Volatility

In trading, we need volatility

The technical definition of volatility goes something like this: “The product of the standard deviation of an asset’s returns and the square root of time.” It measures the statistical dispersion of the returns of, say, a stock price over a specific timeframe. When we talk about volatility, we think about risk and uncertainty — and the possibility of making a loss when you trade.

That’s only half of it. Think of volatility as width not direction. Direction is the trend, width is how much the price moves up and down. A rough guide for the trader is the annualized vol divided by 16 (approximately the square root of 252 trading days), and this gives you the “typical” daily move of a stock. Sixteen percent vol is like a 1% move. Thirty-two percent feel like a 2% daily move in the stock price — this may be crude but it is something which serves as a useful guide.

Volatility is where opportunities are found, and where money can be made. Yes, there is always uncertainty in trading, because trading is a pure expression of capitalism, where loss is always a possibility. The opposite to capitalism is when there is perfect price stability — where nobody will make a loss or a gain. That’s not a market, that’s a price list. Markets exist because different people have different views about the future value of a good or a service. Disagreement is volatility.

Causes of Volatility

A number of things cause volatility: unexpected geopolitical events (like a war or a change in the tariff regime). Reported earnings of a company which perhaps don’t meet expectations (or smash them). A change in a corporate leadership. Thin liquidity and crowded positioning, where a small change in perceptions can cause a rush of the exit. A dash for cash. High volatility frequently marks a regime change or a change in state where market participants become more uncertain than usual as to the outcome and how it will affect their future returns. In these kind of situations, investors and traders often dump their positions and go to cash (which offers nominal certainty — though in a high-inflation environment that may not be the case) or to supposed safe haven assets. Intuitively, that is why prices have an inverse relationship to volatility.

Fear equates to “ Sell! “. Selling lifts realized and implied vol. A big move on a smaller price increases the percentage volatility. It is a useful rule of thumb in equities but not a law of nature — and definitely not applicable to all asset classes. Volatility in a way is an asset class on its own. It is the price of the next move.

Measuring Volatility

For most people when they talk about volatility it’s “historical” or “realized” volatility. How the price has moved in the past. You look back over the last 20 trading days (a common choice) for example and measure how wide that path already was over that period. The blunt instrument is the close-to-close price change.

Range-based estimators of intraday volatility (Parkinson, Garman-Klass, Yang-Zhang) squeeze more information about the price changes during a day using the open-high-low-close (“OHLC”) prices. Yang-Zhang uses the overnight gap (taking into account the closing price of the previous day versus the next day’s opening price). The formulas and worked examples are in the appendix.

The comparison between the indicators is the point: how much information are you using to come up with the estimate? Close-to-close does a poor job, because you are not considering all the values in-between. But all of the above is backward looking anyway. It is the path already trodden. It does not tell you what’s around the corner.

Implied volatility is the other number traders talk about. This is forward looking. It is derived from the price traders are charging for the options they sell on an asset. Volatility, being one of the inputs in option pricing models, is then backed out. When traders expect market turbulence, they bid option premiums up and so, with all other inputs held constant in the pricing model, volatility will rise. High implied vol means high option prices. Low implied vol means low option prices. It is how uncertain is priced in the future.

There is no single option model, but the most storied one is the Black-Scholes-Merton which is the workhorse one for European-style options (exercisable only once at the expiration of the option). Other models are used by traders: binomial and trinomial trees, local-vol and stochastic-vol models, jump-diffusion, SABR models using a vol surface. The implied vol is always the same kind of object. The σ that makes:

You cannot invert Black-Scholes in closed form (you can’t make σ the unknown in the equation). You solve σ numerically using a Newton-Raphson iteration using vega (volatility) until the model price converges with the market price. The call and put formulae are in the appendix.

Exchange-traded options on single stocks

In text books, this is not really dealt with adequately. Listed options on single stocks and ETFs in U.S. markets are American-style. That means that you can exercise your right (not obligation) to buy or sell the underlying at any time until the expiration date of the option. That matters for puts and calls on stocks which pay dividends on certain dates. If you own an American-style call on a stock where the dividend is payable to the holder on a date within the exercise period, you may want to exercise early and receive the stock. That “optionality” within the option itself means that it should trade at a price higher than the European-style option. Black-Scholes European exercise assumes a continuous dividend yield, not a discrete one as in real life.

The standard model and the one the Cboe publishes for theoretical valuation of the options traded on its exchange is the Cox-Ross-Rubenstein binomial tree with discrete dividends. It is a lattice where you step the stock price up or down and at each (daily) node take the greater of the exercise value and the present value of the continuation value and search for the value for volatility which makes the market price equal to the model price.

Traders look at the implied vs. realized vol. Implied is what you pay for the option, and realized is what you get as the underlying contract moves over its life. Over long samples, implied vol has tended to be higher than historical which intuitively can be explained as the value of future uncertainty.

The “Fear Index”: where volatility got its bad name

Most traders have heard of the VIX: the volatility index on the S&P500 index of stocks. It is not the inversion of a single strike, rather a model-free strip of options on the SPX which are translated into a 30-day expected volatility quote. It is called the “Fear Index” because it sis supposed to measure increasing volatility — it jumps when there is a crisis. It is an oscillator, and its “stickiness” around high levels is an indication of prolonged uncertainty. It jumps higher when most people are afraid.

Yet this might not necessarily destroy a trader. What goes up must come down. A market which grinds higher by 1% per day can easily bankrupt a short (the “risk of ruin”). A market which swings 3–4% per day can be the best environment a trader can ever see. A market with swings like these allows the skilled and able trader to profit handsomely. This is the point of capitalism.

Appendix: measuring volatility

(1) Historical or Realized Volatility

This calculation can be adapted to any time frame, including intraday price movements — that is, if you have access to tick data.

(2) Parkinson (High-Low) methodology:

Using this fake intraday stock price data:

• Open (O): 100
• High (H): 104
• Low (L): 98.50
• Close (C ): 102

This measure of intraday volatility was developed by physicist Michael Parkinson, and uses only the High ( H) and Low ( L) of the daily prices. Intuitively, these two measures are the limits of the random price moves during a day, and Parkinson proved that using his method with the high-low range was roughly 5x more efficient than using standard close-to-close returns.

So, the annualized daily volatility is 56.38%

Note, that in the formula, the Parkinson constant

scales the estimator of the squared log High-Low range down so that it is unbiased.

(3) Garman-Klass (OHLC) methodology:

The Garman-Klass method of calculating intraday volatility improves on Parkinson by incorporating the Open ( O) and the Close ( C), and the relative position of the Open price within the range. This improves the efficiency of the estimator to roughly 7.4x the standard close-close estimator.

Here, the GK methodology subtracts a dampening weight and prevents double counting the variance to keep the estimator of the Open-Close range unbiased.

Therefore:

The daily volatility being:

Or 63.45%

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Note the GK volatility is higher because of the close being higher than the open, which adds positive drift to the calculation.

Limitations:

Both Parkinson and Garman-Klass assume zero drift: assets which trend inflate the high-low range which skews the volality estimate higher, as do bad ticks. In addition, Parkinson and GK do not take into account overnight gaps, where the stock closes the previous day at a price considerably higher or lower than the open. As we shall see below, the Yang-Zhang estimator takes this into account.

(4) The Yang-Zhang estimator:

Let’s take the following imaginary price data, using the OHLC for the previous calculations, but adding the price of the previous session close:

• Previous Close (Ct−1​): 98.00
• Open (O): 100
• High (H): 104
• Low (L): 98.50
• Close (C): 102

(i) Calculate the overnight (jump) log return:

(ii) Calculate the Open-Close return:

(iii) Calculate the Rogers-Satchell intraday component:

The Rogers-Satchell intraday component captures intraday path volatility independently of drift (even when prices have a strong upward or downward trend)

The Rogers-Satchell formula (“RS”):

Yang-Zhang blended weighting:

Where:

as n-sample size grows large, k converges to 0.14

The annualized volatility of 68.15% captures the opening gap up in price.

(5) Implied Volatility: Black-Scholes-Merton price of a European Call and Put Option (with continuous dividend yield “q”)

Implied vol is the σ that makes C(σ) or P(σ) equal the observed (market) premium. An interaction on vega is the usual solver:

Where vega:

and the same v for the put. When vega is very small (deep in the money options, tiny time)switch to a bracketed method.

For listed American options, use the Cox-Ross-Rubenstein binomial tree: create an up-down binomial tree using a “trial” , apply the early-exercise check at each node, include scheduled cash dividends (is the value of early exercise greater or less than the continuation value of the tree?), and iterate until the model price equals market price. The which allows the market price to match the model price is the implied volatility of that strike and expiry. Different strikes and expiries will not agree — that disagreement is the vol surface (skew and term structure). Different models will not agree either.

Useful reading:

• Sheldon Natenberg, Option Volatility & Pricing
• Euan Sinclair, Volatility Trading
• John Hull, Options, Futures, and Other Derivatives
• Fischer Black & Myron Scholes, “The Pricing of Options and Corporate Liabilities” (Journal of Political Economy, 1973), and Robert Merton, “Theory of Rational Option Pricing” (Bell Journal, 1973)
• John Cox, Stephen Ross & Mark Rubinstein, “Option Pricing: A Simplified Approach” (Journal of Financial Economics, 1979)
• Michael Parkinson, “The Extreme Value Method for Estimating the Variance of the Rate of Return”(Journal of Business, 1980)
• Mark Garman & Michael Klass, “On the Estimation of Security Price Volatilities from Historical Data”(Journal of Business, 1980)
• L. C. G. Rogers & S. E. Satchell, “Estimating Variance from High, Low and Closing Prices” (Annals of Applied Probability, 1991)
• Dennis Yang & Qiang Zhang, “Drift-Independent Volatility Estimation Based on High, Low, Open, and Close Prices” (Journal of Business, 2000)
• Jim Gatheral, The Volatility Surface
• Cboe VIX white paper (and the American/European theoretical-pricing methodology note) — How a listed-options venue actually turns prices into vol: model-free index vol vs binomial inversion on American stock options.
• Espen Haug, The Complete Guide to Option Pricing Formulas
• Robert Engle on ARCH/GARCH (the 1982 Econometrica paper, or any good survey)
• Demeterfi, Derman, Kamal & Zou, “More Than You Ever Wanted to Know About Volatility Swaps”(Goldman Sachs, 1999)
• Nassim Taleb, Dynamic Hedging

Originally published at https://returnsseeker.substack.com on August 30, 2026.