Risk Latte - Searching for the Most Beautiful Equation in Finance

Searching for the Most Beautiful Equation in Finance

Rahul Bhattacharya
October 6, 2012

Here is a bit of unrecorded history narrated to me a long time ago by someone who was part of it. I had forgotten all about it until recently when on a short flight I read William Poundstone’s book on how to crack Google interviews (which by the way is not possible because you simply cannot read books and crack a job interview at a company like Google, even though you can read books and workbooks and crack many exams in finance).

Many years ago, in an internal job interview, an associate was asked by the Head Swaps trader of a bulge bracket investment bank, “What is the most beautiful equation in Finance? Of course, the veteran trader had no idea that a decade or so later a similar question would be asked by the Google recruiters in the GLAT test (what's the most beautiful equation in mathematics?). The associate, a freshly minted MBA promptly replied "Black-Scholes formula" and when he saw the unimpressed and disappointing look on interviewer's face, he changed his answer to "CAPM"; when he saw the interviewer was still unimpressed he went to add a few more well know theories/models in finance. The swaps trader cut him short and said that the most beautiful equation in finance is V = 0. On paper, in theory, all mathematics behind all these financial products is worthless, without any value. The associate thought this was joke! But it was not, the trader was dead serious.

One a piece of paper the trader quickly jotted down a couple of equations that the associate had mentioned and proved why this was the case. V = 0. And the interview was over.

What the trader had done was to take the Black-Scholes formula, the Capital Asset Pricing Model (CAPM) formula and all the others that the associate had mentioned and rearrange the terms and make a simple algebraic manipulation. Here’s what he did.

The right hand side of the first equation is the rearrangement of the Black-Scholes equation. The right hand side of the second equation is the rearrangement of the CAPM equation. Of course, the terms have been squared and you can easily figure out why. Keep doing the above till you’ve exhausted all equations in finance. The right hand side of all of the above equations will be zero. Therefore, the values of all the Vs will be zero. Thus,

The swaps trader has now retired and runs scuba diving classes somewhere on the West Coast; and does deep sea diving despite the fact that he is over 55 years old.

What the associate did not realize that day was that the swaps trader was borrowing an analogy from Richard Feynman, one of the greatest physicists of our time, who in his famous lectures had said that the most beautiful equation in physics is U = 0 (and proved it). He didn’t actually say that this was the most beautiful equation in physics, only that all physics can be reduced to a simple equation like U = 0. Feynman’s way of approaching this problem was extremely simple and ingenious. The trader had followed Feynman’s recipe to the letter did with the equations of finance exactly what Feynman had done with the equations of physics. Feynman had taken Einstein’s equation, Schrodinger’s equation and a few other important equations in physics and after rearranging and squaring them (in a manner shown above) showed that you ended up with U = 0.

The analogy was totally borrowed from physics but the perspective was the trader’s: all mathematics is worthless unless we know what to do with it.

It is of course a silly thing to say that all finance can be reduced to V = 0 (as silly as saying that all physics boils down to U = 0). But by being silly, the trader was making a profound point. If you go the physics route then in the end you’d reach Feynman’s conclusion. Or, saying it a bit differently, if the ultimate goal of finance is to add value to the society, via transactions in financial products and securities, then the underlying mathematics of that finance should aid and enhance that process otherwise that math is of no value. Admiring Black-Scholes for the inner beauty of that equation or its mathematical elegance is not enough. Black-Scholes has to be put to use to add value to the man whose money is at stake. On a piece of paper, Black-Scholes equation can be reduced to zero.

We can take pride in all the difficult and esoteric mathematics that we borrow from physics and throw at finance to justify the proliferation of complex financial products and financial derivatives. We can keep pushing the mathematical frontiers of quantitative finance in search of more complexity. But we’d be doing a disservice to the discipline if all this does not make the average investor – the guy on the street who trusts the finance gurus, the quants with his life saving – richer or better off.



Reference:
Are You Smart Enough to Work at Google, William Poundstone, Little, Brown and Company (2012).

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