The performance analysis of chart patterns: Monte Carlo simulation and evidence from the euro/dollar foreign exchange market

Walid Ben Omrane; Hervé Van Oppens

doi:10.1007/s00181-005-0007-8

Empirical Economics

January 2006, Volume 30, Issue 4, pp 947–971 | Cite as

The performance analysis of chart patterns: Monte Carlo simulation and evidence from the euro/dollar foreign exchange market

Authors
Authors and affiliations

Walid Ben OmraneEmail author
Hervé Van Oppens

Original Article

First Online: 04 November 2005

171 Downloads
5 Citations

Abstract

We investigate the existence of chart patterns in the euro/dollar intra-daily foreign exchange market. We use two identification methods of the different chart patterns: one built on 5-min close prices only, and one based on both 5-min low and high prices. We look for twelve types of chart patterns and we study the detected patterns through two criteria: predictability and profitability. We run a Monte Carlo simulation to compute the statistical significance of the obtained results. We find an apparent existence of some chart patterns in the currency market. More than one half of detected charts present a significant predictability. Nevertheless, only two chart patterns imply a significant profitability which is however too small to cover the transaction costs. The second extrema detection method provides higher but riskier profits than the first one.

Keywords

Foreign exchange market Chart patterns High frequency data Technical analysis

JEL Classification

C13 C14 F31

This is a preview of subscription content, to check access.

Notes

Acknowledgements

While remaining responsible for any error in this paper, we thank Luc Bauwens, Winfried Pohlmeier and David Veredas (the editors of the special issue) and two anonymous referees for helpful comments which improved the results in the paper. We thank also Eric Debodt, Nihat Aktas, Hervé Alexandre, Patrick Roger and participants at the 20th AFFI International Conference (France) and SIFF 2003 seminar (France). This paper presents the results of research supported in part by the European Community Human Potential Programme under contract HPRN-CT-2002-00232, Microstructure of Financial Markets in Europe.

A. Price curve estimation

Before adopting the Nadaraya–Watson kernel estimator, we tested the cubic splines and polynomial approximations but we conclude empirically that the appropriate smoothing method is the kernel. Because the two first methods carry out too smoothed results and they are not flexible as the kernel method.

From the complete series of the price, P _t (t=1,...,T), we take a window k of l regularly spaced time intervals¹³, such that:

$$P_{{j,k}} \subset {\left\{ {P_{t} |k \leqslant t \leqslant k + l - 1} \right\}},$$

(8)

j=1,...,l and k=1,...,T−l+1. For each window k, we consider the following relation:

$$P_{{j,k}} = m{\left( {X_{{P_{{j,k}} }} } \right)} + \in _{{P_{{j,k}} }} ,$$

(9)

where $\in _{{P_{{j,k}} }}$ is a white noise and $m{\left( {X_{{P_{{j,k}} }} } \right)}$ is an arbitrarily fixed but unknown non linear function of a state variable ${X_{{P_{{j,k}} }} }$. Like Lo et al. (2000) to construct a smooth function in order to approximate the time series of prices P _j,k, we set the state variable equal to time, $X_{{P_{{j,k}} }} = t$. For any arbitrary x, a smoothing estimator of m(x) may be expressed as:

$$\ifmmode\expandafter\hat\else\expandafter\^\fi{m}(x) = \frac{1} {l}{\sum\limits_{j = 1}^l {\omega _{j} (x)P_{{j,k}} ,} }$$

(10)

where the weight ω _j(x) is large for the prices P _j,k with $X_{{P_{{j,k}} }}$ near x and small for those with $X_{{P_{{j,k}} }}$ far from x. For the kernel regression estimator, the weight function ω _j(x) is built from a probability density function K(x), also called a kernel:

$$K(x) \geqslant 0,{\int\limits_{ - \infty }^{ + \infty } {K_{h} (u)du = 1.} }$$

(11)

By rescaling the kernel with respect to a parameter h>0, we can change its spread:

$$K_{h} (u) \equiv \frac{1} {h}K(u/h),{\int\limits_{ - \infty }^{ + \infty } {K_{h} (u)du = 1} }$$

(12)

and define the weight function to be used in the weighted average (10) as:

$$\omega _{{j,h}} \equiv K_{h} {\left( {x - X_{{P_{{j,k}} }} } \right)}{\text{/ }}g_{h} (x)$$

(13)

$$g_{h} (x) \equiv \frac{1} {l}{\sum\limits_{j = l}^l {K_{h} {\left( {x - X_{{P_{{j,k}} }} } \right)}.} }$$

(14)

Substituting (14) into (10) yields the Nadaraya–Watson kernel estimator $\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (x)$ of m(x):

$$\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (x) = \frac{1} {l}{\sum\limits_{j = l}^l {\omega _{{j,h}} (x)P_{{j,k}} = \frac{{{\sum\nolimits_{j = 1}^l {K_{h} {\left( {x - X_{{P_{{j,k}} }} } \right)}P_{{j,k}} } }}} {{{\sum\nolimits_{j = 1}^l {K_{h} {\left( {x - X_{{P_{{j,k}} }} } \right)}} }}}.} }$$

(15)

If h is very small, the averaging will be done with respect to a rather small neighborhood around each of the ${X_{{P_{{j,k}} }} }$'s. If h is very large, the averaging will be over larger neighborhoods of the ${X_{{P_{{j,k}} }} }$'s. Therefore, controlling the degree of averaging amounts to adjusting the smoothing parameter h, also known as the bandwidth. Choosing the appropriate bandwidth is an important aspect of any local-averaging technique. In our case we select a Gaussian kernel with a bandwidth, h _opt,j, computed by Silverman (1986):

$$K_{h} (x) = \frac{1} {{h{\sqrt {2\pi } }}}e^{{ - \frac{{x^{2} }} {{2h^{2} }}}} $$

(16)

$$h_{{opt,k}} = {\left( {\frac{4} {3}} \right)}^{{1/5}} \sigma _{k} l^{{ - 1/5}} ,$$

(17)

where σ _k is the standard deviations for the observations that occur within the window k. However, the optimal bandwidth for Silverman (1986) involves a fitted function which is too smooth. In other words this optimal bandwidth places too much weight on prices far away from any given time t, inducing too much averaging and discarding valuable information in local price movements. Like Lo et al. (2000), through trial and error, we found that an acceptable solution to this problem is to use a bandwidth equal to 20% of h _opt,k:

$$h* = 0.2 \times h_{{opt,k}} .$$

(18)

B. Extrema detection methods

Technical details for both extrema detection methods and projection procedure are presented below:

B.1. M1

M1 is the extrema detection method using the close prices. After smoothing the data by estimating the Nadaraya–Watson kernel function, $\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}$, we compute maxima and minima respectively noted by $max_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}}} $and $min_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}}} $:

$$\begin{array}{*{20}c} {max_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} }} {\left( {X_{{P_{{j,k}} }} } \right)} = {\left\{ {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}\left| {S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{P_{{j + 1,k}} }} } \right)}} \right)} = + 1,S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{P_{{j + 1,k}} }} } \right)}} \right)} = - 1} \right.} \right\}}} \\ {min_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} }} {\left( {X_{{P_{{j,k}} }} } \right)} = {\left\{ {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}\left| {S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{P_{{j + 1,k}} }} } \right)}} \right)} = - 1,S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{P_{{j + 1,k}} }} } \right)}} \right)} = + 1} \right.} \right\}},} \\ \end{array} $$

where S(X) is the sign function, equal to +1 (−1) when the sign of X is positive (negative), and $\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}$ is the first derivative of the kernel function $\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}$. By construction we obtain alternate extrema. We denote respectively by $t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}} \right)}$and $t_{m} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}} \right)}$ the moments correspondent to detected extrema such that:

$$t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}} \right)} = {\left\{ {j\left| {j \in max_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}}} } \right.} \right\}}$$

(19)

$$t_{m} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}} \right)} = {\left\{ {j\left| {j \in min_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{P_{{j,k}} }} } \right)}}} } \right.} \right\}}.$$

(20)

After recording the moments of the detected extrema we realize an orthogonal projection of selected extrema, from the smoothing curve, to the original one. We deduce the corresponding extrema to construct the series involving both maxima, $max_{{P_{{j,k}} }} $ and minima, $min_{{P_{{j,k}} }} $ such that:

$$\begin{aligned} & max_{{P_{{j,k}} }} = max{\left( {P_{{t_{M} (\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (X_{{P_{{j,k}} }} )) - 1,k}} {\text{,}}P_{{t_{M} (\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (X_{{P_{{j,k}} }} )),k}} ,P_{{t_{M} (\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (X_{{P_{{j,k}} }} )) + 1,k}} } \right)} \\& min_{{P_{{j,k}} }} = min{\left( {P_{{t_{M} (\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (X_{{P_{{j,k}} }} )) - 1,k}} {\text{,}}P_{{t_{M} (\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (X_{{P_{{j,k}} }} )),k}} ,P_{{t_{M} (\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} (X_{{P_{{j,k}} }} )) + 1,k}} } \right)}\;. \\\end{aligned}$$

For each window k we get alternate maxima and minima. This is assured by the bandwidth h which provide at least two time intervals between two consecutive extrema. The final step consists to scan the extrema sequence to identify an eventual chart pattern. If the same sequence of extremum was observed in more than one window, only the first sequence is retained for the recognition study to avoid the duplication of results.

B.2. M2

M2 is the extrema detection method built on high and low prices. According to this method, maxima and minima have to be detected onto separate curves. Maxima on high prices curve and minima on the low one.

Let H _t and L _t (t=1,...,T), be respectively the series for the high and the how prices, and k a window containing l regularly spaced time intervals such that:

$$H_{{j,k}} \subset {\left\{ {H_{t} \left| {k \leqslant t \leqslant k + l - 1} \right.} \right\}}$$

(21)

$$L_{{j,k}} \subset {\left\{ {L_{t} \left| {k \leqslant t \leqslant k + l - 1} \right.} \right\}},$$

(22)

j=1,...,l and k=1,...,T−l+1. We smooth these series through the kernel estimator detailed in Appendix A to obtain $\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}$and $\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}$. We detect maxima on the former series and minima on the latter one in order to construct two separate extrema series $max_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}}} $ and $min_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}}} $ such that:

$$\begin{aligned} & max_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}}} = {\left\{ {\left. {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}\,} \right|S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)} = + 1{\text{ , }}S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)} = - 1} \right\}} \\& min_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}}} = {\left\{ {\left. {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}\,} \right|S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}} \right)} = - 1{\text{ , }}S{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}} \right)} = + 1} \right\}}{\text{ ,}} \\\end{aligned}$$

where S(x) is the sign function defined in the previous Section.

We record the moments for such maxima and minima, denoted respectively by $t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)}$ and $t_{m} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}^{'}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}} \right)}$and we project them on the original high and how curves to deduce the original extrema series $max_{{H_{{j,k}} }} $ and$min_{{L_{{j,k}} }} $, such that:

$$\begin{aligned} & max_{{H_{{j,k}} }} = max{\left( {H_{{t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)} - 1,k}} ,H_{{t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)},k}} ,H_{{t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)} + 1,k}} } \right)} \\& {\text{ }}min_{{L_{{j,k}} }} = min{\left( {L_{{t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{L_{{j,k}} }} } \right)}} \right)} - 1,k}} ,L_{{t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)},k}} ,L_{{t_{M} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{m}_{h} {\left( {X_{{H_{{j,k}} }} } \right)}} \right)} + 1,k}} } \right)}{\text{ }}{\text{.}} \\\end{aligned}$$

However, this method does not guarantee alternate occurrences of maxima and minima. It is easy to observe, in the same window k, the occurrence of two consecutive minima on the low series before observing a maximum on high series. To resolve this problem we start by recording the moments for the selected maxima on high curve, t _M(H _j,k), and minima in low curve, t _m(L _j,k). Then we select, for window k the first extremum from these two series, E _1,k, and its relative moment, $t_{{E_{{1,k}} }} $, such that:

$$t_{{E_{{1,k}} }} = {\text{ }}{\mathop {{\text{min}}}\limits_t }{\text{ }}{\left( {t_{M} {\left( {H_{{j,k}} } \right)}{\text{ }},{\text{ }}t_{m} {\left( {H_{{j,k}} } \right)}} \right)}$$

(23)

$$E_{{1,k}} = {\left( {\left. {{\left\{ {max_{{H_{{j,k}} }} } \right\}} \cup {\left\{ {min_{{L_{{j,k}} }} } \right\}}} \right|j = t_{{E_{{1,k}} }} } \right)}.$$

(24)

If we meet a particular case such that a minimum and a maximum occur at the same first moment, then we retain arbitrarily the maximum. To build the alternate series, we have to know the type of the last extremum introduced into the series. If it is a maximum (minimum) then the next extremum has to be a minimum (maximum) selected from the low (high) series such that:

$$\begin{aligned} &\,\,\,\,\,\,\,E_{{j,k|E_{{(j - 1),k}} \in {\left\{ {maxH_{{j,k}} } \right\}}}} = {\left\{ {\left. {min_{{L_{{j,k}} }} \,} \right|\,j = min{\left( {t_{m} {\left( {L_{{j,k}} } \right)}} \right)},t_{m} {\left( {L_{{j,k}} } \right)} > t_{{E_{{(j - 1),k}} }} } \right\}} \\& E_{{j,k|E_{{(j - 1),k}} \in {\left\{ {\min L_{{j,k}} } \right\}}}} = {\left\{ {\left. {max_{{H_{{j,k}} }} \,} \right|\,j = min{\left( {t_{M} {\left( {H_{{j,k}} } \right)}} \right)},t_{M} {\left( {H_{{j,k}} } \right)} > t_{{E_{{(j - 1),k}} }} } \right\}}\,\,, \\\end{aligned}$$

where E _j,k is the extremum detected on original series.

Finally, the obtained series is scanned by the recognition pattern algorithms to identify an eventual chart pattern.

C. Definition of chart patterns

C.1. Inverse head and shoulders (IHS):

IHS is characterized by a sequence of five extrema E _i (i=1,...,5) such that:

$$IHS \equiv \left\{ {\,\,\begin{array}{*{20}l} {{E_{1} < E_{2} } \hfill} \\ {{E_{3} < E_{1} ,E_{3} < E_{5} } \hfill} \\ {{{\left| {p{\left( {E_{2} ,E_{4} } \right)}} \right|} \leqslant tg(10)} \hfill} \\ {{{\left| {p{\left( {E_{1} ,E_{5} } \right)}} \right|} \leqslant tg(10)} \hfill} \\ {{0.9 \leqslant \frac{{V_{{t_{{E_{1} }} }} {\left( {E_{2} ,E_{4} } \right)} - E_{1} }} {{V_{{t_{{E_{5} }} }} {\left( {E_{2} ,E_{4} } \right)} - E_{5} }} \leqslant 1.1} \hfill} \\ {{1.1 \leqslant \frac{h} {s} \leqslant 2.5} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{{E_{2} }} - t_{d} }} {{t_{f} - t_{{E_{4} }} }} \leqslant 2} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{{E_{4} }} - t_{{E_{2} }} }} {m} \leqslant 2} \hfill} \\ {{{\left( {P_{{t_{{max}} }} - P_{{t_{d} }} } \right)} \geqslant \frac{2} {3} \times h} \hfill} \\\end{array} } \right.$$

where

h is the height of the head: $h = V_{{t_{{E_{3} }} }} {\left( {E_{2} ,E_{4} } \right)} - E_{3}$
s is the height average of the two shoulders: $s = \frac{{{\left( {V_{{t_{{E_{1} }} }} {\left( {E_{2} ,E_{4} } \right)} - E_{1} } \right)} + {\left( {V_{{t_{{E_{5} }} }} {\left( {E_{2} ,E_{4} } \right)} - E_{5} } \right)}}} {2}$
$P_{{t_{{max}} }}$ is the highest price observed into the time interval [t _d−(f−d), t _d]: ${P_{{t_{{max}} }} = max{\left( {P_{t} } \right)}\left| {t_{{d - {\left( {f - d} \right)}}} } \right. \le t \le t_{d} }$
t _d is the starting time for the pattern
t _f is the ending time for the pattern
t _d−(f−d)=t _d−(t _f−t _d)
t _f+(f−d)=t _f+(t _f−t _d)
m is the average time which the shoulders take for their total completion

C.2. Double top (DT)

DT is characterized by a sequence of three extrema E _i (i=1,..,3), such that:

$$DT \equiv \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} > E_{2} } \hfill} \\ {{\frac{{E_{1} - E_{2} }} {{V_{{t_{{E_{2} }} }} {\left( {E_{1} ,E_{3} } \right)} - E_{2} }} = 1} \hfill} \\ {{\frac{{E_{3} - E_{2} }} {{V_{{t_{{E_{2} }} }} {\left( {E_{1} ,E_{3} } \right)} - E_{2} }} = 1} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{{E_{2} }} - t_{d} }} {{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 2}} \right. \kern-\nulldelimiterspace} 2}} \leqslant 2} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{f} - t_{{E_{2} }} }} {{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 2}} \right. \kern-\nulldelimiterspace} 2}} \leqslant 2} \hfill} \\ {{{\left( {P_{{t_{d} }} - P_{{t_{{min}} }} } \right)} \geqslant \frac{2} {3} \times {\left( {E_{2} - V_{{t_{{E_{2} }} }} {\left( {E_{1} ,E_{3} } \right)} - E_{2} } \right)}} \hfill} \\\end{array} } \right.$$

C.3. Double bottom (DB)

DB is characterized by a sequence of three extrema E _i (i=1,...,3), such that:

$$DB \equiv \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} < E_{2} } \hfill} \\ {{\frac{{E_{2} - E_{1} }} {{E_{2} - V_{{t_{{E_{2} }} }} {\left( {E_{1} ,E_{3} } \right)}}} = 1} \hfill} \\ {{\frac{{E_{2} - E_{3} }} {{E_{2} - V_{{t_{{E_{2} }} }} {\left( {E_{1} ,E_{3} } \right)}}} = 1} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{{E_{2} }} - t_{d} }} {{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 2}} \right. \kern-\nulldelimiterspace} 2}} \leqslant 2} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{f} - t_{{E_{2} }} }} {{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 2}} \right. \kern-\nulldelimiterspace} 2}} \leqslant 2} \hfill} \\ {{{\left( {P_{{t_{{max}} }} - P_{{t_{d} }} } \right)} \geqslant \frac{2} {3} \times {\left( {E_{2} - V_{{t_{{E_{2} }} }} {\left( {E_{1} ,E_{3} } \right)}} \right)}} \hfill} \\\end{array} } \right.$$

C.4. Triple top (TT)

TT is characterized by a sequence of five extrema E _i (i=1,...,5) such that:

$$TT \equiv \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} > E_{2} } \hfill} \\ {{{\left| {p{\left( {E_{1} ,E_{5} } \right)}\,} \right|}\, \leqslant tg(10)} \hfill} \\ {{{\left| {p{\left( {E_{2} ,E_{4} } \right)}\,} \right|}\, \leqslant tg(10)} \hfill} \\ {{0.9 \leqslant \frac{h} {{E_{1} - V_{{t_{{E_{1} }} }} {\left( {E_{2} ,E_{4} } \right)}}} \leqslant 1.1} \hfill} \\ {{0.9 \leqslant \frac{h} {{E_{5} - V_{{t_{{E_{5} }} }} {\left( {E_{2} ,E_{4} } \right)}}} \leqslant 1.1} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{{E_{2} }} - t_{d} }} {{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 3}} \right. \kern-\nulldelimiterspace} 3}} \leqslant 2} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{{E_{4} }} - t_{{E_{2} }} }} {{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 3}} \right. \kern-\nulldelimiterspace} 3}} \leqslant 2} \hfill} \\ {{\frac{1} {2} \leqslant \frac{{t_{f} - t_{{E_{4} }} }} {{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 3}} \right. \kern-\nulldelimiterspace} 3}} \leqslant 2} \hfill} \\ {{{\left( {P_{{t_{d} }} - P_{{t_{{min}} }} } \right)} \geqslant \frac{2} {3} \times h} \hfill} \\\end{array} } \right.$$

C.5 Triple bottom (TB)

TB is characterized by a sequence of five extrema E _i (i=1,...,5) such that:

$$TB \equiv \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} < E_{2} } \hfill} \\ {{{\left| {p{\left( {E_{2} ,E_{4} } \right)}\,} \right|}\, \leqslant tg(10)} \hfill} \\ {{{\left| {p{\left( {E_{1} ,E_{5} } \right)}\,} \right|}\, \leqslant tg(10)} \hfill} \\ {{0.9 \leqslant \frac{h}{{V_{{t_{{E_{1} }} }} {\left( {E_{2} ,E_{4} } \right)} - E_{1} }} \leqslant 1.1} \hfill} \\ {{0.9 \leqslant \frac{h}{{V_{{t_{{E_{5} }} }} {\left( {E_{2} ,E_{4} } \right)} - E_{5} }} \leqslant 1.1} \hfill} \\ {{\frac{1}{2} \leqslant \frac{{t_{{E_{2} }} - t_{d} }}{{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 3}} \right. \kern-\nulldelimiterspace} 3}} \leqslant 2} \hfill} \\ {{\frac{1}{2} \leqslant \frac{{t_{{E_{4} }} - t_{{E_{2} }} }}{{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 3}} \right. \kern-\nulldelimiterspace} 3}} \leqslant 2} \hfill} \\ {{\frac{1}{2} \leqslant \frac{{t_{f} - t_{{E_{4} }} }}{{{{\left( {t_{f} - t_{d} } \right)}} \mathord{\left/ {\vphantom {{{\left( {t_{f} - t_{d} } \right)}} 3}} \right. \kern-\nulldelimiterspace} 3}} \leqslant 2} \hfill} \\ {{{\left( {P_{{t_{{max}} }} - P_{{t_{d} }} } \right)} \geqslant \frac{2}{3} \times h} \hfill} \\ \end{array} } \right.$$

C.6. Rectangle top (RT)

RT is characterized by a sequence of six extrema E _i (i=1,...,6) such that:

$$RT = \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} > E_{2} } \hfill} \\ {{{\left| {p{\left( {E_{1} ,E_{5} } \right)}\,} \right|}\, \leqslant 0.001} \hfill} \\ {{{\left| {p{\left( {E_{2} ,E_{6} } \right)}\,} \right|}\, \leqslant 0.001} \hfill} \\ {{\frac{{V_{{t_{{E_{3} }} }} {\left( {E_{1} ,E_{5} } \right)}}} {{E_{3} }} = 1} \hfill} \\ {{\frac{{E_{4} }} {{V_{{t_{{E_{4} }} }} {\left( {E_{2} ,E_{6} } \right)}}} = 1} \hfill} \\ {{{\left( {P_{{t_{d} }} - P_{{t_{{\min }} }} } \right)} \geqslant \frac{2} {3} \times h} \hfill} \\\end{array} } \right.$$

C.7. Rectangle bottom (RB)

RB is characterized by a sequence of six extrema E _i (i=1,...,6) such that:

$$RB = \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} < E_{2} } \hfill} \\ {{{\left| {p{\left( {E_{2} ,E_{6} } \right)}\,} \right|}\, \leqslant 0.001} \hfill} \\ {{{\left| {p{\left( {E_{1} ,E_{5} } \right)}\,} \right|}\, \leqslant 0.001} \hfill} \\ {{\frac{{E_{3} }}{{V_{{t_{{E_{3} }} }} {\left( {E_{1} ,E_{5} } \right)}}} = 1} \hfill} \\ {{\frac{{V_{{t_{{E_{4} }} }} {\left( {E_{2} ,E_{6} } \right)}}}{{E_{4} }} = 1} \hfill} \\ {{{\left( {P_{{t_{{max}} }} - P_{{t_{d} }} } \right)} \geqslant \frac{2}{3} \times h} \hfill} \\ \end{array} } \right.$$

C.8. Broadening Top(BT)

BT is characterized by a sequence of five extrema E _i (i=1,...,5) such that:

$$BT = \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} > E_{2} } \hfill} \\ {{E_{3} > E_{1} ,E_{4} < E_{2} ,E_{5} > E_{3} } \hfill} \\ {{{\left( {P_{{t_{d} }} - P_{{t_{{min}} }} } \right)} \geqslant \frac{2} {3} \times h} \hfill} \\\end{array} } \right.$$

C.9. Broadening bottom(BB)

BB is characterized by a sequence of five extrema E _i (i=1,...,5) such that:

$$BB = \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} < E_{2} } \hfill} \\ {{E_{3} < E_{1} ,E_{4} > E_{2} ,E_{5} < E_{3} } \hfill} \\ {{{\left( {P_{{t_{{max}} }} - P_{{t_{{td}} }} } \right)} \geqslant \frac{2} {3} \times h} \hfill} \\\end{array} } \right.$$

C.10. Triangle Top (TRIT)

TRIT is characterized by a sequence of four extrema E _i (i=1,...,4) such that:

$$TRIT \equiv \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} > E_{2} } \hfill} \\ {\begin{aligned} & p{\left( {E_{1} ,E_{3} } \right)} \leqslant tg( - 30) \\& 0.9 \leqslant \frac{{{\left| {p{\left( {E_{1} ,E_{3} } \right)}} \right|}}} {{p{\left( {E_{2} ,E_{4} } \right)}}} \leqslant 1.1 \\\end{aligned} \hfill} \\ {{t_{f} \leqslant t_{{E_{1} }} + 0.75 \times {\left( {t_{{int}} - t_{{E_{1} }} } \right)}} \hfill} \\ {{{\left( {P_{{t_{{E_{1} }} }} - P_{{t_{{min}} }} } \right)} \geqslant \frac{2} {3} \times h} \hfill} \\\end{array} } \right.$$

where t _int is the moment of support and resistance lines intersection: $t_{{int}} = \min _{t} {\left( {V_{t} {\left( {E_{1} ,E_{3} } \right)} \leqslant V_{t} (E_{2} ,E_{4} ),t > t_{{E_{4} }} } \right)}\,.$.

C.11. Triangle bottom(TRIB)

TRIB is characterized by a sequence of four extrema E _i (i=1,...,4) such that:

$$TRIB \equiv \left\{ {\;\;\begin{array}{*{20}l} {{E_{1} < E_{2} } \hfill} \\ {\begin{aligned} & p{\left( {E_{2} ,E_{4} } \right)} \leqslant tg( - 30) \\& 0.9 \leqslant \frac{{{\left| {p{\left( {E_{2} ,E_{4} } \right)}} \right|}}} {{p{\left( {E_{1} ,E_{3} } \right)}}} \leqslant 1.1 \\\end{aligned} \hfill} \\ {{t_{f} \leqslant t_{{E_{1} }} + 0.75 \times {\left( {t_{{int}} - t_{{E_{1} }} } \right)}} \hfill} \\ {{{\left( {P_{{t_{{max}} }} - P_{{t_{{E_{1} }} }} } \right)} \geqslant \frac{2} {3} \times h} \hfill} \\\end{array} } \right.$$

where t _int is the moment of support and resistance lines intersection:

$$t_{{int}} = \min _{t} {\left( {V_{t} {\left( {E_{2} ,E_{4} } \right)} \leqslant V_{t} (E_{1} ,E_{3} ),t > t_{{E_{4} }} } \right)}\,.$$

References

Allen H, Taylor MP (1992) The use of technical analysis in the foreign exchange market. J Int Money Financ 11:304–314CrossRefGoogle Scholar
Andrada-Felix J, Fernandez-Rodriguez F, Sosvilla-Rivero S (1995) Technical analysis in the Madrid stock exchange, Foundation for Applied Economic Research (FEDEA) Working Paper, No. 99–05Google Scholar
Béchu T, Bertrand E (1999) L'analyse technique: pratiques et méthodes, Economica, gestionGoogle Scholar
Blume L, Easley D, O'Hara M (1994) Market statistics and technical analysis: the role of volume. J Finance 49(1):153–181CrossRefGoogle Scholar
Brock W, Lakonishok J, LeBaron B (1992) Simple technical trading rules and the stochastic properties of stock returns. J Finance 47:1731–1764CrossRefGoogle Scholar
Chang PHK, Osler CL (1999) Methodical madness: technical analysis and the irrationality of exchange-rate forecasts. Econ J 109:636–661CrossRefGoogle Scholar
Cheung Y-W, Wong CY-P (1999) Foreign exchange traders in Hong Kong, Tokyo and Singapore: a survey study. Adv Pac Basin Financ Mark 5:111–134Google Scholar
Chordia T, Roll R, Subrahmanyam A (2002) Evidence on the speed of Convergence to Market Efficiency, UCLA Working paperGoogle Scholar
Dacorogna MM, Müller UA, Nagler RJ, Olsen RB, Pictet OV (1993) A geographical model for the daily and weekly seasonal volatility in the foreign exchange market. J Int Money Financ 12:413–438CrossRefGoogle Scholar
Dacorogna M, Gençay R, Müller UA, Pictet OV (2001a) Effective return, risk aversion and drawdowns. Physica A 289:229–248CrossRefGoogle Scholar
Dacorogna M, Gençay R, Müller UA, Olsen RB, Pictet OV (2001b) An introduction to high frequency finance, AcademicGoogle Scholar
Danielsson J, Payne R (2002) Real trading patterns and prices in the spot foreign exchange markets. J Int Money Financ 21:203–222CrossRefGoogle Scholar
Dempster MAH, Jones CM (1998a) Can technical pattern trading be profitably automated? 1. Channel, Centre for Financial Research, Judge Institute of Management Studies, University of Cambridge, working paperGoogle Scholar
Dempster MAH., Jones CM (1998b) Can technical pattern trading be profitably automated? 2. The head & shoulders, Centre for Financial Research, Judge Institute of Management Studies, University of Cambridge, working paperGoogle Scholar
Detry PJ (2001) Other evidences of the predictive power of technical analysis: the moving averages rules on European indexes, EFMA 2001 Lugano MeetingsGoogle Scholar
Dooley M, Shafer J (1984) Analysis of short-run exchange rate behavior: March 1973 to 1981. Floating exchange rates and the state of world trade and payments, 43–70Google Scholar
Fiess N, MacDonald R (2002) Towards the fundamentals of technical analysis: analysing the information content of high, low and close prices. Econ Model 19:353–374CrossRefGoogle Scholar
Gençay R (1998) The predictability of security returns with simple technical trading rules. J Empir Finance 5:347–359CrossRefGoogle Scholar
Gençay R (1999) Linear, non-linear and essential foreign exchange rate prediction with simple technical trading rules. J Int Econ 47:91–107CrossRefGoogle Scholar
Gençay R, Stengos T (1997) Technical trading rules and the size of the risk premium in security returns. Stud Nonlinear Dyn Econ 2:23–34CrossRefGoogle Scholar
Gençay R, Ballocchi G, Dacorogna M, Olsen R, Pictet OV (2002) Real-time trading models and the statistical properties of foreign exchange rates. Int Econ Rev 43:463–491CrossRefGoogle Scholar
Gençay R, Dacorogna M, Olsen R, Pictet OV (2003) Foreign exchange trading models and market behavior. J Econ Dyn Control 27:909–935CrossRefGoogle Scholar
Jensen MC (1970) Random walks and technical theories: some additional evidence. J Finance 25(2):469–482CrossRefGoogle Scholar
LeBaron B (1999) Technical trading rule profitability and foreign exchange intervention. J Int Econ 49:125–143CrossRefGoogle Scholar
Levich RM, Thomas LR (1993) The significance of the trading-rule profits in the foreign exchange market: a bootstrap approach. J Int Money Financ 12:1705–1765CrossRefGoogle Scholar
Levy RA (1971) The predictive significance of five-point patterns. J Bus 41:316–323CrossRefGoogle Scholar
Lo AW, Mamaysky H, Wang J (2000) Foundations of technical analysis: computational algorithms, statistical inference, and empirical implementation. J Finance 56:1705–1765CrossRefGoogle Scholar
Lui YH, Mole D (1998) The Use of Fundamental and Technical Analyses by Foreign Exchange Dealers: Hong Kong Evidence. J Int Money Financ 17:535–545CrossRefGoogle Scholar
Lyons RK (2001) The microstructure approach to exchange rates. MITGoogle Scholar
Menkhoff L (1998) The noise trading approach – questionnaire evidence from foreign exchange. J Int Money Financ 17:547–564CrossRefGoogle Scholar
Murphy JJ (1999) Technical analysis of the financial markets, New York Institute of FinanceGoogle Scholar
Neely CJ (1999) Technical analysis in the foreign exchange market: a layman's guide, Federal Reserve Bank of Saint Louis Working PaperGoogle Scholar
Neely CJ, Weller PA (2003) Intraday technical trading in the foreign exchange market. J Int Money Financ 22(2):223–237CrossRefGoogle Scholar
Neely CJ, Weller PA, Dittmar R (1997) Is technical analysis in the foreign exchange market profitable? A genetic programming approach. J Financ Quant Anal 32(4):405–426CrossRefGoogle Scholar
Osler CL (1998) Identifying noise traders: the head-and-shoulders pattern in U.S. equities, Federal Reserve Bank of New York Working PaperGoogle Scholar
Osler C (2000) Support for resistance: technical analysis and intraday exchange rates, Federal Reserve Bank of New York Working PaperGoogle Scholar
Prost AJ, Prechter R (1985) Eliott waves principle, New classics libraryGoogle Scholar
Ready MJ (1997) Profits from technical trading rules, Working Paper, University of Wisconsin-MadisonGoogle Scholar
Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and hallGoogle Scholar
Sweeney RJ (1986) Beating the foreign exchange market. J Finance 41:304–314Google Scholar

Copyright information

Authors and Affiliations

Walid Ben Omrane
- 1
Email author
Hervé Van Oppens
- 1

1.IAG Business SchoolUniversité Catholique de LouvainLouvain-la-NeuveBelgium

The performance analysis of chart patterns: Monte Carlo simulation and evidence from the euro/dollar foreign exchange market

Abstract

Keywords

JEL Classification

Notes

Acknowledgements

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article