Apr

9

 In Outliers, Malcolm Gladwell seems to be making an argument for nurture in the nature v. nurture debate. In particular, he is interested in how culture, parenting, special opportunities, and timing factor into the stories of the wildly successful. The book is largely a compilation of results from various studies as well as the stories of some well-known individuals. For instance, he opens with the peculiar fact that a disproportionate amount of Canadian hockey players are born in January. The explanation being that they are the oldest players who can join the youth league. This, he argues, gives them a relative advantage, which over time, as they are selected for special opportunities, translates into real superiority.

He later goes on to analyze key breaks that allowed for the success of individuals like Bill Gates, e.g., more hours of access to a computer than virtually anyone else in the world at the time. He examines the importance of IQ, parenting, and the economic background of one's parents. He points to the strong correlation between the economic class of one's parents and one's ultimate success, and he argues that we are probably failing to cultivate a lot of human capital by not properly distributing opportunity.

This is an argument that I tend to agree with, as I have seen first-hand the difference that class makes in the distribution of opportunity. I came to grad school from Ohio State, and my roommate came from Princeton. I had better grades and a paper under my belt, but I was given a full-time teaching load whereas he had no teaching (presumably because I didn't have Andrew Wiles write my letter of recommendation). In theory, this gives him more time to work, which puts him further ahead in his research, and all things being equal, he gets the better job. (In fact, he didn't properly use his time and he was a third-round hire, and I was a first-round hire.) His parents both had graduate degrees and lived in a wealthy suburb of Boston, whereas neither of my parents went to college and lived in Appalachia. He went to Princeton, and as far as my family was concerned, that wasn't an option. I think most of us can relate to this sort of thing.

That said, despite my sympathy for Gladwell's argument, he fails to examine these studies for flaws. He's a little too quick to make sweeping generalizations, and he spends a little too much time explaining the obvious. He should have anticipated and responded to some potential criticisms. I'm glad I read the book, because I learned of the existence of KIPP("Knowledge Is Power Program") schools, which are spreading across the country. Their goal is to provide the kind of college-prep to low income students that is available to the affluent. If you go to their website, you can find information about teaching, starting new schools, and donating. I suspect I'm preaching to the wrong audience, but I think it is a really exciting idea.

Jason Thompson writes:

It would seem your experience highlights how class is not the main variable in success, rather it is your hard work and your intellect. Instead of focusing on the details of your roommate's experience vs. yours, examine the big picture. You have accomplished a greater goal than your roommate without all of his inherent advantages as proscribed by material advantage, in-other words meritocracy works! AFA Gladwell/Side-show Bob's assertion that class matters most, significant empirical work by James Heckman or Charles Murray have thrown water on that flame. Rather its clear that IQ leads to greater wealth, that such wealth persists highlights the important of nature (aka genetics). Overtime one should expect the smart folks and their progeny to obtain greater proportions of the spoils of the economy…unless of course the state intervenes.

Mar

8

I am giving a talk on a paper that I read. The problem under consideration is the following: you have risky assets in which you can invest, and you will have to cover a contingent claim at some future time which depends on the future value of the risky assets (e.g., a put option on one of them). You have some initial wealth and wish to invest in the risky assets so that you can cover your contingent claim. The problem is to find the minimal wealth that will allow you to do this. This paper finds a way to determine the minimal wealth that will allow you to hedge your contingent claim with a prespecified probability. (Stochastic Target Problems with Controlled Loss by B. Bouchard, R. Elie and N. Touzi)

I am working with a professor who would like us to apply these techniques to optimal investing with drawdown constraints. If you set out to find the optimal investment strategy subject to the condition that your final wealth is not less than its running max, you will stay out of the market. We want to see what happens if you optimize under the constraint that your final wealth is not less than its running max with some high probability.

Here are the slides of my talk.

Dec

10

 Diffusion-Limited Aggregation (DLA) is what results when you start with a seed particle and then release another particle from far away that moves according to Brownian motion. When it hits the seed, it attaches. You release another particle moving according to Brownian motion, and wait for it to hit the structure and attach somewhere. You keep doing this. The resulting structure is DLA. As mentioned in the Wikipedia entry, this appears naturally in certain mineral deposits and in dielectric breakdown. Here is a fun java simulation. Apparently, this phenomenon is still not well understood. At the colloquium about this I went to yesterday, the speaker said he had been working on this stuff for 20 years and had made little progress. The hope is that they can exploit the connection between DLA and Schramm-Loewner Evolution (SLE)  to better understand the phenomenon, e.g., are such structures self-similar.

Vincent Andres writes:

Here is an interesting book on the topic (by an ex-colleague).

Nov

25

 It's not as though this wasn't obvious or to be expected, but the academic job market already looks like it is going to be brutal this year. I'm not sure why, but anecdotal evidence suggests that last year was pretty bad too. Several students finishing their PhDs in math here at Michigan went on the job market last year and failed to find jobs. One of them in particular was supposed to be very good and was working with one of the best people in his sub-field. What really shocked me is that my weightlifting partner did not find a job. He did his PhD at Harvard and was an NSF postdoc fellow here. He has published in Inventiones Mathematicae (a top tier journal just below the Annals of Mathematics in prestige). These people stuck around for another year and are going back on the job market this year. Most universities post their math positions on MathJobs.org. I noticed several institutions have withdrawn advertisements for positions. Talking to friends in financial math who are looking for industry jobs is even worse. No one I know has been successful in his search. Don't be surprised if the person serving your coffee at Starbucks is a mathematician.

Nov

21

 The second two talks that Ioannis Karatzas gave were considerably more technical. However, there were some interesting ideas. In a recent paper, he and collaborator Daniel Fernholz abandon the assumption that there are no arbitrage opportunities and attempt to construct a more descriptive theory of market behavior, without many of the tools commonly available in mathematical finance.

In particular, some of the work involves measurements of the "internal volatility of the market," which appears as the "excess growth factor." They use certain functions to generate portfolios. Under certain circumstances, if there is sufficient internal volatility in the market, they show that the portfolio generated by the Gibbs entropy function generates an arbitrage opportunity relative to the market. You can find a survey paper here.

In the talk, Karatzas presented this as a young subject with much work left to do.

Nov

19

 The Math Department at the University of Michigan has held an annual lecture series called the Ziwet Lectures since 1936. Past speakers include von Neuman, Kac, Thurston, and about a half a dozen Fields medalists. This year, the speaker is I. Karatzas . He is giving a series of three lectures. Today he discussed Stochastic Portfolio Optimization (the next lectures will be on Volatility and Arbitrage respectively). He spent a lot of time introducing the subject, which was good for me. One assumes that there are n risky assets available, S_1, . . .,S_n, and they evolve according to the stochastic differential equation

where dW_i(t) is Brownian motion. X(t) is one's wealth at time t and p_i(t) is the percent of one's wealth invested in asset i at time t. Denote by p=(p_1,…,p_n) our portfolio. U(x) is a utility function, i.e., any increasing function that is concave down. Our goal is to maximize the expected value of utility at the time T. In other words, we let V(x)=sup{E[U(X(T)]} where the supremum is taken over all possible portfolios given that our initial wealth was X(0)=x. Apparently we are guaranteed existence of such a thing in general, but finding the optimal strategy is not very tractable, so as always, one starts with special cases.
 

If we assume that the utility function is U(x)=log(x) or U(x)=x^{a}/a for 0<a<1, then one can find a reasonable solution. However, the solution depends on having reliable values for sigma_{i,j}(t) for all times t as well as for the interest rate.

If one assumes that all coefficients involved are constant, then we can handle the problem of a general utility function. The solution is characterized by a partial differential equation called the Hamilton-Jacobi-Bellman (HJB ) equation. Because we have assumed U is concave down, we can apply the Legendre transform and linearize the partial differential equation. We can then solve the linearized equation.

Karatzas ended the talk with several open problems.

I am not sure whether this lends itself directly to practical application, but perhaps it inspires some more practical ideas.

Jeff Rollert asks:

Why would one assume the coefficients are constant?

Chris Hammond responds:

One answer is that over a reasonably short time horizon, they would be approximately constant. I think the same question could be asked of the Black-Scholes model. It is assumed that if S is the price of an asset, dS=S*r*dt+S*sigma*dW(t), where r is the expected return on the asset, sigma is its volatility, and W(t) is Brownian motion. More sophisticated models assume that the volatility is also a random variable that changes with time sigma=sigma(t). But it makes sense to start with the simpler, constant, case.

In some situations in math, it is insightful to assume very simple behavior to get a model case and view reality as some sort of perturbation of that.

I am not sure it is a good answer, but I'm trying to learn more about these things, so if I find a more satisfying answer, I'll let you know.

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