Volatility is the unit of risk: how volatility position sizing works

What is volatility position sizing?

Volatility is how far a price typically moves, commonly measured in one of two ways:

• Daily volatility, the standard deviation of daily returns: the typical size of one day's move as a percentage of price. Multiplied by the square root of 252 trading days it becomes a yearly figure, so 1% a day is about 15.9% a year.
• Average True Range (ATR), from J. Welles Wilder's 1978 book. True Range is the largest of today's high minus low, the absolute difference between today's high and yesterday's close, and the absolute difference between today's low and yesterday's close, so it counts overnight jumps. ATR averages True Range over a set number of days. Because ATR is in price units, StockCharts notes, "low-priced stocks will have lower ATR values than high-price stocks." ATR divided by the price, called ATR percent, puts markets on one scale.

A recent reading is useful because volatility is persistent. Harvey and co-authors (2018): "High volatility over the recent past tends to be followed by high volatility in the near future."

Sizing to the stop. The planned loss at the stop, which Vol 1 calls 1R, is units x stop distance x point value (the cash value of a one-point move for one unit). Holding it steady means units = cash allowance / (stop distance x point value). In a common worked example (point value $1 a share), $100 at risk with a $4 stop buys 25 shares. When volatility doubles and the stop moves out to $8, the same $100 buys 12.5 shares.

Scaling to a volatility target. Volatility targeting sets exposure, the size of the position as a share of capital, to target volatility / estimated volatility. With a 10% target and a 20% estimate, exposure is 0.5. This holds the expected day-to-day swing of the position near the target. It says nothing about the loss at any particular price.

Robert Carver, a systematic-trading author, describes keeping the two decisions apart: "Setting our position size and stop loss independently means we can target a particular % of our account at risk, whilst setting our stop differently according to market conditions." In his example, a $15,000 Apple position with a 10% stop and a $3,750 Bitcoin position with a 40% stop each put $1,500 at risk.

How often does noise reach a fixed stop?

A stop-loss order, or stop, becomes a market order when the price reaches a set level. Measured in the market's own daily volatility, a 5% stop is 5 typical daily moves away at 1% daily volatility, 2.5 at 2% and 1.67 at 3%.

For a driftless Brownian price model (a random walk with no trend) with constant volatility and continuous monitoring, the reflection principle gives the chance of touching the stop within T days: 2 x (1 - N(d / (s x sqrt(T)))). Here d is the stop distance, s the daily volatility and N the standard normal cumulative distribution, so d / (s x sqrt(T)) is the stop in T-day volatility units. Over 20 trading days a 5% stop has about a 26% chance of being touched at 1% daily volatility, 58% at 2% and 71% at 3%. A stop set at a fixed number of daily-volatility units has the same chance at any volatility level; at 2 units it is about 65%.

Real prices have fat tails, overnight gaps and volatility that changes during a trade, so a simulation checked the formula on 2,000 synthetic markets, 10 years each, with volatility that clusters (a GARCH process with a long-run level of 20% a year) and fat-tailed overnight gaps. A long trade was entered at the close every 20 trading days (250,000 entries), each with two stops. One is a fixed 5%. The other is k times the volatility estimate at entry, with k = 4.51 chosen so that the median stop over all entries is also 5%. That k gives the two rules the same median width; it is not a suggested setting. Entries were then split into thirds by the volatility estimate at entry.

The fixed 5% stop was reached after 22.3% of calm-third entries and 41.4% of volatile-third entries. At the median it sat 3.6 ATR from the entry in the calm third and 2.3 ATR in the volatile third. The formula, fed each third's median volatility, gives 19.1% and 45.7%. With a 6% yearly upward drift added to the same markets, the fixed stop's rates were 19.4% and 39.1%.

The volatility-based stop narrowed the gap and turned it around: 35.6% in the calm third and 24.8% in the volatile third, against 31.3% from the formula in every third. The reversal comes from the entry estimate: it looks back, and volatility drifts toward its usual level. After calm-third entries, volatility over the next 20 days had a median of 14.5% a year against an estimate of 13.6%; after volatile-third entries, 21.7% against 23.8%.

Planned loss and daily swing in the same entries

A $20,000 position with a fixed 5% stop plans to lose $1,000 at the stop in every market. Its typical daily swing, the position times the daily volatility estimate, still changes with the market. Sizing the volatility-based stop to $1,000 holds both steady, because the stop and the swing come from the same estimate.

Does a stop cap the loss at the stop price?

The SEC's investor bulletin on stop orders describes the stop price as "a trigger that causes the stop order to become a market order." The fill can land well away from the stop when prices move fast, and a brief dip during the day can trigger a stop that fills far below that day's close.

The simulation filled stops kindly: a day that opened below the stop filled at the open, and a touch during the day filled exactly at the stop, with no other slippage. Even so, 10.4% of the fixed stop's stop-outs filled beyond it after a gap. On those, the loss averaged 1.10 times the planned loss. The 99th percentile of all stop-outs was 1.24 times the plan, and the worst was 3.31 times. A position sized to lose $1,000 at the stop, its 1R, lost $1,010 per stop-out on average.

A stop also has a cost when prices have no memory. Kaminski and Lo's working paper states that "under the Random Walk Hypothesis, simple 0/1 stop-loss rules always decrease a strategy's expected return, but in the presence of momentum, stop-loss rules can add value." A 0/1 rule is all in or all out. Using monthly US equity returns from 1950 to 2004 and switching to long-term government bonds while stopped out, some rules added 50 to 100 basis points a month in the stopped-out months.

What does volatility targeting change?

The same 2,000 synthetic markets were also held two ways for 10 years: a fixed exposure of 1, and an exposure set each day to 20% / estimated yearly volatility, capped at 2. The estimate uses returns up to the day before only, so it could have been known at the time. The 20% target equals the market's long-run volatility.

Across 2,000 paths, the median vol of vol, the standard deviation of the rolling 30-day volatility, fell from 6.1 to 3.2 points, and it was lower with scaling in 99.95% of paths. The scaled line still moved: its 10th to 90th percentile range was 16.4% to 24.4% a year, against 12.3% to 26.6% with fixed exposure, because an estimate built from past days trails each change. Days beyond 3 standard deviations fell from 0.83% to 0.44%, and the worst day from -5.8% to -5.1%.

Return and drawdown barely moved. The median Sharpe ratio, yearly return divided by yearly swings, was 0.10 both ways before costs, and scaling had the higher one in 49% of paths. The median maximum drawdown was -51.4% with scaling and -50.0% without, partly because average exposure was 1.15: scaling adds exposure in calm stretches. Daily exposure changes also cost money. Turnover of 8.5 a year at 0.10% per unit came to about 0.85% a year and took the median Sharpe ratio to 0.06.

This synthetic market has no leverage effect, the tendency of volatility to rise when prices fall, so no return gain was expected. That matches the published results below for bonds, currencies and commodities. Volatility also comes in regimes, longer calm and wild stretches, which Vol 5 takes up. Volatility targeting across several positions needs their correlations, the subject of Vol 6.

What does the research say about scaling by volatility?

• Scaling by recent variance. Moreira and Muir (Journal of Finance, 2017) built portfolios that, in their working paper's words, "scale monthly returns by the inverse of their previous month's realized variance". Their 2016 working paper reports, for the US market from 1926 to 2015, "an overall 25% increase in the buy-and-hold Sharpe ratio." The published abstract gives the reason: "changes in volatility are not offset by proportional changes in expected returns."
• Steadier risk in all assets, higher Sharpe ratios mainly in stocks and credit. Harvey and co-authors (2018) studied 60+ assets. Volatility targeting "consistently reduces the likelihood of extreme returns (and the volatility of volatility)" across all of them, while Sharpe ratios rose for "risk assets (equities and credit)" and barely changed for bonds, currencies and commodities. For US equities from 1927 to 2017 the Sharpe ratio rose from 0.40 to between 0.48 and 0.51, and vol of vol fell from 4.6% to 1.8%. They tie the difference to the leverage effect: scaling cuts exposure when volatility rises after falls, which "effectively introduces some momentum into strategies."
• The critique. Cederburg and co-authors (2020) tested 103 equity strategies. Volatility-managed versions "do not systematically outperform their corresponding unmanaged portfolios in direct comparisons", and versions built only from data available at the time "generally earn lower certainty equivalent returns and Sharpe ratios" than the plain portfolios. DeMiguel, Martin-Utrera and Uppal (2024) summarize that critique and Barroso and Detzel's finding that the strategies do not survive trading costs. They then report a multifactor version that, by their account, holds up out of sample and after costs.

Common mistakes

• Treating a percentage stop as a fixed risk. The planned loss stays fixed while the chance of reaching the stop moves with volatility, from 22.3% to 41.4% across the simulated thirds.
• Treating the volatility estimate as known. It looks back, and volatility tends to drift toward its usual level.
• Tuning the ATR length, the stop multiple or the target on the backtest. They are settings too. Picking them by their past result is the overfitting in Vol 2, and Vol 3 covers judging them on data they never saw.
• Expecting volatility scaling to raise returns everywhere. In Harvey and co-authors' study the Sharpe gains came mainly from stocks and credit.