Intrinsic time and direciontal changes

Physical time vs. Intrinsic time

• Physical time
  • point-based
  • daily, hourly,...
• Traditional approachs to observe price movements in financial time series are based on physical time changes
  • It has become difficult and challenging to obeserve using this. Why?
  • Significant patterns of the trading activity are ignored when using physical time

• Intrinsic time
  • event-based
  • What is event?
    • an event is characterized by a fixed threshold $\lambda$
    • is defined as the absolute price change between two local extremal values exceeding a given threshold $\lambda$
    • Directional changes are used to define event
• Physical time vs. Intrinsic time

Physical time	Intrinsic time
point-based	event-based
homogeneous	inhomogeneous
time scales equally spaced	time triggers only at periodic events

• See Figure below:
  • The lines in the graph summarize the price movements using intrinsic and physical time
  • Significant events of trading activity are ignored when we consider changes over time based on fixed time intervals

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Directional-change : definition

Terminology

Extreme point / DCC point

• If the inequality just above holds,

Term	Description	Example in figure
Extreme point	the time at $P_{EXT}$	points $A$, $B$, $C$, $D$, $E$, $F$, $G$
DCC point	the time at $P_{c}$	points $A^{0.1}$, $B^{0.1}$, $C^{0.1}$, $D^{0.1}$, $E^{0.1}$, $F^{0.1}$, $G^{0.1}$

• Note that whilst an extreme point is the end of the one trend, it is also the start point of the next trend
• An extreme point is only recognized in hindsight; precisely at the DCC point

DC event / OS event

• DC means "Directional Change" and OS does "Overshoot"

Term	Description	Example in figure
DC event	-starts with an extreme point -ends with a DCC point	intervals $[A,A^{0.1}]$, $[B,B^{0.1}]$, $\dots$, $[G,G^{0.1}]$
DCC point	-starts at the DCC point -ends at the next extreme point	ntervals $[A^{0.1},B]$, $[B^{0.1},C]$, $\dots$, $[E^{0.1},F]$

• Note that for a given time series and a predetermined threshold, the DC summary is unique

Definition of DC

• A directional-change(DC) event can take one of the two forms
  • a downturn event
  • an upturn event
• Note that a downturn event occurs in upward run and an upturn event occurs in downward run

downward run	upward run
A period between a downturn event and the next upturn event	A period between a upturn event and the next downturn event
The last low price $p_{L}$ is continuosly updated to the minimum of the current market price $p(t)$ and the last low price $p_{L}$	The last high price $p_{L}$ is continuosly updated to the maximum of the current market price $p(t)$ and the last high price $p_{H}$
An $upturn$ event is an event when the absolute price change between the current market price $p(t)$ and the last low price $P_{L}$ is higher than a given threshold $\Delta x_{DC}$, i.e., $p(t)\ge p_{L}(1+\Delta x_{DC})$ If this holds, the point which the price last troughed ($P_{L})$ is the starting point of a upturn event	An $downturn$ event is an event when the absolute price change between the current market price $p(t)$ and the last high price $P_{H}$ is lower than a given threshold $\Delta x_{DC}$, i.e., $p(t)\le p_{H}(1-\Delta x_{DC})$ If this holds, the point which the price last peaked ($P_{H})$ is the starting point of a downturn event

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Implementation in Python

def myDC(data, d=0.2):
    '''
    - Aloud, M., Tsang, E., Olsen, R. & Dupuis, A. (2012). A Directional-Change Event Approach for Studying Financial Time Series. Economics, 6(1), 20120036.
    - data : a list or array-like time series object
    - d : theta value, which is a threshold to decide upturn/downturn event
    '''
    
    p = pd.DataFrame({
    "Price": data
    })
    p["Event"] = ''
    run = "upward" # initial run
    ph = p['Price'][0] # highest price
    pl = ph # lowest price
    pl_i = ph_i = 0

    for t in range(0, len(p)):
        pt = p["Price"][t]
        if run == "downward":
            if pt < pl:
                pl = pt
                pl_i = t
            if pt >= pl * (1 + d):
                p.at[pl_i, 'Event'] = "start upturn event"
                run = "upward"
                ph = pt
                ph_i = t
                # print(">> {} - Upward! : {}%, value {}".format(pl_i, round((pt - pl)/pl, 2), round(pt - pl,2)))
        elif run == "upward":
            if pt > ph:
                ph = pt
                ph_i = t
            if pt <= ph * (1 - d):
                p.at[ph_i, 'Event'] = "start downturn event"
                run = "downward"
                pl = pt
                pl_i = t
                # print(">> {} - Downward! : {}%, value {}".format(ph_i, round((ph - pt)/ph, 2), round(ph - pt,2)))
    return p

[References]

• Bakhach, A., Tsang, E.P., & Jalalian, H.R. (2016). Forecasting directional changes in the FX markets. 2016 IEEE Symposium Series on Computational Intelligence (SSCI), 1-8.

• Aloud, M., Tsang, E., Olsen, R. & Dupuis, A. (2012). A Directional-Change Event Approach for Studying Financial Time Series. Economics, 6(1), 20120036.