Uncertainty Principle in Quantitative Finance and Probabilistic Correlation
January 1, 2009
Is there a fundamental uncertainty principle in the field of quantitative finance?
If there are two assets, say Dollar-Yen and Gold such that both follow random walks and both are correlated with each other then by knowing the variance of one asset, say, Gold and the correlation between the two assets can we determine the variance of the second asset, say, the Dollar-Yen?
Further, can we talk about financial events, say a Lehman bankruptcy or a Madoff fraud, as random variables of certain length (here, the term “length” could be a misnomer and can instead be thought of as “magnitude” or “quantum”) drawn from a probability distributions such that they are correlated with each other and then state the uncertainty principle as follows: if we know the squared “length” of a random event A and the correlation between the events A and B then can we exactly determine the squared “length” of the event B? Squared length will equal the variance.
We are all aware of the famous position-momentum uncertainty principle of quantum mechanics, otherwise known as the Heisenberg’s uncertainty principle. Electrical engineers work every day with the time-frequency uncertainty principle in signal processing. And then there is the most fundamental uncertainty principle in mathematics. This is Cauchy Schwarz inequality. In fact, high school kids learn about it as the triangle inequality when they study Pythagorean Theorem*.
Every grad school student in physics knows that Heisenberg’s Uncertainty principle of Quantum Mechanics can be derived from, and in fact is predicated upon, the Cauchy Schwarz inequality in mathematics. In fact, if we look carefully at a probabilistic Cauchy Schwarz inequality carefully we will see that the uncertainty principle of finance is staring at us.
Probabilistic Cauchy Schwarz inequality states that if we have two random variables then the following inequality holds:
This implies that for two random variables, A and B, drawn from a probability distribution the product of their variances is greater than or equal to the square of the covariance between them.
If these two random variables are asset prices then the variance of asset A times the variance of asset B is always greater than or equal to the square of the covariance between them. Of course, here the assumption is that these two asset prices are strictly random drawn from a probability distribution and that we are talking about a probabilistic correlation.
To test this hypothesis we did a simple spreadsheet exercise. We generated two random normal numbers, i.e. two independent random numbers from a Normal (Gaussian) distribution with a mean of zero and a standard deviation of one.
We then computed the volatility of these two random normal time series and also the correlation between them. The following image captures the slice of the spreadsheet.
Excel Spreadsheet implementation of Cauchy Schwarz Inequality
No matter how many simulations we ran, the result always came out as predicted by Cauchy Schwarz. The product of the variances (square of the standard deviations) of A and B was always greater than (although never ever equal to) the square of the covariance between them. The equality never held, i.e. the product of the variance was never exactly equal to the square of the co-variances. The inequality always prevailed.
In fact, Cauchy Schwarz inequality is the basis for establishing that the correlation between any two random variables is bounded between minus one and plus one, i.e.
In its most general form, for any two vectors u and v in the space R* , the Cauchy Schwarz (CS) inequality states that:
In other words, CS states that the modulus (absolute value) of the inner (dot) product of two vectors, and with arbitrary length is less than or equal to the product of the norms of the vectors. In the above image of the spreadsheet we can see two vectors, one with a length of and the other with . Here the volatilities can be thought of as – and in fact, shown to be – the distance (length) in Cartesian plane. This can be directly established from the statistical formula for standard deviation. If we have two arrows, A and B, such that they are represented by vectors u and v respectively in an XY plane originating from the same point and if is the length of the arrow A and is the length of the arrow B then Cauchy Schwarz inequality states that:
Where, is the inner (dot) product between two arrows (vectors) with an angle between them. The Cosine of the angle expresses the correlation between the two arrows.
Each of these arrows can be thought of a volatility vector with a length l , for a random particle in XY plane and this length is given by the metric distance of a point , which is the tip of the arrow from the point of origin .
Certain important questions remained to be answered such as are the prices that we observe in the markets for financial assets truly random, in a mathematical sense? Even though standard deviation of a random variable can be represented geometrically in an XY Cartesian plane, can the same be said about the implied volatilities of assets that we observe in the market? These questions will lead to a whole new debate altogether.
*Refer to Bart Kosko’s Fuzzy Thinking. This is an excellent text and we strongly recommend this to all our readers.
On Probabilistic Correlation by Christer Carlsson, Robert Fuller and Peter Majlender.There are lots of other related articles and research papers available on the web.
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