An Example of a Novel Way to Understand Math in Real World - Financial Mathematics

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Probably every 7^th grader will be able to do the following mathematical tasks.

Let ‘s assume we are given the following numbers

            1,000,000        1,435,900        1,500,000        1,400,000       

Pair the numbers 1,000,000 and 1,435,900, like this:

           

            (1,000,000 ,  1,435,900)          Pairing done.

Subtract  1.435.900 from 1,500,000 and show the result:

            1,500,000 – 1.435,900 = 64, 100         Subtraction done.

           

            64,100                                                 Result shown.

Subtract 1,435,900 from 1,400,000 and show the result:

            1,400,000 – 1,435,900 = - 35,900      Subtraction done.

            -35,900                                                Result shown.

Would you be surprised that this is mathematics that is thought in undergraduate studies in quantitative finance? Now, of course, that’s not the whole story, because there are other more exotic parts of mathematics that are taught as well, even within the same course. Those other parts are probability, statistics, and stochastic calculus, for instance. Yet, it’s amazing that this simple calculations are found in very important illustrations of some central financial concepts.

So, where is the trouble? Why not teach 7^th grader quantitative finance and financial mathematics, since the student already has required mathematical knowledge? As a matter of fact, it is possible. Why it is not done is a different story. A few wrong turns in math education and you are lost in numerical labyrinth for the rest of your career. I want to rectify that.

Here is the actual explanation where these numbers come from. The following is the excerpt from an excellent, extraordinary book (“Options, Futures, and Other Derivatives”, 5^th edition, John C. Hull)  in which the definition of a forward contract and certain trading technique associated with it are defined.

While mathematicians are satisfied with the starting, almost innocent phrase ”Let’s assume", "we are given", or "suppose”, and start writing numbers on paper a-priori, without any explanation, directly from fundamental axioms (be it ZFC or Peanno), a financial specialist must deal with, has to strictly define logic and provide reasoning where and why these numbers came into consideration. It is not enough to say that “we assume”, or, “the numbers are given”. Why are they given? Who gave them? Where from?

What a mess of descriptions. For instance, payoff  from forward contract. It is only one of many non mathematical concepts, concepts extraneous to mathematical world, concepts never used in the proofs of mathematical theorems. The others are buy, sell, outcome, position, forward contract, trade, bank, treasury! One has to know these definitions, their relationships, logic that applies to them way before even considering to enter number handling generated by these concepts. Postponing the introduction of the role of proof in mathematics teachers may further blur the boundary between pure and applied math.

Payoff appears to be the word of the day! It specifies a numerical procedure to be performed on selected numbers. Also, legal terms are thrown into the payoff and forward contract definition mix as well, like corporations is legally bind, it's obliged to keep its part of the contract, be it buy, or sell the asset. Obliged is an additional property to buy GBP 1,000,000 for $1,435,900. Note how these properties are added as flavours to the numbers 1,000,000 and 1,435,9000. Math here is very simple. two numbers are given! That's it! But, what is behind the definition of "given" is very important in finance.  From mathematical point of view ( i.e from setology point of view) the numbers' existence is guaranteed by fundamental axioms. No explanation necessary. All those classification, including different currency, who owns the currency, who buys and who sells, is outside math! So, these two numbers are linked together through some kind of specific financial logic, which has its own vocabulary and conceptual relationships. Mathematically, it's sufficient to pair these two numbers, like this:

(1 000 000,  1 435 900 )

What is attached to this pair of numbers is the reasoning why we paired them,  and the financial definitions extraneous to mathematical world. It's the world of financial concepts and relationships, and they do not belong to math.

As we continue reading about this forward contract trade, we come to the concept of "spot exchange rate". This is another "number generator" or "number picker". Now, we have a triplet of selected numbers, with the exchange rate definition in the background:

(1 000 000, 1 435 900, 1 500 000)

From math point of view, the number 1,500,000 is added arbitrary, i.e. math does not see the reason where and why this number is coming from. It's given. We suppose it. It's assumed is there. How we can ignore the fact of spot price presence, exchange rate, etc? Because in order to subtract these numbers, the mentioned definitions are irrelevant for the subtraction. Hence, the words "It's given..." immediately isolate pure math operations and numbers from the set descriptions(of sets they belong to) and from the objects definitions they represent count of. Notice how the words "Let's suppose" can blatantly keep you in dark about, sometimes, beautiful logic inisde the field math is applied to, and how it can suck out any pleasure in working with applied mathematics.

Then, there comes the question "how much is forward contract worth?". This particular question dictates which numbers will be picked and which math operations will be performed on them. 

Again, from mathematics point of view, it is specified, without any further explanation, which numbers are in game. For math, it is enough to use word “IF” and start generating numbers and their relationships. This “IF” implies usage of fundamental axioms. Hence, IF you have number 1,500,000 and IF you have number 1,435,000, deduct the second number from the first. That’s what matters to mathematics. To finance, the reasoning why you deduct second number from the first (and not vice versa).

Payoff = S_T – K

Who would expect that the number 1,435,000 will have the following description: it is a six months forward offer quote for USD-GDP currency exchange. Note how this definition has almost nothing mathematical in itself (except number six, for six months). Cardinalities of a set are not part of this definition. You have to know all this just to pick one number! And that knowledge matters. If your non mathematical logic is flawed, you will select a wrong starting number and, even if your subsequent mathematical operations are perfectly accurate, the result will make no sense within the applied field, because the initial number was wrong.

Educators can ask math students the following question: "Give me an example what number 1,435,900 can represent count of". And, student will start searching from his or her experiences what can have that count. it can be 1,435,900 apples, 1,435,900 pears, 1,435,900 oranges, 1,435,900 birds, 1,435,900 rockets, even 1,435,900 thoughts. But, would you expect from a student to give you the following interpretation: "Number 1,435,900 is the number of dollars a bank is offering in a 6-months forward contract, in currency exchange for 1,000,000 British pounds on August 16, 2001" ? Probably not. You don't expect student to know financial concepts at that early age. And, moreover, why would the financial field should be in focus for this example. So, what is the point of asking student to give you example of a number, unless you want to indicate that there have to exist World # 1, in this case financial world, that has its own set of rules, axioms, premises, definitions, whose logic will define to which set the number 1,435,900 belongs? This financially based description apparently can not be deduced by looking at the number only. And, that there is World # 2, world of pure math, which starts with "given" numbers, no matter what is the reasoning of obtaining that number. Distinguishing these two worlds is in the essence of understanding mathematics.

Look at the freedom of how the scenarios in this trade are created: if spot rate falls to 1,400,000. Note the "number picker"! It's "spot rate"! Look how the fundamental axiom that number exist, is disguised in this functional description why and what picks actual number. From math point of view, this is the same as "number is given", 'let's suppose", "let's assume".

The interplay between pure math and our numbers is further advanced with actual logic between numbers' sources. While the sources themselves do not belong to math world, the numbers picked by them do. Hence, the rules between these "sources" (which are extraneous to mathematics) indirectly influence which numbers will be picked and will enter the specified calculation. Also note that the actual math operation is motivated by the things and reasoning outside math. We want to do subtraction because we want to find the payoff! Payoff, as a concept, has nothing to do with math. If it does you will see theorems in mathematics proved using or referencing this concept. But there are no such theorems. So, we have payoff as:

Payoff = 1,400,00 - 1,435,900 = -35,900

So, here, you deal with specifications what each number represents, and financial logic and reasoning structures what to do, what sequence of mathematical operations to perform. Whether the result of this calculation is called "payoff" or "orange with freckles" or "space carrot", mathematics couldn't care less. Math sees only two numbers provided to it and math operation to do. It's up to you to keep track what is counted and why.

In financial application of mathematics, we want to generalize the math operations on specific numbers, using, apparently the English language and financial terms and definitions from the financial domain of counting.  Hence, we will say, the payoff from a long position in a forward contract on one unit of asset is:

S_T – K

Look how we have described the logic and requirements what to do with numbers. This reasoning is completely outside mathematics, and the sequence in subtraction matters to the financial domain. Math just see the numbers and subtraction. What these numbers represent, i.e. which set they belong to, is described in financial terms. We have S_T = spot price and K = forward contractually specified price. The pairing of these numbers, before even any mathematical operation is done on them (in this case it is subtraction), is specified by the definition of a forward contract, by existence and definition of market, concept of spot trading. This is what is required for you to know to pick right cardinalities at the first place, before doing any mathematical operations on them.

You have a number, say, 1,500,000. Pure number. Units are not yet assigned to it. The number will remain the same, but the definition what it represents will change in accordance to some domain rules. In finance, these rules are defined by "buy" "sell" "obliged" "exchange" "asset" "forward" "payoff". These definitions and rules change the ownership of that cardinality, that number. Note how number, and for that matter quantity of dollars it represents, remains unchanged. What is changed is who owns that amount of currency, and that's not a mathematical concept.

It is a "fierce pairing of numbers" and "fierce changing" of descriptions of sets to which those (same!) numbers belong to which is part of the forward definition and many aspects of trading. The similar conclusion applies to other domains of applied mathematics.

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[ applied mathematics, financial mathematics, quantitative finance, math applications, math examples, learning math, ]

Contextual triviality. What Has to be Done When Someone Uses the Word "Logic".

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When someone uses the word "logic"he/she should immediately point out what is initially assumed to be true/false and what truth results are derived, i.e. what can be proved from those initial assumptions.Word logic is nice to use, it can be fancy, it can show you want to be precise in your communication or explanation. However, the axiomatic system should be known and understandable for all the parties that are part of the "logic" communication. Saying that something is logical doesn't mean it is obvious, and it doesn't mean it does not require a proof. If something appear to be "trivial", the logical context of axioms and premises should be clear to all participants who want to accept that "trivial" remark. Contextually "trivial" is OK.

The other day, while browsing mathematics section at Indigo bookstore, I have noticed a book "Logic for Mathematicians" by A. G. Hamilton, (Google books http://goo.gl/17kEj). I was pleased to see that the book reflects my view that logic is an independent discipline from mathematics and that mathematics is only one of the area of the application of logic. While Frege may have integrated both directions of thinking, I am glad that A. G. Hamilton "presented the subject matter without bias towards particular aspects, applications or developments, but an attempt has been made to place it in the context of mathematics and to emphasise the relevance of logic to the mathematician.".

To me, this is important because of my view (most of the posts in this blog) to differentiate clearly the worlds that define what is to be counted, measured etc, from the mathematical world that accepts pure numbers as starting points. Logic is used in both worlds.



[ logic, mathematical logic, math, math concepts, axioms, mathematics, teaching math, teaching mathematics, understanding mathematics, Frege, Hamilton, ]

Real World Example of Natural Numbers

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In real world application of math we have to keep track what we have counting, while pure math cares only about pure numbers.

This is an example of counting rounds in an UFC match (UFC - I do not approve nor like).

[ math, mathematics, math application, axioms, examples of natural numbers, natural numbers, axioms, sets, number theory, ]

Free Will

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In the very moment, when you breath, when oxygen molecules split the molecules of your brain biochemical energy storage, the very state of neural firing, dynamics, neurotransmitters rush and retreat that exist at that moment, is the consequence of your free will, the will riding on neural paths configurations and their energy that you release by thinking in exactly that way..at that moment..

A Few Hints How to Introduce Mathematical Concepts

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Teacher should postpone introduction of strange, exotic mathematical names and labels to, otherwise, most likely, easy to explain and easy to understand concepts in mathematics. Teacher should  first explain mathematical concepts in terms of required sequence of mathematical operations on numbers and on sets involved and then introduce labels or historically accepted names for them. Most of us will agree that many of those names are there for historical reasons, and often they are confusing, misleading, and even intimidating (if Borel, Hausdorff needed so much work to prove that theorem or formulate it, what chances do I have?). To me, it is sometimes better to refer to a theorem by a number first, like Theorem 1, Theorem 2, … and only later assign historical names or other labels to reference them.

Math lectures sometimes look to students like putting misleading, cluttered, over-detailed traffic signs on an easy to use roads and highways.

But, what are we really doing or what we want to do when we say "introducing math concepts through A or B or C real world examples"? Do math concepts need to be introduced through real world examples at all? No, they don't. They can be derived or formulated directly from axioms.  So, what would be the goal of "introducing math" through real world examples? To show that math concepts can be motivated by real life examples, but, at the same time the same math concepts can be derived inside math only, without referencing any real world domain. To me, the mandatory step of introducing some math concepts from real world example is to mandatory show that the same concept can be defined or derived from ZFC axioms. This would clearly show the border and connections, if you wish, at the same time, between applied math and pure math developed from ZFC axioms.

[introducing math, math ], math concepts, math education, mathematics, theorems ]

What is a number, really?

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In this post I would like to introduce a concept of a number. I want to show a method how we can conceptualize a number, to actually understand what is it. This introduction will provide multiple benefits for anyone interested in deeper, fundamental understanding of mathematics. It can provide answers about what exactly is the subject of research in mathematics. The explanation can be a useful and effective starting point for all those creative minds who ask “why mathematics?” “what for we have to calculate all that?”, and “what is the number actually?”.

Let’s say, to start with, that we can conceptually, visually or in our minds, differentiate objects among themselves, and that we can, and then we want, to count them. When we, for whatever reason, group objects in some collections, we can be in situation to determine which collection has more objects, if we want to. Don’t forget, we don’t have any numbers, or names for counts, defined yet.

How we can determine which set, which collection has more elements, objects?

We can compare collections of objects by matching, pairing, elements from one collection with objects, elements from another collection. Pairing has to be, obviously, one to one.  Let’s say, we have a set of apples and set of pencils, as shown in the Picture 1. We want to find out a number of elements in each set.

We can start first by determining which, if any, set has more elements. In order to do that,  we only need to match, to pair, one object from one set with an object  from another set. If no objects are left unmatched then two sets have the same number of elements. As we can see in the Picture 1, by pairing apples and pencils, no objects are left unmatched, hence these two sets have the same number of elements. We do not have the name yet for that count, for that number, but, the good thing is we know what we are talking about! We are talking about certain number of elements, defined by the exact match of two collections. That property is what we are after, that “numberness” is what we are after.

Now, note one very interesting thing. If we replace apples with pears, and pair, match every pear with every pencil, we can see that the match is again achieved! The pairs are again complete and no pears or pencils are left unmatched. Hence, these two collections, two sets, have the same number of elements. Now, let’s introduce another collection of objects, say watches. Pair the watches with pears. As we can see it’s the same count, complete match, hence the same number of elements. Note very, very interesting observation here! No matter what objects we are matching, as long as the match is one to one, and as long as the pairing is complete, we have the same number of elements in two collections.

It is this property, this “numberness”, common in pairing two sets, that we call a number. You see, no matter what kind of objects are in two sets, if they match, that property, is the actual number. That is the concept of a number! It is, at this point, completely arbitrary how we are going to label this numerical concept we have just discovered and defined. Word, symbol, reference for it is completely arbitrary. In English, it is called number five, and the numeric symbol is 5. Label for our new concept is really of very minor significance at this point. The concept we obtained is way more important than the tag we will use to reference it in our speech. Of course, we could start with three object and obtain number three, or seven objects, and obtain number seven, etc. Notice how we, now, have this set property to work with, set property related to its number of elements, the quantity of objects, that we have abstracted from any two sets of objects, that we can call a number, or a count! That’s the actual concept of a number. Notice, also, how “number” in its essence, is not even a noun, but more like an adjective, that describes “quantitative” aspect of two sets, the number of paired elements of two collections.

Picture 1. One example of number conceptualization.

As long as we know that this labels, 1,2,3,…represents that property of one to one pairing between the elements of two sets (with the goal to determine if they have the same number of elements) we are on a good path to work with numbers and quantification.

You may ask at this point “well, I don’t always see two sets when I count objects of one set, I just count them without any pairing with the elements of another set”. Good question! What you actually do, by, say, counting CDs, in your collection, is matching them with the set of natural numbers in your mind, which is completely ok. But, note, you have natural numbers at your disposal to use them for counting other objects, while in our previous explanation we are actually just defining the numbers! We qwere after very definition of a number. Once the numbers are defined, as we did for number five, you can use that number five and others in counting any kind of objects!

Look at that number five, universal count of five, for any kind of objects. Here is one more interesting conclusion. You see, how at this point, we can deal with counts only! We can deal with a count 3 and a count 5, which we call a number 3 and a number 5, regardless which objects they represent count of! If we want to add them, it will always be 3 + 5 = 8, no matter what we have counted! This is exactly what pure math is about! We, now, can use our generalized knowledge that 3 + 5 = 8, and utilize it any real world situation, for instance, if we have 3 cars and we buy 5 more cars we will have 8 cars.

Note that “purity” of math is just related to the fact that we do not care what we have just counted. We were only interested in adding, subtracting, dividing, pure counts, pure numbers we have abstracted from real world objects counting.

If we want to mix pure math and real world scenarios and objects we are counting, it’s easy! We can just put a small letter beside the number, to keep track what we have counted. Hence, in physics we have 3m + 5m = 8m, for distance, length in meters. Then, also we can put 3 apples + 5 apples = 8 apples in agriculture studies. We essentially do two steps here, during the additions of real objects. When we want to add 3m + 5m, we actually separate pure numbers from the meters counted, we enter with these two numbers the world of pure math, where we do calculation of numbers only, 2 + 3 = 8, and then we go back to real world of meters (because we have those small letters to remind us what we have counted) with the result 8, and associate the name of the object, in this case it’s a physical unit of length, or distance, which is meter, (m), to the number 8. And, voila, we have just used pure math in the real world application!

[ applied math, applied mathematics, concept of a number, concept of a set, natural numbers, number, set and number, set theory ]

Mathematical Proof for Enthusiasts - What It Is And What It is Not

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With the term mathematical proof we want to indicate a logical proof, i.e. proof using logical inferences, in the field of mathematics. So, it should really be “a proof in the field of mathematics”. Also, we have to assume, and be fully aware, that proof must be “logical” anyway. There are really no illogical proofs. Proof that appears to be obtained (whatever that means) by any other way, other than using rules of logic, is not a proof at all. Assumptions and axioms need no proof. They are starting points and their truth values are assumed right at the start. You have to start from somewhere. If they are wrong assumptions, axioms, the results will show to be wrong. So, you will have to go back and fix your fundamental axioms.
When you have first encountered a need or a task for a mathematical proof, you may have asked yourself "Why do I need to prove that, it's so obvious!?".

We used to think that we need to prove something if it is not clear enough or when there are opposite views on the subject we are debating. Sometimes, things are not so obvious, and again, we need to prove it to some party.

In order to prove something we have to have an agreement which things we consider to be true at the first place, i.e. what are our initial, starting assumptions. That’s where the “debate” most likely will kick in. In most cases, debate is related to an effort to establish some axioms, i.e. initial truths, and only after that some new logical conclusions, or proofs will and can be done.

The major component of a mathematical proof is the domain of mathematical analysis. This domain has to be well established field of mathematics, and mathematics only. The proof is still a demonstration that something is true, but it has to be true within the system of assumptions established in mathematics. The true statement, the proof, has to (logically) follow from already established truths. In other words, when using the phrase "Prove something in math..." it means "Show that it follows from the set of axioms and other theorems (already proved!) in the domain of math..". Which axioms and theorems you will start the proof with is a matter of art, intuition, trial and error, or even true genius. You can not use apples, meters, pears, feelings, emotions, experimental setup, physical measurements, to say that something is true in math, to prove a mathematical theorem, no matter how important or central role those real world objects pr processes had in motivating the development of that part of mathematics. In other words, you can not use real world examples, concepts, things, objects, real world scenarios that, possibly, motivated theorems’ development, in mathematical proofs. Of course, you can use them as some sort of intuitive guidelines which axioms, or theorems, you will use to start the construction of a proof. You can use your intuition, feeling, experience, even emotions, to select starting points of a proof, to chose initial axioms or theorems in the proof steps, which, when combined later, will make a proof. But, you can not say that, intuitively, you know the theorem is true, and use that statement about your intuition, as an argument in a proof. You have to use mathematical axioms, already proved mathematical theorems (and of course logic) to prove the new theorems.

The initial, starting assumptions in mathematics are called fundamental axioms. Then, theorems are proved using these axioms. More theorems are proved by using the axioms and already proven theorems. Usually, it is emphasized that you use logical thinking, logic, to prove theorems. But, that's not sufficient. You have to use logic to prove anything, but what is important in math is that you use logic on mathematical axioms, and not on some assumptions and facts outside mathematics. The focus of your logical steps and logic constructs in mathematical proofs is constrained (but not in any negative way) to mathematical (and not to the other fields’) axioms and theorems.

Feeling that something is "obvious" in mathematics can still be a useful feeling. It can guide you towards new theorems. But, those new theorems still have to be proved using mathematical concepts only, and that has to be done by avoiding the words "obvious" and "intuition"! Stating that something is obvious in a theorem is not a proof.However, using own intuition to construct a proof or to formulate theorem is definitely useful.

Again, proving means to show that the statement is true by demonstrating it follows, by logical rules, from established truths in mathematics, as oppose to established truths and facts in other domains to which mathematics may be applied to.

As another example, we may say, in mathematical analysis, that something is "visually" obvious. Here "visual" is not part of mathematics, and can not be used as a part of the proof, but it can play important role in guiding us what may be true, and how to construct the proof.

Each and every proof in math is a new, uncharted territory. If you like to be artistic, original, to explore unknown, to be creative, then try to construct math proofs.

No one can teach you, i.e. there is no ready to use formula to follow, how to do proofs in mathematics. Math proof is the place where you can show your true, original thoughts.

[ set, set theory, concept of a set, sets in mathematics, real world, applied math, applied mathematics, axioms, math education, math proof, mathematical axioms, mathematical proof, mathematical theorems, mathematics, theorems, tutoring ]