Calculus: God's Own Language - II
14th June 2014
[Note: This abridged article is excerpted from the upcoming book Quantitative Finance: The Long Road Through Physics by Rahul Bhattacharya.]
Just like physics, without calculus there would be no quantitative finance. Calculus enters quantitative finance due to the philosophical, as well as the mathematical world view that is known as continuous time finance.
Great advance in theory of finance, especially, in understanding the mathematics behind the dynamics of financial derivatives, has been possible because of a philosophical framework of analysis known as continuous time finance. Continuous time finance comprises techniques for pricing and valuing financial derivatives under the notion of continuous trading. One of the assumptions of the continuous time world view was that time, which is a variable that goes into a model to value or price financial securities, was continuous and, as far as stock trading was concerned, flowed continuously. There were no breaks or interruptions in this flow; around the clock a security can be traded. In other words, the trading of financial securities never stopped on the stock exchanges. In the simplest mathematical sense, continuity means that we can draw a line or a curve on a piece of paper without lifting our pencil. However, this simple notion of continuity gives rise to another troubling concept, that of infinite divisibility. If we say that time is continuous then we mean - and imply - that it is infinitely divisible. And within this world view, the same is true for the stock price or the price of any financial asset that evolves over time. That too, needs to be infinitely divisible.
Without such a world view we would not be able to price financial derivatives or solve many of the difficult problems that relates to pricing of securities and financial derivatives Thus both time and the stock price are continuous and are infinitely - i.e. infinitesimally - divisible. Brownian motion - a kind of random process - of which we will learn more later on in this book, which is used to model a stock price process is a special case of a continuous process. In this the stock price, via independent increments, evolves continuously over time as "time" itself flows continuously without any break.
This interrelationship of ideas of mathematical continuity and infinite divisibility is what drives differential calculus. It is a mathematical trick, but one needs to understand the profundity of this trick. This trick helps us to understand the language of God and makes the understanding of the universe possible. Simply put if a variable is continuous, we can keep on slicking it - like we slice an apple - and we will never reach a point, or a slice, beyond which we cannot proceed. No matter how small a slice we cut out there will always be something left. Mathematically speaking, the variable, or the quantity, approaches zero but never becomes zero. Take the price of a stock and keep making smaller slices of it. Common sense, and a healthy dose of reality, tells us that eventually we will reach some minimum value, say, $1, $10 or $0.10 in which that stock price is denominated. We cannot go beyond that limit. The same goes for time during which this stock is traded on any stock exchange in the world. We will measure that time in hours, minutes, seconds, and, even for those technology minded professionals, milliseconds or microseconds. Beyond that point we simply cannot divide time.
However, the notion of continuous time finance says we can sub-divide time, during which a stock trades, into smaller and smaller slices and keep doing that but we'll never reach zero. In other words, the smallest interval of time over which a stock can trade is smaller than any concept of microscopic time that we have (milliseconds, nanoseconds, etc.) but it never becomes zero. If represent as the smallest interval of time then we write, , to arrive at all formulation of laws about finance, where the arrow signifies that the smallest increment of time approaches zero but never ever becomes zero. And the same idea can be applied to the stock price as well. We can slice and dice a stock price into smaller units and carry on the process infinitely till we reach an infinitesimally small value which is greater than zero but which is smaller than any small value that we can think of.
Most of us, who are not familiar with calculus or have not had any formal training in mathematics, may think that this is just mathematical fiction. But this mathematical fiction, which is the backbone of the study of calculus, allows us to find the value of complex financial derivatives and financial securities very easily and express that in form of nice formulas (which is known as "closed form solutions").
No sooner had Isaac Newton and Gotfried Leibniz invented Calculus in the 17th century a deep contradiction occurred in the philosophical framework with which to view our world and this universe. Over the course of the next three centuries this contradiction would at times remain subsumed in the arguments of the philosophers and at times would flare up in the form of great academic debate. Since the time of ancient Greeks man had believed - and rightly so - that all matter in this world, such as an apple, the water in the rivers and ocean, blocks of wood, the earth, the solar system, every material object was divisible into discrete pieces. In other words, if we keep breaking matter down we will reach a finite limit which is the atom. The ancient Greeks believed, and this belief formed part of philosophical thinking over the course of centuries, that matter is not infinitely divisible. In the early twentieth century, of course, the proof of this doctrine came in the form of quantum physics where it was indeed demonstrated that matter comprised of atoms and other finite - tiniest of the tiny, but finite and discrete - elements such as electrons, protons, and other elementary particles. All matter in the universe was finite and yet the laws which described the motion and the interaction between that matter assumed that the space and time in which this matter resided and moved was continuous, i.e. infinitely divisible. In other words, in the language of calculus, space and time in which matter moved and interacted with each other was continuous.
Calculus, the language of physics, the language in which God talks to us, is based on the notion of "infinitesimals". Simply put, an "infinitesimal" is a number that is not zero but is smaller than any finite number we can think of. The ancient Greeks had abhorred the notion of infinitely large and infinitely small numbers. Infinity was an anathema to them for they believed that it was humanly impossible to comprehend numbers - or, quantities - that were infinitely large or infinitely small. Yet, calculus - or, more precisely, differential calculus, which became the lingua franca of the physicists and philosophers to explain the dynamics of the Newtonian world - could not proceed without the notion of infinitesimals.
Today, high school and college students around the world learn about this notion of infinitesimal the moment they get introduced to the subject of differential calculus (and the concept of Limits, which forms part of differential calculus). And from thereon, it just becomes part of our thinking. When we calculate the slope of a curve given by any function, , where, the letter before the parenthesis tells us that the value of depends on the value of , we make use of the quantity, where, , symbolizes differentiation and this quantity captures a the change in value of , with respect to a "very small change" in the value of . This "very small change", also termed as an "infinitesimally small change" in the value of is at the heart of the issue. In fact, technically, is arrived at by taking limits of the change in the value of as we measure the change in the value of . It is written using the Greek symbol, (called "delta") and write . In other words, we arrive at the expression of , the slope of the curve, only when we make , which means goes to zero. Note the phrase, " goes to zero". It never ever becomes zero but it approaches zero. This is defined as an infinitesimally small change in the value of . And, only at such infinitesimally small level the change in the value of is captured accurately with respect to the change in the value of .
But can we indeed break up time into "infinitesimally" small slices such that it is smaller than any smallest unit of time that we can imagine, such as a hundredth of a second, a millisecond, a nanosecond, etc., and yet it is not zero? In other words, cans a slice (or, a unit) of time be vanishingly small but not zero? Again, can we say the same for a slice of space, or, as in the case of a financial asset can we make its value so small that it is smaller than any real number we can think of (never mind, the absurdity of such a number when expressed as a unit of currency or value) and yet it is not zero? No, of course not. Yet, the key mathematical machinery that underlies the mechanics of asset pricing posits dictates a framework of analysis - the framework of calculus - where space and time are indeed treated as "infinitesimals".
- Penrose, Roger, The Road to Reality, Vintage Books, 2004
- Bhattacharya, Rahul, Quantitative Finance: The Long Road Through Physics, CFE School Publishing, 2014
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