# 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?

• Have you tried looking at historical time series of prices of some stocks to test whether they bounce as you describe? Also, do dividends matter? Jan 27 at 18:01
• Not sure why the downvote but yes ... I am basing this question on historical data for some asset. Ten is admittedly a big number but nevertheless I think this question deserves a proper answer. Whoever downvoted this, please let me know the reason so that I know the reason for it, and might even take down the question if that reason is valid. Jan 27 at 18:24
• not sure I understand your notation, what does $\lfloor\cdot\rfloor$ means? if it is the positive part, the absolute value being positive already, I do not get it. I do not understand the sup over $j$ neither since it is in the sum sign: this is a dummy variable, no? Jan 28 at 10:57
• The symbol $\lfloor.\rfloor$ is the floor function ($\lfloor 1.1 \rfloor =1$, $\lfloor 0.1 \rfloor =0$ and $\lfloor -0.9 \rfloor = -1$ etc.) ie it is the highest integer closest to its value but lesser than that value. As regards to the $\sup$ over all $j$, I have added more information in the post. I hope that makes it clearer. Jan 28 at 17:27
• unfortunately it is not really better, you probably want to get the supremum over all the possible partition in $k+1$ intervals of $n$ integers? in such a case you need to define the set of all the partitions, such that it is made of operators allowing to write properly your sum. I think that your difficulties come essentially from the fact that your problem is ill-posed at this stage. Jan 29 at 14:32