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More on Princeton Companion to Mathematics aka ‘PCM’.
Quite a few of the single Companion chapters are available from the authors’ own web sites… I gave links to Terry Tao and Tim Gowers blogs below.
Gordon Slade wrote the chapter ‘Probabalistic Models in Critical Phenomena‘
Its a nice overview of whats been happening in this area, with sections about Critical Phenomena, Branching Processes, Random Graphs, Percolation, Ising Model, Random Cluster model and also introduces SLE the Stochastic-Loewner-Evolution work by Oded Schramm, Wendelin Werner etc.
If you’ve never heard of this stuff heres a motivator of sorts, by yours truly –
Found a great entry on Terry Taos blog : a guide to strengthening math inequalities, with examples and tricks, entitled “Amplification, Arbitrage and the Tensor Power trick”.
At the end of that article the comments digress towards the relevance of Math Journals, and this is taken up in a linked blog entry – “more on Journals”
I repeat here the comment I made on that blog, namely –
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.
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.
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