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Chapter 9. Mathematical Tools

The mathematicians are the priests of the modern world.

Bill Gaede

Since the arrival of the so-called Rocket Scientists on Wall Street in the ’80s and ’90s, finance has evolved into a discipline of applied mathematics. While early research papers in finance came with few mathematical expressions and equations, current ones are mainly comprised of mathematical expressions and equations, with some explanatory text around.

This chapter introduces a number of useful mathematical tools for finance, without providing a detailed background for each of them. There are many useful books on this topic available. Therefore, this chapter focuses on how to use the tools and techniques with Python. Among other topics, it covers:

Approximation
Regression and interpolation are among the most often used numerical techniques in finance.
Convex optimization
A number of financial disciplines need tools for convex optimization (e.g., option pricing when it comes to model calibration).
Integration
In particular, the valuation of financial (derivative) assets often boils down to the evaluation of integrals.
Symbolic mathematics
Python provides with SymPy a powerful tool for symbolic mathematics, e.g., to solve (systems of) equations.

Approximation

To begin with, let us import the libraries that we need for the moment—NumPy and matplotlib.pyplot:

In : import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

Throughout this discussion, the main example function ...

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