# Share Price Volatility

## Volatility: Three Summary Ideas

• Physically, “volatility” is the wiggling of share prices or index values, as illustrated by Shiller’s figure, above. The wiggling is nonlinear crowd dynamics. It is not a mere Gaussian process. Three crowd-related theorems are listed on the main page of the website: Cobweb, System Dynamics, and Behavioral Finance.
• In classic portfolio theory, ”volatility” is equated directly with “risk” using (incomplete) statistical models.
• Some trading professionals regard “volatility” as “profit opportunity,” not as “risk.”

We explore volatility in seven small steps:

## Portfolio Theory – Classic

From Modern Portfolio Theory and from Markowitz Model:

Modern Portfolio Theory intends to do this:

The mean-variance framework for constructing optimal investment portfolios was first posited by Markowitz in 1952. It uses the variance of asset prices as a proxy for risk. The theory compares the expected (mean) return of a portfolio with the variance of the same portfolio. The theory attempts to model risk in terms of the likelihood of losses. The framework has some limitations.

## Portfolio Theory – Criticisms

• The model of financial markets does not match the real world.
• The expected values cannot capture the true dynamic features of actual data in real time.
• The risk, return, and correlation measures are based on expected values, so investors use past data to estimate key parameters for the model.
• MPT attempts to model risk in terms of the likelihood of losses, but says nothing about why those losses might occur. There is no attempt to explain an underlying structure to price changes.
• By comparison, many engineering approaches to risk management are specific about causes, the effects, and the associated risk level for the system under study.
• MPT assumes that returns follow a Gaussian distribution. Actual returns have a bell-like shape, but are usually not Gaussian. Volatile spikes can be large.

## Volatility (Variance) – Description

• Volatility is a statistical measure of the dispersion of returns around the mean price for a given security or market index. Volatility is often measured as either the standard deviation or the variance between returns from that same security or market index.
• Volatility often refers to the amount of uncertainty or risk related to the size of changes in a security’s value.  A higher volatility means that the price of the security can change dramatically over a short time period in either direction. A lower volatility means that a security’s value does not fluctuate dramatically, and tends to be steadier.
• Volatile assets are often considered riskier because the price is expected to be less predictable.
• Volatility is a variance measure, bounded by a specific period of time.
• Historical volatility is based on historical prices and represents the degree of variability in asset returns.
• EXAMPLE STEP-BY-STEP VARIANCE CALCULATION SHOWN IN A TABLE.
• The final step is to calculate the standard deviation (the square root of the variance calculation).

The final calculation (standard deviation) yields \$2.87.  This figure gives traders some idea how far the price might deviate from the average price. The range is regarded as a measure of risk. Not as an opportunity to earn.

The theoretical premise is that prices follow a normal distribution, approximately, and in turn, certain percentages of the returns calculations will fall within 1 or 2 or 3 standard deviations of the mean value. In practice, something else happens instead.

## Volatility (Variance) – Equation

From Investopedia’s definition of variance: [Scroll down to “The Formula for Variance Is”]

• Variance intends to compare the relative performance of each asset in a portfolio.
• Variance measures how far each number in the set is from the mean.
• Variance measures variability from the average or mean.
• Variance treats all deviations from the mean the same regardless of their direction.
• Variance gives added weight to outliers, the numbers that are far from the mean.
• Squaring outlier numbers can skew the data (from the statistical point of view).
• Standard deviation is often used instead of variance. Easier to work with.

## Volatility – Criticisms

From the definition of volatility in finance and from the Black-Scholes model:

• Predictive power of sophisticated models is little better than simple “past volatility” methods.
• Some professionals ignore or dismiss volatility forecasting models.
• Few theoretical models explain how volatility comes to exist.
• Volatility does not measure the direction of price changes, merely their dispersion from the average.
• The Black-Scholes equation assumes predictable constant volatility (not observed in real markets).

## Volatility – Schools of Thought Today

1. From MIT OpenCourseWare: search on these example titles.

Observe on right sidebar more links to MIT OpenCourseWare.

1. From InstituteofTrading: “ex Goldman Sachs Trader Tells Truth about Trading – Part 1” (YouTube, length 12 min). This is an interview with Managing Partner Anton Kreil on Mar 26, 2013. Total of 6 videos.

Highlights:

• Volatility is a large price change in a short time; a great opportunity to gain in a short time.
• Trading outlook, from different positions in the trading system (conflicts of interest).
1. Managers known for trading volatility opportunities.
Mark Spitznagel, of Universa Investments, Nassim Nicholas Taleb, Paul Britton of Capstone Holdings Group, Andrew Feldstein of Blue Mountain Capital Management, and Nelson Saiers of Saiers Capital.
1. Book: “The Black Swan,” Random House, ISBN 978-1-4000-6351-2.  Nassim Nicholas Taleb (2007).

## Volatility – Modeling The Dynamics

A Gaussian system model is tractable only if one can linearize the nonlinear dynamics – a tough task.

Linearizing. Bernard Widrow (Stanford, 1986) designed the algorithm that separates a mother’s heartbeat and her fetus heartbeat. He needed two correlated series, and two uncorrelated series.  He converted the nonlinear system into a linear system with additive Gaussian noise.   He has successfully linearized a number of nonlinear systems.  He has personally examined stock market data in detail. The markets are a nonlinear system.

A herd model is consistent at a macro level (per the theorems  Cobweb, System Dynamics, and Behavioral Finance).  A “swarm” model (borrowed from nature) is mathematically consistent for the details.

Feedback: adaptive DSP is consistent. Finite calculus methods and numerical precision are important.

Time scales: what we “see” depends on the sample rate we choose.

The Fourier transform of stock and index data shows that one can distinguish between Gaussian white noise, Brownian motion, and the information in the data itself. This is clearly visible on the front page of the website. Statistical methods cannot do that.

Volatility does not necessarily indicate “risk.”

• The “risk” is proportional to our inability to see the information in the data itself.
• The “opportunity” is proportional to our ability to see the information in the data itself.
• The larger the variance at some time step, the larger the opportunity.
• “Outliers” tend to return to some “mean value” in a short time period.
• Gain on the up movement, and then gain on the down movement.
• Larger returns overall, and lower correlation with the market averages.
• Tractability and robustness versus “hedging.”

Statistical procedures welcome mathematical havoc.

• The S&P index, viewed over a long time period, does not exhibit ergodic behavior around some fixed mean value. This is clearly visible on the front page of the website.
• Differencing is the only way to force the data to approximate an “ergodic” constraint.
• Differencing removes lower frequency information and retains high frequency noise (R W Hamming, 1989).
• Comparisons of differenced data will increase “spurious correlations” (G Udny Yule, 1926).