A Quantized Measure of Intraday Price Path Length Relative to Range

Summary

The document proposes measuring how much a stock price moves within a day by summing absolute changes between selected observations, rounding each change down in units of a chosen price increment, and normalizing the total by the day’s range. It asks whether this quantity could remain large over long periods even when the stock stays inside a limited daily range. The increment might correspond to an options strike spacing, and the price series could use minute closes.

This is a proposed measure and an open mathematical question, not a validated trading signal. The text supplies no empirical results or proof of a bound. Its formulation also leaves details to clarify, including the index notation and what constraints govern the selected observations. The idea distinguishes total traversed price path from overall high-to-low range, but its usefulness depends on defining the sampling, quantization, and supremum precisely before comparing values across days or securities.

Key ideas

• The proposed measure totals quantized absolute price changes across selected observations.
• It normalizes that total by the observed price range.
• The question is whether repeated movement can make the measure persistently large within a bounded range.
• The document provides neither a proof nor empirical evidence for a bound.
• Sampling rules and index definitions need clarification before implementation.

Tags

# Bound on path length of a stock price


# Bound on path length of a stock price












Consider a time series $(S_i)$ representing a stock price (say close prices of one minute candles). Let $\Delta$ be a quantization step (could be the price step in the strike prices of the corresponding options) and let $$K \triangleq \sup_j \cfrac{ \left ( \sum_{j=1}^{r} \lfloor \lvert S_{i_j}-S_{i_{j-1}}\rvert / \Delta \rfloor \right )} {(R / \Delta)}.$$ Here, the indices $i_j$ come from the index set $(i_0,i_2,...,i_k)$ such that $1 \leq i_0 < i_1 < ... < i_k \leq n$ assuming there were $n$ close prices each day. To evaluate $K$, we just need the supremum of this expression for all possible values of $j$ given the constraints on the indices as shown above. I tried to formulate the numerator based on the standard expression from analysis for the path length of a function with the difference of having quantized the absolute difference by $\Delta$.

Basically I have tried to define a measure $K$ that measures the discretized path length of the stock price normalized by the range $R$ of the stock price which is the difference between the max and min bounds of that stock price series during that day.

Is it possible for us to have $K > 10$ (ten is arbitrary here but it can be any big number) consistently every day for years together? Put in other words, can the stock move between a limited price range each day but cross strike prices a lot of times throughout the day consistently for years together? If not, what is the constraint due to which it cannot do that?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.