*Inner Price*

Any collection of historical prices typically assumes two values: the best bid (buy) and the best ask (sell). These prices are collected from the complete order books specific for the given exchange. The order books contain all clients orders submitted in the market at the given moment. The non-zero price difference between the best offers on the sell and buy sides, called spread, indicates the level of a market's liquidity (Bessembinder 1994; Dyhrberg et al. 2018; Menkhoff et al. 2012). It also has a direct connection to realised volatility (Bollerslev and Melvin 1994), and is an indicator of the transaction cost (Hartmann 1999). Another role of the spread is to show the extent of uncertainty the market has on the fair price of the traded asset. The size of the spread constantly changes over time, together with the level of uncertainty. This fact does not allow us to employ only bid or ask prices to study properties of the whole market at the micro level since some part of the information is in the risk to be lost while analysing intraday data. The average of these two values (mid-price) is also not the best alternative since it does not keep the knowledge on the size of the current spread. Therefore, an alternative measure should be chosen to apply the directional-change algorithm to the real data analysis.

The trend-dependent concept of inner price was selected to resolve the spread issue. Inner price, specific for the given moment of time, is defined as the bid or the ask price depending on the direction of the current trend. In other words, it is the price where the spread is deduced. The following example demonstrates details of the concept. According to the directional-change algorithm, one should wait for the price increase by *δ* percents from the local price minimum to register a new directional change if the current trend is downward. In this case, the value of the inner price coincides with the best price on the offer side of the order book, that is, the ask price. A new intrinsic event will tick only when the distance between the latest bid price and the inner price reaches the size of the chosen directional-change threshold *δ*. Alternatively, the inner price takes the value of the best bid price, and the distance is measured between the newest ask and the extreme if the current mode is upward.
