Archive for the ‘Mathematics’ Category

Interesting quant blog

June 17, 2012

A quick post to show a link to the blog of Tim Johnson at Heriot-Watt Unviersity.

Lots of writing on derivatives topics, from models to regulations, plus plenty on markets and the crisis.

Worth a look. Click here to visit.

Are our models too complex?

June 17, 2012

Gillian Tett is a well-respected writer for the Financial Times and frequently picks up the topic of complexity in financial markets.

In a recent article (see here) she makes a case that the era of number crunching is over, and that the world of investments is back again firmly in the domain of human relationships and evaluations.

Once upon a time we would measure credit risk with a numbers like survival probability stripped from CDS market prices. Once upon a time we were all happy to value a transaction with models that almost no-one other than the quants understood.

Her view is that these times are gone.

Galois Theory for dummies

April 21, 2012

This post is a LeanPost: it will be developed further depending on feedback from my readers.

See my note here on what a LeanPost is.

Update, January 2014: Dear Reader, for the last year I have been struggling to find time to work on this blog, and this article is particularly in need of further work. The start is good, and people have commented that they like it, but for me to continue I need to devote quite a lot of energy to my Galois notes and I think this is not going to happen soon (I am very busy at PrismFP you see). In the meantime, let me just add one link to an article which I think is about 80% the right kind of explanation of Galois theory. Here it is (in my opinion it could be improved, but I cannot do that at the moment so I leave you only with the link). Here is another presentation which is good too. End of update.

The most popular undergraduate text (in the UK at least) for a course in Galois Theory is by Ian Stewart from the University of Warwick (see here). If you google for “Galois Theory filetype:pdf” there are a few presentations available. A couple that sit around the top of the search results are by:

  • Brent Everitt from the Mathematics department at York University (see here),
  • Miles Reid at Warwick too (see here).

My principle complaint at all these texts is that they make exactly the same mistake (though Miles Reid does try his best to avoid it):

they work towards the main result rather than starting from it.

That’s often the case for lots of mathematical texts, but for Galois Theory it is particularly frustrating because there is actually a central theorem.

I say:

“Tell us what the main theorem is, give us an idea of how it works, and then go through all the details of the proof”.

Not the other way around!

This post is my effort at An Introduction to Galois Theory that starts at the interesting bit.


Smile, it’s Volga!

April 9, 2012

Armed with the Hardy Decomposition for option prices, it now becomes much easier to understand why the smile exists.

To be clear, options trader might use the smile to manage supply & demand, but here we discuss the mathematical basis for smile – which is important if you want to understand how to generate smile in a monte-carlo model.


Calculating option prices in your head

April 5, 2012

We all know that option prices are calculated with the Black-Scholes formula, using a volatility, time-to-maturity, strike and forward. Typically you just chuck them all into your computer and let it spit out the number.

Trouble with this is how do you get an intuition for prices, especially when you are looking at options trades like conditional steepeners or calendar spreads?

Recently I set myself the problem of getting some simple way to cross-check the numbers coming out of my PC, and to get some intuition for the way an option’s value decays to its intrinsic value as the forward moves (which is what you definitely need for options trading).

In this post I show you a simple approximation I have found which does a pretty stellar job of accurately telling you the price of an option at any strike, and which you can calculate in your head!


Why does the yield curve slope upwards?

January 26, 2012

In this post I give a short, but I think rather usefully direct reason for why the yield curve should slope upwards. All it requires is for you to put yourself in the shoes of an investor that has to lock up their money in a bond for a fixed amount of time (and a very simple piece of algebra).

Yes, you can find plenty of papers which give complicated economic reasons for why this should be the case, but I tend to think that it is the simple insights that explain most of what you see happening in financial markets.


What is the risk-neutral measure?

January 24, 2012

Here is a short list of the most common ‘big-concept’ questions that I was asked throughout my years as a quant (whether coming from people on the trading floor, in control functions, or from newcomers to the team), in no particular order:

What is the risk-neutral measure?

What is arbitrage-free pricing?

What is a change of numeraire?

What is the ‘market price of risk’?

I don’t know of a single book on financial mathematics that attempts to give answers to these questions for a reader that is not familiar with stochastic calculus (which most traders are not, of course). Over a series of posts I will put down my own user-friendly answers to these questions, and we’ll end up with a grand Guide to Financial Derivatives Pricing for a Non-Technical User!

This post covers the first: what is the risk-neutral measure?


What was Fermat’s Non Proof?

January 12, 2012

No one in the mathematical world believes that Fermat actually had a valid proof of his famous Last Theorem (click here for the Wikipedia article).

But I’d be interested to see what his non-proof looked like.

Looking at why something is broken is often a good way to get insight into a new research direction. Think of how the mistake in Euler’s proof of Fermat’s theorem for n=3 led to the discovery of the concept of unique factorization: click here to read about this in the interesting Fermat’s Last Theorem blog).

Does anyone know of attempts to best-guess Fermat’s non proof?

Here’s one effort

After a bit of googling I came across the following website which claims to give an elementary proof: click here to see it.

Experimentation in Art & non-deterministic grammars

December 31, 2011

One of my favourite abstract painters is Richard Diebenkorn. Click here to see an Art Blog which has a post on his most famous Ocean Park Series. Click here for a link to a blog showing one of the Dibenkorn’s canvases in an NY apartment.

In these sorts of works it is fascinating how you can see a process of experimentation that resulted in the final canvas — tones of colour are covering earlier lines and colours, and the development of the composition is clearly shown as part of its completeness. The brushstrokes and decisions made by the artist are intuitive, and the artist works hard to push paint around until a pleasing result is achieved.

Here comes the maths.


Intuition for the forward FX equation

December 29, 2011

Every quant knows the expression that defines a forward FX rate on date t with maturity T:

where B_f is the foreign discount factor and B_d is the domestic discount factor. But what is the best way to explain this intuitively? Here is my suggestion.