T3 — Tillson T3

Six chained EMAs recombined with volume-factor coefficients: as smooth as a long average but with much of the lag removed. vfactor 0 collapses it to a plain triple EMA chain; 1 maximises the lag correction (and the overshoot).

Signature

T3(period, vfactor, on=<source>)

Argument	Type	Required	Notes
period	int Param or literal	Yes	EMA length used six times. Default 5, bounds 1–200.
vfactor	float Param or literal	Yes	Lag-correction strength. Default 0.7, bounds 0.0–1.0.
on=	symbol, OHLCV field, or indicator	No	Source. Defaults to the trade symbol’s close.

Formula

With $e_1 \ldots e_6$ the chained EMAs and $b$ the volume factor:

$$ c_1 = -b^3,\quad c_2 = 3b^2 + 3b^3,\quad c_3 = -6b^2 - 3b - 3b^3,\quad c_4 = 1 + 3b + b^3 + 3b^2 $$ $$ T3 = c_1 e_6 + c_2 e_5 + c_3 e_4 + c_4 e_3 $$

Warmup

The chain seeds from the first value and converges over roughly 6n bars — the longest warm-up of any average in the library.

Reading the value

t3.value
t3.crossed_above(other)

Example

class T3Trend(Strategy):
    """A smooth trend line that still turns quickly."""
    period = Param(5, min=1, max=200)
    trade = Symbol()
    sizing = FixedCapital()

    def execute(self):
        t3 = T3(self.period, 0.7, on=self.trade)
        if self.trade.close > t3.value and self.trade.is_flat:
            self.buy(self.trade, reason="Above T3")
        if self.trade.close < t3.value and self.trade.is_long:
            self.sell(self.trade, self.trade.position, reason="Below T3")

Notes

• Tim Tillson, TASC (1998).
• Proven by an assert-via-trade run on SPY daily (converged to an independent six-EMA replication within 0.05 after 40 bars; zero alarm trades).
• Six chained EMAs seeded from the first value: expect early bars to differ from long-history implementations by a decaying amount.
• A different recipe from pandas_ta’s t3(10, 0.7), not a different scale: measured on a 300-bar tape ours stays about 2.2 % away and never converges to it, because the construction (window offset, seed, weight normalisation) is its own.