Unfortunate business names

Tonight, I noticed the following sign of a cycle shop in Uppsala:

From where I was sitting, I read it as SKITOTAL. Depending on where you separate the word it may be something stinky.

One can understand why IKEA in Thailand didn't notice that Swedish product names meant something offensive or sensitive in Thai (see Wall Street Journal article here) but one wonders how come that a Swedish company didn't notice the stinkiness of its name in Swedish.

Holy fish

Talking about animals, and having read this about fish, I remembered last September's story about the holy Swedish salmon or Jesus fish, as it was baptized.

Holy Swedish Salmon

A fisherman in Gothenburg caught this salmon and started shouting out of religious joy. A certain Mr Ludwigson was summoned and verified the holiness of the fish: "I've been in this business for 30 years and have never seen anything like it." The salmon was initially scheduled for a holy supper. But then this was deemed a sacrilege. So a tour of it was planned. First in Malmö were it was put on display in the supermarket. Then in Gothenburg for public viewing at the Nordstan shopping center. I don't know what happened after that.

Swedes are supposed not to be too religious. But, especially in Uppsala, there are many fundametalist Christians, e.g., pentecostals, and mormons and what have you. Once in a while, they appear at the center of town and if you happen to pass by you feel like you're in a bible belt. They preach and shout and try to convert and proselytize and so on. Once in a while you also find some crazy religious acts too. The worst of those fundamentalist Christians are in par with their American peers. In fact, they are the same. This is religion imported to Sweden from the Bible belt.

What's next in the holy fish business? The Atlantic flounder is commonly known as Helgeflundra in Swedish (i.e., Holy Flounder). Despite its name, there is no known report of it appearing with a holy sign. Perhaps it'll come next. This is my prophesy.

Emails from a disturbed individual

Everyone in academia has, from time to time, received mails from people who make extraordinary claims. Recently, I received at least 2 emails from a person called Thierno M. Sow claiming:

The goal of [my paper] is to prove the ABC Conjecture, the Beal Conjecture, the Goldbach Conjecture, the Riemann Hypothesis and the Twin Primes Infinity.

Take a look at the paper he sent me (and others) to see that he is, most likely, a disturbed individual. I am not a psychologist, but it is rather clear that this person suffers from some disorder. His paper contains "mathematics" and "chemistry" and claims to have solved some of the most difficult problems. All in 12 pages. Oh yes, it is also devoted to the memory of Nelson Mandela!

Page 1 of Sow's paper

Page 11 of Sow's paper

Human trap

Listen to the BBC broadcast here, and read this article from the UK Column.

Quotations (as collected by Paul Chernoff)

Direct link: http://math.berkeley.edu/~chernoff/quotations.html

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

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

$f(S')$

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.

Informative signposts

I found this picture on the Internet.The signposts for Alexandoupoli/Feres/Soufli do the opposite of providing too little information.
The first post warns the driver to take the exit in 1000 m.
There is a second post 100 m further down.
Just in case the driver is absent-minded, there is a reminder that the exit is in 600 m.
And then there is another one 400 m before the exit.
Probably there is a final one at the exit, but the picture gets out of focus and I can't tell.
(Who said that Greek signposts are non-informative?)
Enjoy:

Gaddafi

Riots in Libya too.
Which reminded me of Gaddafi's signature.
There is an ancient proverb saying that "your signature is your mirror".

Scientific Peer Review

Ah, the reviewers!

An unavoidable evil.

Vietnamese restaurant

Last week, in Bellevue (Washington State, USA), I saw a Vietnamese Restaurant with a pretty original name: What the Pho'. I was driving but I had to stop to take a photo:

People say it's quite successful and the food is good. I wonder how much the name helps. It certainly helped this store.

Creationist mathematics

I wrote a couple of days ago about creationists and the like. Today I want to give an example of their scientific stupidity by pointing out one of their latest articles on infinity. I first read it in Recursivity.

Creationists have a presence on the Internet via the site "Uncommon Descent". People like Dembski often write stupid articles trying to prove that gods (they call them intelligent designers) exist. In this article, Robert Sheldon writes:

For example, take the number line from 1 to ∞. It’s infinite of course. But now divide every number by the largest number on the line, and we have mapped the entire number line into the fractions between 0 and 1. 

What the hell does he mean by "divide by the largest number"? He may have heard that the transformation x → 1/x maps (0,1] onto [1, ∞) but he hasn't quite understood this high school concept.

So the rational numbers contain the entire integer number line between 0 and 1, and the rational numbers go up to infinity too. 

Excuse me? The rational numbers contain what? The... integer (?) line? And how does this follow from the above?

Then the rational numbers are at least ∞² bigger. (Yup, I’m being sloppy, because Cantor also showed how to map x²–>x, so instead of calling it ∞², he called it ℵ₀ cardinality where integers and rational numbers have the same size infinity.)

Here he is, again, completely off. So much so that it smells from quite far.

[The cardinality of the irrrationals] really is bigger [than the cardinality of the rationals], and Cantor called it ℵ₁ cardinality.

Mr Sheldon, go do your homework before you write anything on infinities.

Now if you are like Cantor's left wing critics, then arbitrary things must be random. It is a peculiar property of atheists that they all worship the god of Chance. It would seem possible that they might worship Lady Luck instead, but no, Xaos, Random Chance, must remain the king of the modernist pantheon. So this contingency drives them bonkers.

This is so funny! Left wing critics of Cantor? Atheists worshipping the god of Chance? What about communists? Homosexuals? Liberals? Punks? People with tattoos? (I think the latter folk don't like Cantor either.)

This is the kind of "mathematics" typically used by creationists. The religious folk will, at the same time, applaud them because, in their eyes, they are oh so sophisticated!

For another example of creationist maths look here.

Creationists and similarly-minded religious imbeciles should leave mathematics for mathematicians and scientists. Instead, they should just go to their church to light a candle or whatever their particular religion tells them to do.

Don't talk to aliens

I'm spending a few months at Cambridge. I couldn't miss it therefore but be informed of the latest announcement of Professor Stephen Hawking, namely:

“To my mathematical brain, the numbers alone make thinking about aliens perfectly rational. The real challenge is to work out what aliens might actually be like.”

“We only have to look at ourselves to see how intelligent life might develop into something we wouldn’t want to meet. I imagine they might exist in massive ships, having used up all the resources from their home planet. Such advanced aliens would perhaps become nomads, looking to conquer and colonise whatever planets they can reach.”

“If aliens ever visit us, I think the outcome would be much as when Christopher Columbus first landed in America, which didn’t turn out very well for the Native Americans.”

However, today, another fact became known to me which, surely, and beyond any doubt, provides the definitive explanation about aliens:

The Gwizls, an alien species far more advanced than humans, missed their invasion of planet Earth and all disappeared in the depths of sea by lack of knowledge of the basic properties of water (H₂O)

The announcement was made by Professor Pierre Brémaud (visiting fellow of the Newton Institute for Mathematical Sciences). Proof below: