. For example, the derivative of abs(x) at point x = 0 is set to 0, as the left derivative is -1 and the right derivative is 1.

As this definition is not mathematically accurate, and as this creates problems with higher order derivatives, XAD provides a set of smoothed math functions which are differentiable at all points and can be used as a replacement for the original function. They are implemented to provide accurate derivatives outside a small area around the discontinuity, and approximate the original function using a spline within this area.

As an example, the smooth_abs function is illustrated in the figure below (with c = 0.001):

smooth_abs function

Note that discontinuities may be hidden in conditional constructs in the original code. In order to benefit from the smoothed math functions, the conditionals need to be replaced by functions. For example:

# original code
y = 0.0
if value > strike:
    y = value - strike

# equivalent smoothed code
y = math.smooth_max(0.0, value - strike)
// original code
double y = 0;
if (value > strike)
    y = value - strike;

// equivalent smoothed code
double y = smooth_max(0, value - strike);

A reference of all provided smoothed math functions is given in Smoothed Math Functions.