compute kth central moments in one pass

Continuing on the same topic as my previous post, its nice to be able to gather all the kth order moments in a single pass.

Last time I mentioned the boost/accumulators example, but you will have noticed two issues if you use that.  Firstly, moment<k> tag will give you the kth simple moment relative to zero, whereas we often want the kth central moment of a sequence relative to the mean.  Secondly, although boosts accumulator is well written it does seem to take a while to compile [~ 12 seconds for code using moment<12>].

After some playing around Ive got a faster simpler approach, where the inner loop accumulates kth powers of the element.  After you’ve run the sequence through, you can then easily extract variance, and all the kth central moments.  So in adding the more general case of kth moments, Ive made the particular variance case simpler.  That often seems to happen in programming and math!

algebra

First a bit of math and then the code.  We want to express the kth central moment in terms of the k basic moments.

First, lets define the basic moment as -

We rearrange the binomial expansion -

So we have the kth central moment given as a weighted sum of the kth simple moments -

which shows that all we need to accumulate as we walk across the sequence is the kth simple powers .

Notice the variance is now handled as a special case where k=2.  Likewise k=0 corresponds to n, the element count and k=1 is the sum of elements.

c++ impl

Heres a basic impl of the above expression -

Slide talk – “Libor Market Models : the reasons behind the success”

I found some very interesting (promo?) slides from SocieteGenerale discussing WHY the Libor Market Model LMM [aka BGM/J, an instance of HJM] has become so popular, despite its significant imperfections.

But are we fooling ourselves that we can still put “the wrong number in the wrong formula to get the right price”?

I think they miss the point, basically you need some parameters to make a good model fit the environment [to paraphrase Dermans definition of useful 'model' is that it allows you to price something from other, somehow related, market observables].  Sure, you need several parameters to get enough flexibility to calibrate to the market.

very punchy talk by emanuel derman

Dermans autobiography ‘My Life as a Quant‘ is a lovely nontechnical read.  It follows from his childhood in South Africa, his desire to study physics and later disillusionment with postdoc study, through developing symbolic math software at Bell Labs, to his career as a rocket scientist on Wall St and top quant at Goldman Sachs.  He comes across as a refined and honest intellectual with a real passion for his craft and the journey makes for a good story.

I just discovered an online video of a very punchy talk he gave at NYU on whats wrong with models [.mov format ~90Mb ~10mins].

This is awesome and he doesnt hold back.  His main point is that although Black Scholes is a great advance, an elegant useful model, its only gonna give you a ballpark figure…

Unlike QED or other physics theories which are deeply descriptive of nature and can be accurate to 8 decimal places in predictive power, quantitative models interpolate.  He makes an analogy with models like Black Scholes – Imagine you need to approximate the value of a manhattan apartment if you only have the square foot market value of a lower east side apartment for comparison… ok, I got the NY areas mixed up, but its clear we value one instrument in terms of another, making all sorts of reasonable ad hoc adjustments, such as higher comparative building maintenance service charges etc..  and factor all that into an ‘implied’ price per square foot price.

Visual DSLs in finance?

It seems to me the time is ripe for Visual DSLs to appear in the financial quantitative domain – you can imagine the pure housekeeping involved in having 2500 trade strategies operating across 3500 tick feeds.. hmmm.

So you might be asking what the hell is a DSL?  A DSL or Domain Specific Language can mean many things, but usually falls into 2 classes – Visual and Textual.

Say I want to create a web app that generates some HTML page dynamically… regardless of whether I program in Ruby or Lisp I really want a nicer way to open and close HTML tags than

print('<html>\n') ... print(</html>\n')

… a textual DSL gives you a language that is a much closer match to the problem .. I want to just write something like