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by Charles Brucaliere for Forbes via Google Gemini AI
In a year when the stock market has primarily been driven by the whims of President Donald Trump’s tariff policies and social media posts, it’s even harder than usual to forecast whether the market will move up or down in the next week, or even day. The S&P 500 index crashed 10% in a two-day span after Trump’s already infamous press conference announcing his “Liberation Day” reciprocal tariffs on 90 countries on April 2, then rallied 9.5% a week later when he backtracked on most of them……..Continue reading….
By: Hank Tucker
Source: Forbes
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Critics:
Much research has been devoted to modelling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place. Roll (1984) shows that volatility is affected by market microstructure. Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of adverse selection, they adjust their trading ranges, which in turn increases the band of price oscillation.
In September 2019, JPMorgan Chase determined the effect of US President Donald Trump’s tweets, and called it the Volfefe index combining volatility and the covfefe meme. Volatility does not measure the direction of price changes, merely their dispersion. This is because when calculating standard deviation (or variance), all differences are squared, so that negative and positive differences are combined into one quantity.
Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time. For example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. Ignoring compounding effects, this would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule).
A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a normal distribution; in reality stock price movements are found to be leptokurtotic (fat-tailed). Although the Black-Scholes equation assumes predictable constant volatility.
This is not observed in real markets. Amongst more realistic models are Emanuel Derman and Iraj Kani’s and Bruno Dupire’s local volatility, Poisson process where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of stochastic volatility. It is common knowledge that many types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all.
In foreign exchange market, price changes are seasonally heteroskedastic with periods of one day and one week. Periods when prices fall quickly (a crash) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible bubble) may often be followed by prices going up even more, or going down by an unusual amount. Most typically, extreme movements do not appear ‘out of nowhere’; they are presaged by larger movements than usual or by known uncertainty in specific future events.
This is termed autoregressive conditional heteroskedasticity. Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again. Measures of volatility depend not only on the period over which it is measured, but also on the selected time resolution, as the information flow between short-term and long-term traders is asymmetric .As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.
The risk parity weighted volatility of the three assets Gold, Treasury bonds and Nasdaq acting as proxy for the Marketportfolio seems to have a low point at 4% after turning upwards for the 8th time since 1974 at this reading in the summer of 2014. Volatility as described here refers to the actual volatility, more specifically:
Actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price.
Actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past.
Near synonymous is realized volatility, the square root of the realized variance, in turn calculated using the sum of squared returns divided by the number of observations.
Actual future volatility which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an option). Now turning to implied volatility, we have:
Historical implied volatility which refers to the implied volatility observed from historical prices of the financial instrument (normally options)
Current implied volatility which refers to the implied volatility observed from current prices of the financial instrument
Future implied volatility which refers to the implied volatility observed from future prices of the financial instrument
For a financial instrument whose price follows a Gaussian random walk, or Wiener process, the width of the distribution increases as time increases. This is because there is an increasing probability that the instrument’s price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.
Since observed price changes do not follow Gaussian distributions, others such as the Lévy distribution are often used. These can capture attributes such as “fat tails”. Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc. Using a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations.
Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down. To annualize this, you can use the “rule of 16”, that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252).
This also uses the fact that the standard deviation of the sum of n independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables. However importantly this does not capture (or in some cases may give excessive weight to) occasional large movements in market price which occur less frequently than once a year.
The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation σ, the expected value of the magnitude of the observations is √(2/π)σ = 0.798σ. The net effect is that this crude approach underestimates the true volatility by about 20%.
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