Notes on Black Swan Pond Boundaries in a Recent DataViz

The purpose of this short note is to walk through some of the equations used for the "Black Swan Ponds"in a recent visualization I have been playing with. This was done to double-check some things, and serve as a point of reference in case someone finds an issue with the equations that are being used or their implementation.

The visualization is available here:

Black Swan Pond Boundaries
Determination of boundaries of Black Swan Ponds.
In this case, using 3 standard deviations as boundary,
with σ=0.5 and t=0.083

What I am doing is highlighting the region where the stock price after time $t$ is at least $N$ standard deviations from the mean - following "standard" assumptions regarding the stock price' behavior being "geometric Brownian motion".

I don't think that there is a definitive definition of the value for $N$ which constitutes a Black Swan. Just that it is "extremely unlikely", relative to standard methods of quantifying probability. I don't think that one wants to necessarily predict/estimate probabilities of them, rather than simply seeing where this region might lie relative to "normal" assumptions - "normal" in both the statistical and non-statistical senses.

Assuming that the stock price follows a geometric Brownian motion process, ultimately this means that the distribution of stock price follows a lognormal distribution with arithmetic(!) mean and variance given by (see, e.g., wikipedia),

$$ \begin{eqnarray} E(S_t) &=& \text{arithmetic mean} = S_0 * e^{\mu t} \nonumber \\ Var(S_t) &=& \text{arithmetic variance} = S_0^2 * e^{2 \mu t} \left({ e^{ \sigma^2 t} - 1 }\right) \nonumber \end{eqnarray} $$ where $$ \begin{eqnarray} S_0 &=& \text{initial stock price} \nonumber \\ t &=& \text{time to expiration} \nonumber \\ \mu &=& \text{drift} \nonumber \\ \sigma &=& \text{volatility} \nonumber \\ \end{eqnarray} $$

In the case at hand, $\mu=0$, so this ultimately reduces to $$ \begin{eqnarray} E(S_t) &=& \text{arithmetic mean} = \text{m} = S_0 \nonumber \\ Var(S_t) &=& \text{arithmetic variance} = \nu^2 = S_0^2 \left({ e^{ \sigma^2 t} - 1 }\right) \nonumber \end{eqnarray} $$ where $$ \begin{eqnarray} S_0 &=& \text{initial stock price} \nonumber \\ t &=& \text{time to expiration} \nonumber \\ \sigma &=& \text{volatility} \nonumber \\ \end{eqnarray} $$ Then, if, $N$ is how many standard deviations constitute "far enough from the mean" to "be" a Black Swan, then the boundaries of the "Black Swan Ponds" are given by

$$ \begin{eqnarray} \text{Upper Bound for Lower Black Swan Pond} &=& m-N*\nu \nonumber \\ &=& S_0 - N * S_0 * \sqrt{\left({ e^{ \sigma^2 t} - 1 }\right) } \nonumber\\ \text{Lower Bound for Upper Black Swan Pond} &=& m+N*\nu \nonumber \\ &=& S_0 + N * S_0 * \sqrt{\left({ e^{ \sigma^2 t} - 1 }\right) } \nonumber \end{eqnarray} $$

These are the bounds I am currently using in the visualization, with $N=3$ based on an informal review of a few things on the internet (feel free to suggest a better value).

Are Three Standard Deviations Enough?

One thing that has struck me here is that the Black Swan Ponds can visually appear to be a bit "closer" than one might expect when using only three standard deviations. Thus, while this might need to be tweaked a bit, the primary purpose of the ponds is to simply try to integrate into the viz information on the potential impacts of the tails that are implied by "normal" assumptions.

Popular Posts