# Intelligent Dynamic Backlash Agent: A Trading Strategy Based on the Directional Change Framework
## 2 Directional Changes: An Introduction  
In this section, we explain how market prices are summarized based on the DC concept. Under the DC framework, the market is represented as alternating uptrends and downtrends. The basic idea is that the magnitude of price change during an uptrend, or a downtrend, must be at least equal to a specific threshold _theta_. Here, _theta_ is a percentage that the observer considers substantial (usually expressed as a percentage). For example, Figure 1 depicts a price's drop between points A and A\({}^{0.1}\). This price drop is equal to the selected, hypothetical, threshold of 0.1%. In this case, we say that we have a DC downtrend that starts at point A. Any price change less than the identified threshold, _theta_, will not be considered as a trend when summarizing price movements [1; 2]. Under the DC framework, each uptrend is followed by a downtrend and vice versa. The detection of a new uptrend, or downtrend, is a crucial task. The detection of a new downtrend, or uptrend, is a two-step algorithmic approach:  
_Step 1_: If the market is currently in a downtrend, let \(P_{EXT}^{next}\) denotes the lowest price in this downtrend. The value of \(P_{EXT}^{next}\) may change as the price movement continues. We use Table 1 to exemplify this note. For example, at time 20:55:00, in Table 1, the mid-price is 1.48260. The lowest price observed between time 20:55:00 and 20:58:00 is 1.48230 which was observed at time 20:56:00. Therefore, the value of \(P_{EXT}^{next}\), at time 20:58:00 is 1.48230. However, as the price's movement continues, at time 21:01:00, the mid-price becomes 1.48180. In this case, the lowest price observed between point time 20:55:00 and time 21:01:00 becomes \(P_{EXT}^{next}=1.48150\) (which was observed at time 21:00:00). Similarly, if the market is currently in uptrend, then \(P_{EXT}^{next}\) would refer to the highest price in this uptrend. _Step 2_: Let \(P_{c}\) be the current price. We say that the market switches its direction from a downtrend to an uptrend if \(P_{c}\) becomes greater than \(P_{EXT}^{next}\) by at least \(theta\) (where \(theta\) is the threshold predetermined by the observer). Similarly, we say that the market switches its direction from an uptrend to a downtrend if \(P_{c}\) becomes less than \(P_{EXT}^{next}\) by at least \(theta\). The detection of a new DC uptrend or a new DC downtrend is a formalized inequality, as shown in Equation (1). For example, in Table 1, at time 21:05:00, the current price, \(P_{c}\), is 1.48310. At time 21:05:00, the \(P_{EXT}^{next}\) is 1.48150 (which was observed at time 21:00:00). In this case, the magnitude of price's change between \(P_{c}\) and \(P_{EXT}^{next}\) is \(>0.1\%\). Thus, the inequality in Equation (1) holds and we can confirm the observation of a new DC uptrend. In other words, at time 21:05:00, we can confirm the observation of a new DC uptrend which has started at time 21:00:00. If the inequality in Equation (1) holds, then the time at which the market traded at \(P_{EXT}^{next}\) is called an \"extreme point\" (e.g., point B in Table 1) and the time at which the market trades at \(P_{c}\) is called a DC confirmation point, or DCC point for short (e.g., point B\({}^{0.1}\) in Table 1). By definition, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time** & **Mid-Price (\(P_{c}\))** & \(P_{EXT}^{next}\) & **Point** \\ \hline
20:55:00 & 1.48260 & 1.48260 & \\
20:56:00 & 1.48230 & 1.48230 & \\
20:57:00 & 1.48240 & 1.48230 & \\
20:58:00 & 1.48260 & 1.48230 & \\
20:59:00 & 1.48200 & 1.48200 & \\
21:00:00 & 1.48150 & 1.48150 & B (Extreme point) \\
21:01:00 & 1.48180 & 1.48150 & \\
21:02:00 & 1.48170 & 1.48150 & \\
21:03:00 & 1.48159 & 1.48150 & \\
21:04:00 & 1.48280 & 1.48150 & \\
21:05:00 & 1.48310 & 1.48150 & \(\mathbb{B}^{0.1}\) (DCC point) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The progress of the value of \(P_{EXT}^{next}\) during the period from 20:55:00 and 21:05:00. According to Figure 1 this period refers to a downtrend.