Quant Toolbox > 35. Signals > Technical signalsMomentum

Momentum

The momentum signal is defined as

Smomt ≡ ewmaτHL

w (t,∆X·) (35.5)

Exponentially Weighted MovingAverage (EWMA)

ewmaτHLw (t, x·) ≡ 1

γw

∑w−1i=0 e

− ln(2)τHL

ixt−i (3a.185)

series of the incrementsof a risk driver Xt (1.6)

∆X· ≡ {. . . ,∆Xt−1,∆Xt,∆Xt+1, . . .}

Momentum signal in equities trading

When w ≈ ∞, the EWMA approximation (3a.186) implies

Smomt+1 ≈ e

− ln(2)τHL Smom

t + (1− e−ln(2)τHL )∆Xt (35.6)

⇓

Smomt+1 ≈ e

− ln(2)τHL Smom

t + εt+1 (35.7)

Xt follows a random walk (2.8)

AR(1) process →

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Quant Toolbox > 35. Signals > Technical signalsMomentum

Momentum

The momentum signal is defined as

Smomt ≡ ewmaτHL

w (t,∆X·) (35.5)

Exponentially Weighted MovingAverage (EWMA)

ewmaτHLw (t, x·) ≡ 1

γw

∑w−1i=0 e

− ln(2)τHL

ixt−i (3a.185)

series of the incrementsof a risk driver Xt (1.6)

∆X· ≡ {. . . ,∆Xt−1,∆Xt,∆Xt+1, . . .}

Momentum signal in equities trading

When w ≈ ∞, the EWMA approximation (3a.186) implies

Smomt+1 ≈ e

− ln(2)τHL Smom

t + (1− e−ln(2)τHL )∆Xt (35.6)

⇓

Smomt+1 ≈ e

− ln(2)τHL Smom

t + εt+1 (35.7)

Xt follows a random walk (2.8)

AR(1) process →

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Quant Toolbox > 35. Signals > Technical signalsFilters

Filters

The linear time-invariant filter (34.33) can be used to build a signal definedas

SLTIt ≡ [h ∗X]t =

∑τ∈ZhτXt−τ (35.9)

causal input response matrices:hτ = 0 if τ < 0 (34.35)

{Xt ≡ (X1,t, . . . , Xn̄,t)′}t∈R

n̄ observable processes

Filters leverage strong results from spectral theory, see Chapter 34.

Momentum as linear time-invariant filter

The momentum signal Smomt (35.5) is a linear time-invariant filter (35.9)

where the input response reads

hτ =

{1γwe− ln(2)τHL

τ if 0 ≤ τ ≤ w − 1

0 otherwise(35.10)

with γw ≡∑w−1τ=0 e

− ln(2)τHL

τ .

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Quant Toolbox > 35. Signals > Technical signalsCointegration

Cointegration

The z-score (2.132) of a cointegrated process Yt can be used as cointegrationsignal

Scointt ≡ Yt − µ

σ(35.11)

unconditional meanµ ≡ E{Yt}

unconditional st. deviationσ ≡ Sd{Yt}

The time required before we can hope to cash in any profit is represented bythe half-life τHL (2.133), see Section 2.7.

• Scointt > 0 ⇒ Yt is expected to decrease;• Scointt < 0 ⇒ Yt is expected to increase.

Cointegration signals are used in adaptive execution algorithms (10.46).

In particular, the cointegration signals in high-frequency are defined incommon activity time ∆Aκ (1.106).

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Quant Toolbox > 35. Signals > Technical signalsCointegration

Cointegration

The z-score (2.132) of a cointegrated process Yt can be used as cointegrationsignal

Scointt ≡ Yt − µ

σ(35.11)

unconditional meanµ ≡ E{Yt}

unconditional st. deviationσ ≡ Sd{Yt}

The time required before we can hope to cash in any profit is represented bythe half-life τHL (2.133), see Section 2.7.

• Scointt > 0 ⇒ Yt is expected to decrease;• Scointt < 0 ⇒ Yt is expected to increase.

Cointegration signals are used in adaptive execution algorithms (10.46).

In particular, the cointegration signals in high-frequency are defined incommon activity time ∆Aκ (1.106).

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

Quant Toolbox > 35. Signals > Technical signalsCointegration

Cointegration signal

• microprice (1.95) series of the AMZN and GOOG stocks;• time binning (1.109) for common activity time ∆Aκ (1.106):

∆a ≡ ∆q =10,000;• the cointegrated series are obtained via eigenvectors (see Table 2.6).

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update