Chart Patterns

Question 1

(a) Upper line: $u(t)=120-0.5t$. Lower line: $l(t)=100+0.3t$. Height $h(t)=u(t)-l(t)=(120-0.5t)-(100+0.3t)=20-0.8t$. (2 marks: correct lines + subtraction) Apex when $h(t^\*)=0 \Rightarrow 20-0.8t^\*=0 \Rightarrow t^\*=25$ days. (2 marks)

(b) Widest point is at $t=0$: height $=20$. (1) Breakout price = upper line at $t=10$: $u(10)=120-0.5(10)=115$. (2) Measured-move target (upward) = breakout price + pattern height = $115+20=135$. (2)

(c) Downside target $T=N-(H-N)$. Expand: $T=N-H+N=2N-H$. (2 marks for algebra) Condition for positivity: $T>0 \Rightarrow 2N-H>0 \Rightarrow H<2N$. So the head must be less than twice the neckline level (equivalently the drop $H-N$ must be smaller than $N$). (2 marks)

(d) (7 marks — indicative content)

• Measured-move magnitude scales with pattern height; if average bar range (volatility) is a large fraction of pattern height, the trendlines are poorly defined and the breakout price / height estimate carries large error → lower reliability. (2)
• Low intra-pattern volatility (contracting range, as in triangles/pennants 3.5.4/3.5.5) with a sharp volume-backed breakout tends to give cleaner, more reliable targets. (2)
• A rectangle/range (3.5.6) measured over noisy wide bars gives ambiguous height; a clean narrow-bodied consolidation gives a crisp height. Reference to continuation vs reversal (3.5.11): continuation patterns forming after a strong trend with declining volatility are statistically more reliable. (2)
• Coherent linking of ≥2 subtopics and a stated rule. (1)

Question 2

(a) (6 marks)

def find_local_extrema(prices, k):
    maxima, minima = [], []
    n = len(prices)
    for i in range(n):
        lo, hi = max(0, i-k), min(n-1, i+k)
        window = prices[lo:hi+1]
        others = prices[lo:i] + prices[i+1:hi+1]
        if others and prices[i] > max(others):
            maxima.append(i)
        elif others and prices[i] < min(others):
            minima.append(i)
    return maxima, minima

Marks: window bounds/boundary handling (2), strict comparison excluding self (2), returns both lists (2).

(b) (7 marks)

def is_double_top(peaks, prices, tol):
    for a, b in zip(peaks, peaks[1:]):
        pa, pb = prices[a], prices[b]
        # relative price agreement
        if abs(pa - pb) / max(pa, pb) <= tol:
            # require a trough (lower low) between them
            trough = min(prices[a+1:b]) if b > a+1 else None
            if trough is not None and trough < min(pa, pb):
                return True
    return False

Tolerance condition: $\dfrac{|P_a-P_b|}{\max(P_a,P_b)}\le \text{tol}$ (2), consecutive peaks (2), intervening trough below both peaks (2), returns bool (1).

(c) (5 marks)

def measured_target(pattern_type, neckline, extreme):
    height = abs(extreme - neckline)
    if pattern_type in ("double_top", "head_shoulders"):
        return neckline - height   # downside reversal
    elif pattern_type in ("double_bottom", "inverse_hs"):
        return neckline + height   # upside reversal
    raise ValueError("unknown pattern")

Sign handling: top/head patterns are bearish reversals → subtract height below neckline; bottom/inverse patterns are bullish reversals → add height above neckline. extreme is the head/peak (tops) or trough (bottoms). Marks: correct formula (2), correct sign per pattern (2), explanation (1).

(d) (4 marks) False positive: two similar peaks in a strong uptrend continuation (e.g., a bull flag or minor pullback) can mimic a double top without a genuine reversal. Filter: require declining volume on the second peak and a volume surge on neckline break — reversal patterns typically show volume divergence, while continuation patterns keep trend-direction volume. Marks: valid failure mode (2), volume filter tied to continuation vs reversal (2).

Question 3

(a) (6 marks, 1 each)

• Rising wedge (uptrend): reversal (bearish) — converging up-slanting lines with weakening momentum.
• Falling wedge (downtrend): reversal (bullish) — converging down-slanting lines, momentum fades.
• Bull flag: continuation — brief counter-trend consolidation within an uptrend.
• Inverse head-and-shoulders: reversal (bullish) — three troughs, middle deepest, at a bottom.
• Ascending triangle (uptrend): continuation — flat top, rising lows, breaks up.
• Rounding bottom: reversal (bullish) — gradual U-shaped base.

(b) (6 marks)

• Cup-and-handle: measured move = cup depth projected up from breakout of the cup's rim/resistance. Target $=R+D_{cup}$.
• Inverse H&S: measured move = head depth (neckline $-$ head) projected up from neckline break. Target $=N+D_{head}$.
• The claim holds only if both the projection base (rim $R$ vs neckline $N$) and depth ($D_{cup}=D_{head}$) coincide. Equal depth alone is insufficient because the breakout/projection reference levels differ ($R\neq N$ in general). Condition: $R=N$ and $D_{cup}=D_{head}$ ⇒ identical targets; otherwise the targets differ by $R-N$. (claim as stated is FALSE without the reference-level condition — award full marks for identifying this).

(c) (6 marks)

• Entry rule: enter on close beyond the range boundary (long above resistance, short below support) with a confirmation filter (e.g., close $>$ boundary by $\ge$ tol).
• Target rule: measured move = range height (resistance − support) projected in breakout direction.
• Stop rule: place stop at the opposite boundary (or mid-range).
• Statistic: over $N$ historical rectangle breakouts, compute the fraction hitting target before stop, or mean $R$-multiple; compare to a random-entry benchmark.
• Null hypothesis $H_0$: rectangle breakout targets are hit no more often than a random-timing benchmark (expected hit-rate = baseline); reject if observed hit-rate significantly exceeds it (e.g., binomial/one-sided test, $p<0.05$). Marks: entry (1), target (1), stop (1), statistic (1.5), null hypothesis (1.5).

[
  {"claim":"Triangle height h(t)=20-0.8t is zero at apex t*=25","code":"t=symbols('t'); h=(120-Rational(1,2)*t)-(100+Rational(3,10)*t); ts=solve(h,t)[0]; result = (h==20-Rational(4,5)*t) and (ts==25)"},
  {"claim":"Breakout price 115 and measured target 135","code":"bp=120-Rational(1,2)*10; height=20; target=bp+height; result = (bp==115) and (target==135)"},
  {"claim":"HS downside target simplifies to 2N-H and positive iff H<2N","code":"N,H=symbols('N H'); T=N-(H-N); result = (simplify(T-(2*N-H))==0)"},
  {"claim":"measured_target sign logic: double_top below neckline, inverse_hs above","code":"neck=100; ext_top=130; ext_bot=70; dt=neck-abs(ext_top-neck); ihs=neck+abs(neck-ext_bot); result = (dt==70) and (ihs==130)"}
]