Friday, December 16, 2011

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

Probably every 7th 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 7th 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”, 5th 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 = ST – 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:

ST – 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 ST = 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, ]