Fundamentals of Deterministic Calculus -applications to Finance (Part 1)
A) Background
Deterministic Calculus is a key topic in mathematics. Most of us have learnt this topic at undergraduate/postgraduate levels. Some of us also get the opportunity to apply these concepts to job roles. A good foundation of concepts of deterministic calculus plays a significant role in understanding stochastic calculus and know how exactly it is different from stochastic calculus.
That being said, one should not undermine the importance of deterministic concepts because many of them can be readily applied to industry applications. Also, considering the plethora of topics that are studied under deterministic calculus, I have decided to publish a two-part article on this, as it will allow us to keep the document crisp and an easy read. In this Part 1 of the article series, we shall review the following along with their applications to finance:
· Functions and their common types
· Limits and continuity
· Convergence
· Derivative
· L’Hôspital rule
B) Functions
Let A and B be two sets, let f define a rule associating every element x of A to exactly one element y of set B. Such a rule is called a mapping or a function. Note, for our discussion, we are only considering functions with one-to-one mapping.
Mathematically, f: A -> B, equivalently, y = f (x)
Further, if A and B are themselves collections of functions, then f is called an operator.
In general, function is a very lucid concept which may be applied to understand the behavior of various assets/processes encountered in quant finance.
Types of Functions
1. Exponential function
An exponential function can be represented as y = ex
This function is generally used for discounting cash flows / payoff in continuous time.
Finance application: Simulation of stock price in a risk neutral world eg: S(T) = S(0). exp(rt)
2. Logarithmic function
A logarithmic function is defined as an inverse of the exponential function represented as ln(y) = x
This function is frequently used for calculating daily returns from a given asset price time series.
Finance application: Day-over-day return calculation eg: log(S(t+1)/S(t)) Or y = ln(S) is a starting point for arriving at the expression for simulation of stock price path using Ito’s.
Further, as exponential and log functions are anti-derivatives of each other, we frequently switch between these two forms while solving equations.
3. Trigonometric function
Trigonometric functions also find applications in finance. These functions can come handy while working on polar coordinates, Fourier transforms etc.
A simple example of a trig function, y = sin x
Finance application: Fourier transform method may be applied for pricing of options. This technique involve consumption of trig function as a part of the numerical method
4. Functions of Bounded Variation
Assume a time interval [0, T] which we partition into n sub-intervals by selecting t i, with i=1,2,3…n
[t i — t i-1] is the length of each subinterval. Assume f(t) be a function defined over [0,T].We represent the sum as:
We can have an infinite number of partitions. It is important that this summation be of bounded variation. This ensures that the function is smooth and does not represent too many irregularities. As we will see later, having smooth functions makes it easy to perform operations including differentiation and Reimann integrals.
A function with a bounded variation is represented as:
Loosely speaking, V0 measures the length of the trajectory followed by f(.) as t moves from 0 to T.
The bounded variation roughly says that the function does not blow up. For readers familiar with stochastic calculus, this idea may subtly hint towards mean squared convergence.
5. Functions of unbounded variation
In contrast, when there are functions of unbounded variation, these make modelling difficult as the function does not remain tractable and may show tendency to blow up. Below is a pictorial depiction of one such function.
In stochastic calculus too, if we encounter a function that shows this kind of a behaviour, we make appropriate transformations and assumptions to try and keep the function in finite space i.e., not allowing to collapse to zero neither blow up to infinity.
C) Limits and Continuity
Limits play a very crucial role in modelling. For example, consider a continuous time system, wherein time step dt is very very small, we represent it formally as:
Continuity is another related concept. We generally work on functions / processes that exhibit continuity. For example, consider a root finding exercise, wherein estimation of a root becomes feasible when functions do not have any disconnect i.e., they show continuity.
Finance application:
Limits are a central concept when we solve for gradients (think derivatives) which enable us to measure sensitivity of say an option to a small change in the underlying asset. Evaluation of such gradients would not have been feasible in the absence of limits.
Continuity property of functions demonstrates the fact that there are no discontinuities in the function evolution. That’s why we can say a certain function is continuous
D) Convergence
Consider a simple sequence given as: x0, x1, x2, ……. Xn
The sequence may be anything — numbers, functions, operations etc. The logic of convergence of a sequence has to do with the value of xn as nà∞
Convergence plays an important role in ensuring that a certain process remains finite and tractable. This idea forms an important component when we perform analysis of stochastic processes.
Finance application: The idea of Quadratic variation studied under stochastic processes relies on the concept of mean squared convergence which is in-line with the general concept of convergence.
E) Derivative
In simple words, a derivative attempts to explain the variation observed in a dependent variable when the independent variable changes by a certain amount.
A derivative is also defined as a rate of change. However, one point to note is we consider smaller increments in the independent variable. Formally, we define the derivative as:
2nd order derivative is just the rate of change of the first order derivative
For readers who come from an engineering background, we can visualize the 1st order derivative as speed, while the 2nd order derivative may be thought of as acceleration.
A note on how to take limit for the above: Focus on the equation for dy/dx. The limit taken may either be a right hand limit Or left hand limit. In a deterministic calculus setting, we should get an equivalent result.
Finance application:
Bond risk sensitivity: Bond duration and bond convexity measure the interest rate sensitivity of the bond price w.r.t. change in yield:
F) L’Hospital’s rule:
L’Hospital’s rule is a mathematical method for solving limits of indeterminate forms, which are expressions that cannot be easily solved in limits. It can be applied to a variety of situations.
More formally, the rule says that if the limit of f(x) / g(x) is indeterminate, then the limit can be found by evaluating the limit of derivative of f(x) and g(x). Mathematically
Note, this rule can be applied multiple times in a single case, as long as the indeterminate form is still present after each application.
This completes the list of concepts that we had set out to cover in Part 1 of this article series. The concepts discussed above are the foundation for understanding more advanced concepts in applications maths including stochastic calculus which is a building block of quantitative finance.
What to expect in Part 2 of this article
Very soon I will complete the Part 2 of this article series. In that, we will cover a few more concepts that are fundamental to the understanding of deterministic calculus with application to finance, namely:
· Integration
· Taylor Series expansions
· Partial derivatives
and more…
Similar to the style that we used in this article, in Part 2 as well, I will keep the notation as simple as possible so that it remains an easy read for a wider audience.
Thank you for reading!
