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The math of kinks

How to engineer piecewise linear functions

But, why?

I was reading Professor Donald G. Saari's "Mathematics of Finance: An Intuitive Introduction" and I came across section 2.3.2 on designing financial portfolios of puts and calls. Before continuing, I encourage you to check it out here!

The section highlights how to compose piecewise linear payoff functions using three types of options:

  1. all call options,
  2. all put options, and
  3. a mixed combination of both.

But, upon closer inspection, the chapter only covers how to draw "mixed" option graphs composed of puts from the left and calls from the right of a desired price interval. What if we want to draw more complex piecewise linear functions with multiple "kinks"?