August 2014


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Books on derivative pricing come in all shades and colors. Some books give a brief introduction of derivatives at a leisurely pace and then suddenly the content becomes very mathematical. There are some books that have theorems and lemmas all through. There are some books that talk about risk-neutral pricing giving very little intuition about the concept. In the gamut of books available, I think this book stands out for a couple of reasons. The fact that we all live in incomplete markets is addressed right from the beginning. This immediacy has the reader’s attention right away. If the markets are incomplete, i.e there are more state variables than the instruments, how does one hedge an exposure ? Can there be a perfect hedge ? If not, how does one compare between two or more hedging options? These are very practical questions for an options trader. An option trader intuitively knows that a perfect option hedge that is taught in a grad school is an idealistic scenario that holds good under a ton of assumptions. Real world is messy. Pick up any book where Black Scholes is derived; in 9 out 10 books, you will see measure theory as a prerequisite to understanding the content. This book though, does not to have a math heavy prerequisites as most of the book can be read with linear algebra, elementary calculus and probability knowledge. So, in a way, this book can be read by a wider audience. Even though this book uses many numerical simulations, the author also believes that

There are computations one can do with pen and paper that even the fastest computers cannot perform.

Even though the book has 13 chapters and organized logically, the author suggests that one can take several trails for first, second or third pass through the book. One can follow discrete finance trail by working through Chapters 1, 2, 5, 6; One can go over the continuous finance trail by working through Chapters 8, 9, 10, 11 and 12; One can go over risk management trail by reading through Chapters 3,4 and 13. If you are like me who likes to read stuff that combines intuitive arguments and math, you might want to go over the entire book cover to cover.

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This book is definitely a class apart from the usual books on math finance. It uses a practitioner’s view to explain many concepts that are tricky to understand at the first go. The use of matrix computations, Fourier transforms, optimization methods, etc. makes this book more appealing to a practitioner rather than a theoretician. Great reference for someone working as a desk quant .

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imageTakeaway:

We see/hear/talk about “Information”  in many contexts. In the last two decades or so, one can also go and make a career in the field of “Information” technology. But what is “Information” ? If someone talks about a certain subject for 10 minutes in English and 10 minutes in French, Is the “Information” same in both the instances?. Can we quantify the two instances in someway ? This book explains Claude Shannon’s remarkable achievement of measuring “Information” in terms of probabilities. Almost 50 years ago, Shannon laid out a mathematical framework and it was an open challenge for engineers to develop devices and technologies that Shannon proved as a “mathematical certainty”. This book distils the main ideas that go in to quantifying information with very little math and hence makes it accessible to a wider audience. A must read if you are curious about knowing a bit about “Information” which has become a part of every day’s vocabulary.

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This book is mainly targeted at high school / college kids who feel their learning efforts are not paying off, teachers who are on the look out for effective instruction techniques, parents who are concerned with their child’s academic results and want to do something about it.

The author of the book, Dr. Barbara Oakley, has an interesting background. She served in the US army as a language translator before transitioning to academia. She is now a professor of engineering at Oakland University in Rochester, Michigan. In her book, she admits that she had to completely retool her mind. A person who was basically in to artsy kind of work had to read hard sciences to get a PhD and do research. Needless to say the transition was a frustrating experience.  One of her research areas is neuroscience where she explores effective human learning techniques. The author claims that her book is essentially meant to demystify some of the common notions that we all have about learning.

This book is  written in “personal journal” format,i.e. with images, anecdotes, stories etc. It is basically a collection of findings that are scattered in various places such as academic papers, blogs, pop science books. So, this book does the job of an “aggregator” , ,much like a Google search, except that the results are supplemented with comments and visuals.

Some of the collated findings mentioned in the book are  :

1) Focused vs.. Diffused mode of thinking : Tons of books have already been written on this subject. The book provides a  visual to remind the reader the basic idea behind it.

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In the game “pinball,” a ball, which represents a thought, shoots up from the spring-loaded plunger to bounce randomly against rows of rubber bumpers. These two pinball machines represent focused (left) and diffuse (right) ways of thinking. The focused approach relates to intense concentration on a specific problem or concept. But while in  focused mode , sometimes you inadvertently find yourself focusing intently and trying to solve a problem using erroneous thoughts that are in a different place in the brain from the “solution” thoughts you need to actually need to solve the problem. As an example of this, note the upper “thought” that your pinball first bounces around in on the left-hand image. It is very far away and completely unconnected from the lower pattern of thought in the same brain. You can see how part of the upper thought seems to have an underlying broad path. This is because you’ve thought something similar to that thought before. The lower thought is a new thought— it doesn’t have that underlying broad pattern. The diffuse approach on the right often involves a big-picture perspective. This thinking mode is useful when you are learning something new. As you can see , the diffuse mode doesn’t allow you to focus tightly and intently to solve a specific problem— but it can allow you to get closer to where that solution lies because you’re able to travel much farther before running into another bumper.

2)  Spaced repetition : This idea has lead a massive research area in the field of cognitive psychology. The book nails it with the following visual :

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Learning well means allowing time to pass between focused learning sessions , so the neural patterns have time to solidify properly. It’s like allowing time for the mortar to dry when you are building a brick wall, as shown on the left. Trying to learn everything in a few cram sessions doesn’t allow time for neural structures to become consolidated in your long-term memory— the result is a jumbled pile of bricks like those on the right.


3) Limited short term memory :
Experiments have shown that you can at max hold 4 items in your working memory. This means the key to making sense of stuff lies in effective storage and retrieval of concepts/ideas from your long term memory than trying to cram everything in to working memory(which will anyway vanish quickly)

4) Chunking : From KA Ericsson (academician behind the notion of “deliberate practice{ ) to Daniel Coyle (pop science book author)  -  all have emphasized this aspect. Again a visual to summarizes the key idea :

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When you are first chunking a concept, its pre-chunked parts take up all your working memory, as shown on the left. As you begin to chunk the concept, you will feel it connecting more easily and smoothly in your mind, as shown in the center. Once the concept is chunked, as shown at the right, it takes up only one slot in working memory. It simultaneously becomes one smooth strand that is easy to follow and use to make new connections. The rest of your working memory is left clear. That dangling strand of chunked material has, in some sense, increased the amount of information available to your working memory, as if the slot in working memory is a hyperlink that has been connected to a big webpage.

5) Pomodoro to prevent procrastination : Knowledge scattered around various blogs and talks are put in one place. The idea is that that you do work in slots of (25min work + 5 min break).image


6) { (Recall + Test > Reread) , ( Interleave + Spaced repetition > massed practice )  }
– These ideas resonate through out the book “Make it Stick”. This book though summarized the ideas and supplements them with this visuals such as :

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Solving problems in math and science is like playing a piece on the piano. The more you practice, the firmer, darker, and stronger your mental patterns become.

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If you don’t make a point of repeating what you want to remember, your “metabolic vampires” can suck away the neural pattern related to that memory before it can strengthen and solidify.


7) Memory enhancement hacks :
Most of the ideas from “Moonwalking with Einstein” and other such memory hack books are summarized for easy reading

8) Reading / engaging in diverse material pays off : This has been a common trait amongst many people who do brilliant stuff. Pick up any person who has accomplished something significant, you will find they have varied interests.

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Here you can see that the chunk— the rippling neural ribbon— on the left is very similar to the chunk on the right. This symbolizes the idea that once you grasp a chunk in one subject, it is much easier for you to grasp or create a similar chunk in another subject. The same underlying mathematics, for example, echo throughout physics, chemistry, and engineering— and can sometimes also be seen in economics, business, and models of human behavior. This is why it can be easier for a physics or engineering major to earn a master’s in business administration than someone with a background in English or history. Metaphors and physical analogies also form chunks that can allow ideas even from very different areas to influence one another. This is why people who love math, science , and technology often also find surprising help from their activities or knowledge of sports, music, language, art, or literature.

9) Adequate sleep is essential for better learning : This is like turning the lights off on the theatre stage so that artists can take a break, relax and come back for their next act. Not turning off the mind and overworking can only lead us to an illusion of learning, when in fact all we are doing is showcasing listless actors on the stage(working memory).

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Toxins in your brain get washed away by having an adequate amount of sleep everyday.

The book can easily be read in an hour or two as it is filled with lot of images/ metaphors/ anecdotes and recurrent themes. The content of this book is also being offered in the form of  4 week course at Coursera

Lady Luck favors the one who tries

– Barbara Oakley

Matrix Algebra Theory  Computations and Applications in Statistics

We often come across mathematical expressions represented via matrices and assume that numerical calculations exactly happen the way expressions appear. Let’s take for example

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These are the well known “normal equations” to compute regression coefficients. One might look at this expression and  conclude that the code that computes beta inverts the Gramian matrix XTX and then multiplies the inverse with XTy. Totally false. Why? The condition number of the Gramian matrix XTX equals square of the condition number of X. The higher the condition number of the matrix, the more numerically unstable is the solution.This is the recurrent theme of the book by James E. Gentle.

The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different

The book is a magnum opus (~ 500 pages) on linear algebra. It is divided in to three parts. The first part deals with the theoretical aspects of matrices. The second part highlights various applications where matrices become almost your best friends. The third part of the book deals with numerical linear algebra.

At some point in one’s learning path, every stats guy needs to know about the basic difference between Real numbers in mathematics and the floating point numbers (numbers that are stored in a computer). Simple mathematical laws like associativity (a+b)+c = a +(b+c) do not hold good in the space of floating point numbers. In fact the set of floating point numbers in a computer is a complicated finite mathematical structure. Basic mathematical laws do not hold true. As statisticians or practitioners, one need not know in great depth about such things but I guess one must have a passing familiarity with these principles. I mean you can easily spend at least 3 to 4 months understanding every aspect of what an SVD implementation entails. But it is not necessary for everyone to get in to such a level of detail.  In that sense, an end user can speed read the last part of the book because the technology to do all the complicated stuff is mature and is available in MATLAB, R, Octave, Python etc. However if you slow down and read through the content, you will develop a ton of appreciation to the software that helps you do SVD/Eigen/QR factorizations using one line of R / Python / MATLAB code.

The first two parts of the book bring out a ton of amazing things about Matrices. There are close to 650 equations in the entire book , all of which are explained with out using words such as theorems, lemmas, etc. At the same time, I don’t think there is even a single equation where the rationale behind it is swept under the carpet.

If you read this book cover to cover in absolute rapt attention, my guess is,  you will start noticing matrices, classes of matrices and matrix operations EVERYWHERE, in whatever applied work you do. You can’t help but notice them. That’s the beauty of this book.