22 March 2012

Quotations (as collected by Paul Chernoff)

The very last one is from a poem by Antonio Machado (scroll at the very bottom of the front page of my blog to see it).

And here is the one above it:
A good mathematical joke is better, and better mathematics,
than a dozen mediocre papers. -- J.E. Littlewood

21 March 2012

"Proof" of Fermat's Last Theorem

Via Recursivity, I became aware of a certain "journal" titled Journal of Mathematical and Computational Science, which publishes articles like this:
A simple proof of the [sic] Fermat's last conjecture and the connection with the Goldbach conjecture, by Ikorong Anouk Gilbert Nemron.
I offer it to my readers as a prime example of a pitiful paper, of no substance at all, which finds its place in a "journal", after, supposedly, having been peer-reviewed by one or more referees chosen by the editorial board. The people at the editorial board should be completely ashamed of letting junk, like the paper above, being published.

But, as a colleague of mine remarked, anything, absolutely anything can be published nowadays, somewhere. The problem with certain journals is that there is zero quality control. I urge my readers to click on the link above and have a laugh. There is material for laugh for everyone.You don't need to know mathematics (neither does the author) to have a laugh. 

20 March 2012

Equinox, or how to define things properly

Google reminded me of Equinox today.
Equinox is defined as a point on the trajectory (and hence a point in time) of the Earth around the Sun at which the line L joining the Sun and the Earth and the axis A around which the Earth revolves are perpendicular to one another. Since A remains, approximately, fixed in space, it turns out that there are exactly two equinoxes (a simple consequence of the intermediate value theorem for continuous functions).

Compare this simple definition with the one given on Wikipedia:

An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator. The term equinox can also be used in a broader sense, meaning the date when such a passage happens. The name "equinox" is derived from the Latin aequus (equal) and nox (night), because around the equinox, the night and day have approximately equal length.
At an equinox, the Sun is at one of two opposite points on the celestial sphere where the celestial equator (i.e. declination 0) and ecliptic intersect. These points of intersection are called equinoctial points: classically, thevernal point and the autumnal point. By extension, the term equinox may denote an equinoctial point.
An equinox happens each year at two specific moments in time (rather than two whole days), when there is a location (the subsolar point) on the Earth's equator, where the center of the Sun can be observed to be vertically overhead, occurring around March 20 and September 22 each year.

Such a simple concept, but such a convoluted definition. No wonder that many people have little understanding of trivial facts, such as the equinox.

Of course, we should not forget that the definition above will not satisfy some creationists, for whom the Earth does not move, because--they claim--the Vatican believes the same thing, and because their religious texts say so (blessed are the poor in spirit, for theirs is the kingdom of heaven; Matthew 5:3).

P.S. The Earth's rotational axis, denote by A above, is not really fixed but moves very very slowly. So slowly that it takes 26 thousand years to complete a cycle. Today, A points towards the star Polaris (the commonly known Northern Star), but 10 thousand years ago it did not. This phenomenon is known as precession of the equinoxes because, as a result of it, the equinoxes change very very slowly too. It was described by the ancient astronomer Ptolemy, about 2000 years ago, who attributed it to Hipparchus who was born 200 years before Prolemy.

8 March 2012

The theorem of option pricing made EZ

I am writing this to convince an analyst friend of mine that the so-called theorem of option pricing has nothing to do with probability and that, philosophically, is very simple.

I will prove the fundamental theorem of option pricing in a trivial case.

Suppose there is a box which transforms the dollars you put in into something of different value. For example, I put 1 dollar in the box and this becomes either 10 dollars or 0.01 dollars. The problem is that I don't know what the output of the box is and also I know nothing about the probability of the outcome. All I know is that 1 dollar turns magically into something else: either 10 dollars or 1 cent.

More generally, suppose that the box takes a token that is valued at $S$ dollars and spits out another token that is valued $S'$ dollars which could be higher or lower than $S$. To be concrete, and also keep things simple, let's say that $S'$ is either $(1+b)S$ or $(1-a)S$. If we put $u$ tokens in the machine, then the machine will spit out exactly the same number tokens all of which will be valued the higher price or all at the lower price. We allow the number of tokens to be any positive number, for example 2/3 of a token is possible. Assume that $0 < a < 1$ and $b >0$.

Now, me being a smartass, tell you the following: "Listen buddy, the machine makes money, not all the time, but sometimes. I give you the following option: You won't have to do anything. I will operate the machine for you. If it makes money I will give you some. If not, you won't get anything."Oh, great", you reply, "go ahead". "Well," I say, "you know, you have to pay me a bit now, so that you get the benefits later." "How much," you ask. "We'll figure it out", I reply.

To make things general let's say that our contract is a certain function
meaning that if the machine turns changes the value of one token to $S'$ dollars then I will give you $f(S')$ dollars.

My rationale is as follows. I'm not a sucker. I won't risk anything at all. I will charge you $X$ dollars and, with this, I will buy $u$ tokens, costing me $uS$ dollars, and put the difference $c = X-uS$ aside. I will put the $u$ tokens in the machine and the machine will change the value of each token to $S'$. In the end, I will have $uS'$ dollars from the machine, plus $c$ aside, which means that I wil have
$Y = uS' + c$ dollars
and since I am a gentleman, I will have to fulfil my promise, meaning that
$Y = f(S')$.
Since $Y-X = u(S'-S)$, we see that
$X+u(S'-S) = f(S')$
must be fulfilled. And this leads to two equations with two unknowns, $X$ and $u$. The equations are:
$X+ubS = f((1+b)S)$,     if the price goes up,
$X-uaS = f((1-a)S)$,    if the price goes down.
Subtracting the second from the first gives
$u = \frac{ f((1+b)S)- f((1-a)S)}{(a+b)S}$.
Putting this back into the second equation, we find
$X = \frac{a}{a+b} f((1+b)S) + \frac{b}{a+b}  f((1-a)S)$.
I observe that my solution is good, because $u \ge 0$ and because both $u$ and $X$ depend on nothing else (not on my astrologer, neither on my mood) except the price $S$ of the token. So I tell you that: I will charge you $X$ dollars. (If $uS$ turns out to be larger than $X$, then I will temporarily borrow $c$ dollars and return them at the end.)

That is all.

Now that you have learned the above, you can create a dictionary of jargon:
  1. Market: it is the box you see above in the picture.
  2. Share: the token.
  3. Stock: a set of tokens.
  4. Bond: the quantity $c$; with $c$ positive (respectively, negative) interpreted as buying (respectively, selling).
  5. Portfolio: the pair $(u,c)$.
  6. Hedging strategy: it refers to the number of tokens $u$.
  7. Option: the function $f$.
  8. Price: the variable $X$.
  9. Completeness: it refers to the fact that there is a unique solution $(u,X)$ to the system of equations. (If $S'$ takes not two, but three values, completeness is lost.)
  10. Arbitrage: the absence of arbitrage is that I make no money. 
  11. Transaction cost: I may charge you an extra fee.
  12. Equivalent martingale measure: You can think of a random variable $R$ taking value $a$ with probability $b/(a+b)$ or value $b$ with probability $a/(a+b)$ (these probabilities constitute the probability measure), write $S'=(1+R)S$ and rewrite the equation for $X$ as $X= E[f(S')] = E[Y]$ (one says that $(X,Y)$ is a martingale).
Who could have ever thought that there is such a rich dictionary behind a simple equation?

By the way, what theorem have we proved? Cast in the fancy terminology, we have proved a theorem saying that, in our complete market with no arbitrage, any option can be priced fairly by using a unique hedging strategy which specifies our portfolio in terms of shares of stock and bonds.

In reality we have proved that I lure you to put your money in the magic box, that I have no risk of losing anything, and that it is you who bears all the risk. However, by charging a bit more than the fair price $X$, by doing the same not just with you but with a few thousand other people whom I attract by designing fancy options $f$, I surely make some money.


What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant