The Strange Equation of physics that turned our world Upside Down
March 7, 2010
The math used in the study of financial derivatives and other areas of quantitative finance is the same math that is used in physics. The world of quantitative finance is filled with physicists, some bright and some failed. And one can easily argue that the celebrated Black-Scholes partial differential equation for option pricing should perhaps be considered a part of applied physics.
Yet, as my old boss, who taught me option trading, kept saying, “this is not physics”.
The world of physics, which I have now abandoned for good, is much stranger than any world we can imagine. And whether we like it or not, we cannot live without it. Every time I pull up those Excel spreadsheets with fancy calculations on matrices and eigenvectors and what have you to solve some problem in quantitative finance, it reminds me of a time when I thought that the bizarre world of quantum mechanics would somehow redeem the intellectual sinner in me.
In 1925, Werner Heisenberg, Max Born and Pascal Jordan wrote a strange language in which a large portion of physics would be written. It was the language of Quantum Mechanics.
We are talking about that very strange equation (which Max Born himself called the "strange formula") which Heisenberg wrote down in 1925 and showed it to Max Born.
Say, you have two measurable quantities, and (these are variables) then can you have an equation whereby:
Where, is an extremely small number, but not zero? That is, can the expression be non-zero where both and are variables and measurable quantities? This was apparently nonsense. High school algebra tells us that should always be zero.
Further, if we square both sides of the above equation then can the right hand side, i.e. be negative? This was a truly bizarre question, even to Max Born. The square of any number is positive, so how can be negative. What Heisenberg was saying was that take variables, and which are measurable quantities, and then subtract from and you'll not get zero. It will be an extremely small number. Secondly, square the expression and you'll get a negative number.
Actually, the above is not at all nonsensical or bizarre, as Max Born very quickly realized.
Even if we have got no clue about quantum mechanics the above should reveal to us if we think a bit hard. What if and are matrices and not numbers? And what if on the right hand side of the above equation contains and imaginary number, i.e. , where ?
If and are matrices, instead of just numbers, then of course it is not necessary that should be zero. In fact, if and are two arbitrary, square matrices of dimension then most of the time we will have , where, would mean a null matrix, who all elements are zero. This is the non-commutative property of square matrices.
Further, if is extremely small, but contains an imaginary number, , such that, say, , then and the squaring the equation on both sides will give us:
Well, we needn't bother about quantum mechanics, even though it changed our whole world in a way we can never fully fathom. Let quantum mechanics be the domain of geniuses like Heisenberg, Born, Pascal, Schrödinger and many others.
But we cannot ignore matrices and imaginary numbers, even if we are risk managers, traders and "quants" in banks and financial institutions. Matrices are ubiquitous in every area of quantitative finance, whether it is portfolio theory and optimization, stochastic calculus, principal component analysis, market risk estimation, etc. And a very good understanding of matrices and all advanced matrix operations is essential to the study of financial engineering. Non-commutative algebra does matter if you are working with volatilities, correlation, covariance, cholesky and eigenvector matrices.
Imaginary numbers are also important though this may really throw some finance practitioners off guard. Where could possibly one use imaginary numbers in finance, which is essentially concerned with the study of money, financial assets and their movements? Even if we use fairly advanced math to solve problems in quantitative finance, isn't the use of imaginary numbers stretching one's imagination a bit too far?
Not at all! If you are trying to find closed form solutions for something called the Heston's model of stochastic volatility or if you are using Fourier Transforms to value financial derivatives, you'd land up staring at imaginary numbers. The moment you start working characteristic functions of a probability distributions you are confronted with imaginary numbers.
The world of quantitative finance - or what we call the study of financial engineering - may not be as strange as physics. But it sure is strange!
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