# Autocall pricing: what does "Lipschitz continuous parameterization" mean?

I've been reading through this research paper (A Monte Carlo Pricing Algorithm For Autocallables That Allows for Stable Differentiation by T. Alm, B. Harrach, D. Harrach, M. Keller) about a method for valuing Autocalls. I've understood everything up to page 12 where they introduced a rotation matrix in order to "obtain a (Lipschitz) continuous parameterization of the bounds". I've searched for dozens of sources explaining this concept but to no avail. I have two questions:

• Can someone please explain what does it mean to have a Lipschitz-continuous parameterization of said bounds (or at least point to some literature that explains it) ?

• What problem does this rotation solve, what's the intuition behind it?

Thanks.

• Do you know what lipschitz continuity is? That would be a ace to start. Oct 5 '20 at 1:41
• I do understand it but not to a very advanced level. I understand the Lipschitz Constant, how it relates to the Mean Value Theorem, and how the graph of a Lipschitz-Continuous function lays outside of a double cone along with every secant line of the function.The intuition I got is that the function doesn't vary too much. Oct 5 '20 at 11:01
• Try wikipedia as a start, very nicely explained there. Lipschitz continuity is the second "strongest" type of continuity for functions after differentiable-continuity: en.wikipedia.org/wiki/Lipschitz_continuity Oct 5 '20 at 11:06
• I've read the article but I didn't find it to answer those two questions Oct 5 '20 at 11:22

An example: Consider the XY-plane. Let $$Y(x)$$ be a function of $$x$$. This function can be seen as describing the upper bound of the area below the graph. This function can then have the Lipschitz property, which means that for all $$x_1$$ and $$x_2$$, we have $$|Y(x_1)-Y(x_2)| < K|x_1 - x_2|$$ for some constant $$K>0$$ independent of the $$x_1$$ and $$x_2$$ chosen.
If you are not so familiar with Lipschitz continuity, you can interpret this that the function is quite nice. It is continuous and even more. You might for example note that all difference quotients are bounded. Take any $$x$$ and some small $$h>0$$, and choose $$x_1 = x+h$$ and $$x_2=x$$.
$$|Y(x+h)-Y(x)| < Kh \Leftrightarrow \frac{|Y(x+h)-Y(x)|}{h} < K$$ So if the function has a derivative in some point its absolute value is bounded by $$K$$. The absolute value of slope between two points on the graph is always bounded by $$K$$
You have the corresponding definition in several dimensions if you have functions of several variables and/or vector valued functions: If $$Y$$ is a function from the parameters where the output is vector of bounds, the Lipschitz condition is more or less the same but with vector norms used: $$|| Y(x_1) - Y(x_2) || < K ||x_1 - x_2||$$ for all $$x_1, x_2 \in \mathbb{R}^n$$ and $$Y(x_1), Y(x_2) \in \mathbb{R}^m$$ for some dimensions $$n$$ and $$m$$.