## Dimensional analysis in stochastic finance

Yesterday I was reading through Ole Peters’s lecture notes on ergodicity, a topic that I got interested in thanks to my extensive use of Utility Theory in my work nowadays. And I had a revelation – that in standard stochastic finance, mean returns and standard deviation of returns don’t have the same dimensions. Instead, it’s mean returns and the variance of returns that have the same dimensions.

While this might sound counterintuitive, it is not hard to see if you think about it analytically. We will start with what is possibly the most basic equation in stochastic finance, which is the lognormal random walk model of stock prices.

$dS = \mu S dt + \sigma S dW$

This can be rewritten as

$\frac{dS}{S} = \mu dt + \sigma dW$

Now, let us look at dimensions. The LHS divides change in stock price by stock price, and is hence dimensionless. So the RHS needs to be dimensionless as well if the equation is to make sense.

It is easy to see that the first term on the RHS is dimensionless – $\mu$, the average returns or the drift, is defined as “returns per unit time”. So a stock that returns, on average, 10% in a year returns 20% in two years. So returns has dimensions $t^{-1}$, and multiplying it with $dt$ which has the unit of time renders it dimensionless.

That leaves us with the last term. $dW$ is the Wiener Process, and is defined such that $dW^2 = dt$. This implies that $dW$ has the dimensions $\sqrt{t}$. This means that the equation is meaningful if and only if $\sigma$ has dimensions $t^{-\frac{1}{2}}$, which is the same as saying that $\sigma^2$ has dimensions $\frac{1}{t}$, which is the same as the dimensions of the mean returns.

It is not hard to convince yourself that it makes intuitive sense as well. The basic assumption of a random walk is that the variance grows linearly with time (another way of seeing this is that when you add two uncorrelated random variables, their variances add up to give the variance of the sum). From this again, variance has the units of inverse time – the same as the mean.

Finally, speaking of dimensional analysis and Ole Peters, check out his proof of the Pythagoras Theorem using dimensional analysis.

Isn’t it beautiful?

PS: Speaking of dimensional analysis, check out my recent post on stocks and flows and financial ratios.

While flipping TV channels last evening (an activity I seldom undertake nowadays) I came across this new advertisement for Myntra.com:

I watched this advertisement 2-3 times, and to me the clincher seemed to be the fact that you can return goods to Myntra and get your cash back the same day.

The intention of the advertisement is clear – for someone who is uncomfortable with buying clothes online (like the woman in this advertisement), the fact that you can return the stuff and get your money back immediately can be a huge incentive to try.

The problem, however, is with the overall message it conveys. One of the biggest problems with online retail in India is the high rate of returns. Returns create friction in several ways – from the logistics cost to reversing payments to possible fraud to possible damage of goods. From this perspective, returns are undesirable behaviour as far as retailers are concerned.

In this context, it’s rather bizarre that Myntra is putting out an ad that promotes the use of returns. While it might be a decent incentive to attract new customers and expand the market, the problem is that it encourages your existing customers (who are likely to transact more than new customers) to misbehave!

In other words, Myntra’s latest ad actually encourages undesirable behaviour from customers! I find it quite puzzling.

PS: On the other hand, Myntra’s competitor Amazon is actually making returns less friendly. If you return an electronic product now, you can only get a replacement, and not your money back.

## Understanding Stock Market Returns

Earlier today I had a short conversation on Twitter with financial markets guru Deepak Mohoni, one of whose claims to fame is that he coined the word “Sensex”. I was asking him of the rationale behind the markets going up 2% today and he said there was none.

While I’ve always “got it” that small movements in the stock market are basically noise, and even included in my lectures that it is futile to fine a “reason” behind every market behaviour (the worst being of the sort of “markets up 0.1% on global cues”), I had always considered a 2% intra-day move as a fairly significant move, and one that was unlikely to be “noise”.

In this context, Mohoni’s comment was fairly interesting. And then I realised that maybe I shouldn’t be looking at it as a 2% move (which is already one level superior to “Nifty up 162 points”), but put it in context of historical market returns. In other words, to understand whether this is indeed a spectacular move in the market, I should set it against earlier market moves of the same order of magnitude.

This is where it stops being a science and starts becoming an art. The first thing I did was to check the likelihood of a 2% upward move in the market this calendar year (a convenient look-back period). There has only been one such move this year – when the markets went up 2.6% on the 15th of January.

Then I looked back a longer period, all the way back to 2007. Suddenly, it seems like the likelihood of a 2% upward move in this time period is almost 8%! And from that perspective this move is no longer spectacular.

So maybe we should describe stock market moves as some kind of a probability, using a percentile? Something like “today’s stock market move was a top 1%ile  event” or “today’s market move was between 55th and 60th percentile, going by this year’s data”?

The problem there, however, is that market behaviour is different at different points in time. For example, check out how the volatility of the Nifty (as defined by a 100-day trailing standard deviation) has varied in the last few years:

As you can see, markets nowadays are very different from markets in 2009, or even in 2013-14. A 2% move today might be spectacular, but the same move in 2013-14 may not have been! So comparing absolute returns is also not a right metric – it needs to be set in context of how markets are behaving. A good way to do that is to normalise returns by 100-day trailing volatility (defined by standard deviation) (I know we are assuming normality here).

The 100-day trailing SD as of today is 0.96%, so today’s 2% move, which initially appears spectacular is actually a “2 sigma event”. In January 2009, on the other hand, where volatility was about 3.3% , today’s move would have been a 0.6 sigma event!

Based on this, I’m coming up with a hierarchy for sophistication in dealing with market movements.

1. Absolute movement : “Sensex up 300 points today”.
2. Returns: “Sensex up 2% today”
3. Percentile score of absolute return: “Sensex up 3%. It’s a 99 %ile movement”
4. Percentile score of relative return: “Sensex up 2-sigma. Never moved 2-sigma in last 100 days”

What do you think?