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Fundamentals of Deterministic Calculus — applications to Finance (Part 2)
Background
This article is in continuation to Part 1 of this article series that I had written a few days back. For readers who would like to go through Part 1 of this article, the following is the link to the same:
Fundamentals of Deterministic Calculus — with applications to Finance (Part 1)
In this Part 2 of the article, we will discuss the following three items with its applications to finance.
· The Integral
· Partial differentials
· Taylor Series expansions
I will follow the same style as Part 1, wherein I discuss the mathematical concept first and then explain a financial application for the same.
A. The Integral
In this section we will understand the Reimann integral. For any continuous function, we cannot apply the Σ (i.e., summation) symbol. The reason being, in continuous time, we have an infinite number of outcomes that we attempt to combine. Since we are taking a sum of an uncountable number of outcomes, we need to switch to another metric that will allow us to combine the outcomes, and this takes to the concept of integration. An integral allows us to calculate the area under the curve for a continuous function.
Mathematically represented as:
However, if the function to be integrated is not smooth, then it can make the above approximation difficult to solve. For instance, consider a non-smooth function shown as below:
Solving the integral for a non-smooth function as above may not be as accurate as a smooth function. Further, it will take more computational effort to compute the areas of the rectangles under the function. In this case, the areas of rectangles must be very small in order to ensure that we work on smooth sections of this non-smooth function…
