Volatility and Confidence Intervals in Pricing

A fair value estimate is a distribution, not a number. Volatility is how you measure its width, and the width is what determines your spread.

From uncertainty to spread

Suppose your fair value is 100.00 and you expect to hold any resulting position for about 30 seconds before offsetting it. If the instrument's volatility implies a standard deviation of 1.5 cents over 30 seconds, then a fill exposes you to roughly that much risk.

Quoting a spread narrower than your uncertainty means you are consistently trading inside your own error bars, which loses money on average. The spread has to cover the expected adverse move over the holding period, plus adverse selection, plus costs.

This gives a usable heuristic:

It opens on a quote that is inside its own floor, which is the state to recognise. Then quadruple the holding time: the floor only doubles, because the square root means offsetting twice as fast buys considerably less than it seems to.

Scaling with time

Volatility scales with the square root of time when returns are independent:

so a 1% daily volatility is about $1\% \times \sqrt{252} \approx 16\%$ annualised, and conversely an instrument with 32% annualised volatility moves about 2% in a typical day.

For a market maker the relevant horizon is seconds, and the same scaling applies downward. This is worth being fluent in, since it converts a headline volatility figure into the number that actually matters: how far can this move while I am holding it. The full derivation is in the statistics course.

Realised versus implied

Realised volatility is measured from past prices. It is backward-looking, and it says nothing about a scheduled event tomorrow.

Implied volatility is extracted from option prices and is the market's forward-looking estimate. Where options exist, implied volatility is usually the better input for sizing a spread, because it already incorporates known upcoming events.

The gap between the two is itself a traded quantity, and it is the core of options market making.

Volatility is not constant

Two facts about real volatility change how you use it.

It clusters. High-volatility periods follow high-volatility periods. A recent spike is genuine evidence that the next hour will also be volatile, which means recent data should be weighted more heavily than distant data. This is what models like EWMA and GARCH formalise.

It is predictable in a way returns are not. You cannot forecast direction, but you can forecast magnitude reasonably well. This asymmetry is why volatility is the quantity market makers model most carefully.

Practical tools

Rolling standard deviation of recent returns, with a window matched to your horizon.

Exponentially weighted volatility, which decays the influence of older observations and responds faster to regime changes.

Bands around a moving average, which give a quick visual sense of whether the current price is unusual relative to recent behaviour, though they are descriptive rather than predictive.

The response to rising volatility is mechanical: widen the spread, reduce quoted size, and check existing inventory, since the position you already hold just became more dangerous. See spread widening.