Estimating High-Low Range Under Geometric Brownian Motion

Summary

The document asks whether an analytical expression exists for the expected high-low price range of an asset following geometric Brownian motion over a sampling interval. The author reports numerical simulations suggesting that the range scales approximately with volatility and a power of elapsed time, and says the relationship remained robust across the volatility levels examined. The estimate is offered as a simulation-based observation, not as a derived formula.

The motivation is to build real-time average true range measures from higher-frequency prices. The author recognizes that actual markets do not follow geometric Brownian motion exactly and asks whether an analytical solution is available. No answer, derivation, or validation appears in the supplied document, so it does not establish the proposed scaling as a general result. Any practical use would need to account for sampling frequency, market microstructure effects, and departures from the assumed process.

Key ideas

• The question concerns the expected high-low range under a geometric Brownian motion model.
• The author reports a simulated power-law scaling involving volatility and elapsed time.
• The reported relationship is a numerical observation rather than an analytic derivation.
• The proposed application is real-time average true range estimation from higher-frequency prices.
• The document provides no answer and acknowledges that real market behavior can differ from the model.

Tags

# High/Low range under GBM - Analytic solution?


# High/Low range under GBM - Analytic solution?












Does anybody know of an analytical solution to the expected high / low range for an asset that follows a GBM process over sampling frequency dt? I have ran numerical simulations and find that the scaling rule is approximately: $$ 1.8\sigma * T^{0.522}$$

That's robust (r^2 = 0.98) against vols up to 50%.

I am trying to generate rules to provide real-time measures of the ATR using higher frequency price data, obviously real life differs from the GBM result but I am still curious if this problem has an analytic solution?

Thanks.

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