Recently, during a Numerical Analysis lecture, our lecturer noticed that what we should have mastered by now remained elusive to many. As a challenge, he assigned us a paper: "Examine the importance of calculus, specifically in the financial industry."As the 'comrades' we are we, all laughed. I barely think most of us will actually write this paper but I took the challenge. Have I ever really looked into why I study complex Calculus?: From Calculus 1 to Calculus 3 and now Ordinary Differential Equations, I wondered when and how I'd apply this mathematical knowledge.

Today, I embark on a journey to uncover the essence of calculus, who devised this mental puzzle, and its practical applications. In this article, join me in discovering the reality of mathematics, particularly calculus, and its real-world implications within the financial realm. With this, I aim to complete my assignment and offer a piece for my weekly article series.

So what is calculus?

Well i’m sure if this question went around on the street as the random content creators do with random questions, this would get the most convoluted responses, and for sure very far from the true definition of calculus. As you read this, you may anticipate a complex definition for something that has, at some point, kept you awake while studying and perhaps even led to frustration. I can relate.

Well, i'm happy to share that calculus is simply the study of change. As straightforward as it sounds, it delves into the mathematics of change. The word calculus actually means “small pebble or stone” in Latin and is named after the small rocks used by the Romans for counting. It's amusing how a name tied to "small" introduces one of mathematics' most intricate concepts.

It has two branches; integrals and derivatives. Integrals show the accumulation of discrete values of a given function over a range while derivatives define the rate of change at a specific point. We'll see more on that!

What does it do?

Well now that we grasp that calculus is just the study of change, what really does it do and maybe how does it do it? Calculus simply measures the rate of change that occurs in almost every phenomenon in the universe. The operative word is measure because then now we can actually quantify. By this, it can help us determine how fast something is changing at any given point. A good example, speed of a car. Yes, that equation you learned in primary school—speed equals distance divided by time—marked your inaugural encounter with calculus.

How does it do it?

Well calculus hinges on three core elements: limits, derivatives, and integrals.. Worry not, I shall walk you through what these terms mean. A limit is a fundamental concept that describes the behavior of a function as it gets closer and closer to a particular point.

Okay, let me explain this better, now when a boy is driving his toy car most at times the car can go so fast that it bumps into a wall. Now, if the he wants to drive the car as fast as possible and every time he does so the boy doesnt want to hit the wall he’ll keep driving it faster and faster and the more he does the closer he comes to a speed that is fast enough not to hit the wall. That speed is simply the limit.

A derivative therefore is a representation of the rate of change in a function at a particular point. Very good, back to our toy. Now the toy is on a winding road and the derivative of a function is like figuring out how fast the car is going at one specific spot on the road. This will happen if that one spot is closely monitored, measuring how much speed changes if moved just a tiny bit.

Finally we have integrals. An integral represents the accumulation or total of quantity over an interval. For this one we’ll have a jar of sweets and you want to know the number of sweets in it. To find out, you take the jar lid and several similar ones, which if you’ll notice are close to the same size as the jar’s opening. What you now do is fill each with the sweets and count how many sweets can fit in each lid and add up all those counts. That result number of sweets is almost like the integral. It tells you how many sweets could possibly be there in the jar.

So where do we actually use it?

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Well calculus is used every single day in a number of fields. This is a quick search up on google and you’ll have them there. So since i have a vendetta with this article lets focus on the financial industry. Here is what i have come across:

1. In Banking we have:

- Interest calculation where calculus is used to calculate compound interest, which determines the growth of savings accounts, loans, and investments over time.
- Risk assessment where calculus helps banks assess credit risk by analyzing historical data to predict the likelihood of loan default.

2. In Investment Services we have:

- Portfolio optimization where calculus is used to optimize investment portfolios by finding the mix of assets that maximizes returns while minimizing risk.
- Option pricing where the Black-Scholes model, based on calculus, is used to price options and derivatives, crucial for trading and risk management.

3. In Insurance we have:

- Actuarial science where actuaries use calculus to model and calculate insurance premiums, estimate future liabilities, and assess the financial health of insurance companies.
- Risk analysis where calculus helps insurers analyze and price various types of insurance policies, such as life insurance, health insurance, and property and casualty insurance.

4. In Real Estate we have:

- Mortgage financing where calculus is used to calculate monthly mortgage payments, determine amortization schedules, and assess the financial feasibility of real estate investments.
- Property valuation where calculus helps appraisers estimate the value of real estate properties based on factors like location, size, and market conditions.

5. In Capital Markets we have:

- Stock price analysis where calculus is used to analyze stock price movements, calculate price volatility, and develop trading strategies.
- Bond pricing where calculus helps determine bond prices, yields, and the impact of interest rate changes on bond portfolios.

6. In Financial Technology (FinTech) we have:

- Algorithmic trading where FinTech companies use calculus in the development of trading algorithms that execute high-frequency trades and optimize investment strategies.
- Risk management tools where calculus is applied in developing risk assessment and modeling tools for online lending platforms and insurance technology (InsurTech) companies.

7. In Private Equity and Venture Capital we have:

- Investment Valuation where calculus is used to assess the valuation of startups and private companies, helping investors determine equity stakes and investment terms.
- Exit Strategies where calculus aids in analyzing exit strategies such as mergers, acquisitions, or initial public offerings (IPOs) to maximize returns.

8. In Regulatory Bodies we have

- Market Surveillance where regulators employ calculus to monitor financial markets for irregularities, fraud, and insider trading.
- Risk Regulation where calculus-based risk assessment models inform regulatory policies aimed at maintaining financial stability.

In a world where calculus may appear as an abstract entity, we must recognize its role as the cornerstone for understanding change within our universe. From gauging a car's speed to deciphering intricate financial trends, calculus emerges as a potent tool that unveils deeper insights into our surroundings. I am happy now I know why I study calculus, and maybe it is not so bad after all.