# Smoothness

Our eyes initially evolved underwater while our ancestors were fish and only later adapted to seeing things above.

We are now terrible at seeing things when dunked under, however that's not the point.

Water blocks out most of the red part of the electromagnetic spectrum, which is why we can't see infrared.

#### Ugliness vs Truthiness

Just as our fishy eyesight is hindered unbeknown to ourselves, the way we tend to think and visualise financial data can be greatly improved.

Part of the problem is that descriptions of financial phenomena often fill several dimensions, for which you can use the 'object-plot' duality to simplify things.

Another issue is a lack of smoothness. Choppiness is jarring.

Ignore what any of these plots mean - which are more appealing?

You have probably ordered them in ascending appeal.

Smoothness counts.

Unfortunately financial data is not smooth, that's why moving averages are so popular.

They may or may not be OK indicators, but the real appeal is smoothing out past data and presenting a narrative. However moving averages remove a lot of important information.

#### Quantum Fish

In the smoothness classification lingo, from top to bottom, the plots go from C0 (differentiable no where) to C2 (twice differentiable everywhere).

Modern finance is built upon stochastic analysis - or 'how to use calculus on C0 time series'. Perhaps we can interpret stochastic analysis in some fashion which retains the information but gives us the beauty back?

If we had evolved from fish who swam in a quantum soup rather than a classical physical ocean, how would we see everything around us?

With 'quantum vision' we would see many possible futures.

Fish aren't renowned for their long memories, but quantum fish probably have none at all. Why worry about the past much at all when you're swimming in a quantum soup?

Conjuring futures and possible parallel universes is the job of financial analysts.

Simply sorting the returns, gives us this plot.

It shows us the upsides and downsides plus the years which gives us a historical context to each point of data. Plus we could fit a nice smooth curve to the data.

How about a histogram?

We could easily fit a smooth positively skewed curve to this data without losing much fidelity and building upon past events we can get a sense of possible future events. E.g. we are 10 times more likely to gain more than 30% than lose more than 30%.

We have stopped thinking about the data as a linear narrative, but have used it to sketch possible futures. In the process we have removed the choppiness in return for smoothness.

#### Conclusion

Our inexperience with stochastic processes on a day to day basis leads us to view data in a linear deterministic way which is both visually jarring and misleading.

As well as being pleasing to the eye, smoothness is indispensable in stochastic analysis and making intuitive sense of financial data.

Now, how about a quantum Sushi roll?