Wilson's School Maths: May 2012

Sunday 20 May 2012

How Facebook became the world's biggest social network - animation

Facebook is hitting the stock exchange with its IPO later this week - and the latest estimates increase its potential valuation, making it the biggest floatation ever. But how did the social network get so big - and can it possibly get any bigger?

Source - the guardian

Wednesday 2 May 2012

The wettest drought ever

April was the wettest on record with Engalnd receiving 75% more rain than it does on average. Yet we remain in drought!

This graph which shows the percentage above or below the average level of rainfall for each month since the start of 2010 shows why one wet month may not solve the problem.

Source - http://www.bbc.co.uk/weather/feeds/17903057

Tuesday 1 May 2012

Black-Scholes: The maths formula linked to the financial crash

BBC News Magazine recently featured a formula that is apparently related to the financial crash - the Black-Scholes model.

$C(S,t)=N(d_1)~S-N(d_2)~K e^{-r(T-t)}\,$
$d_1=\frac{\ln(\frac{S}{K})+(r+\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}$
$d_2=\frac{\ln(\frac{S}{K})+(r-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}} = d_{1}-\sigma\sqrt{T-t}.$

$\begin{align} P(S,t) &= Ke^{-r(T-t)}-S+C(S,t)\\ &= N(-d_{2})~K e^{-r(T-t)}-N(-d_{1})~S. \end{align}\,$

See the Wikipedia article for details. A-level students will notice the following:

it's multivariable
it's got exponentials
it's got logarithms
it's hard!

Here is an excerpt from the BBC article:

Black-Scholes was first written down in the early 1970s but its story starts earlier than that, in the Dojima Rice Exchange in 17th Century Japan where futures contracts were written for rice traders. A simple futures contract says that I will agree to buy rice from you in one year's time, at a price that we agree right now.

By the 20th Century the Chicago Board of Trade was providing a marketplace for traders to deal not only in futures but in options contracts. An example of an option is a contract where we agree that I can buy rice from you at any time over the next year, at a price that we agree right now - but I don't have to if I don't want to.

Options allow a trader to have a delicious risk-free portfolio You can imagine why this kind of contract might be useful. If I am running a big chain of hamburger restaurants, but I don't know how much beef I'll need to buy next year, and I am nervous that the price of beef might rise, well - all I need is to buy some options on beef.

But then that leads to a very ticklish problem. How much should I be paying for those beef options? What are they worth? And that's where this world-changing equation, the Black-Scholes formula, can help.

from http://www.bbc.co.uk/news/magazine-17866646

Wilson's School Maths