Parabolic SAR Strategy

Description

Parabolic SAR (Stop and Reversal) strategy is a trading algorithm which role is to predict market trend change and buy / sell an asset in a specific market conditions. It works well in a bull or bear markets. If a market is neither bear nor bull then SAR coefficient does not have to be helpful and SAR coefficient might be ‘belated’.

Values of all the variables defined in the following paragraphs, except $l_t \text{ and } h_t$ , are not defined for $t = 1$ .

Lets define by $h_t$ ( $l_t$ ) the market high (low) at the end of a day $t$ . Moreover, $O_t$ is a strategy’s opinion about market trend at the day $t$ . It has value $+1$ if the opinion is uptrend, and $-1$ otherwise. It will be formally defined later. To define it precisely (by a math expression) we need first define Extreme Point ( $EP$ ), Acceletarion Factor ( $AF$ ) and Stop and Reverse ( $SAR$ ) variables.

A variable $EP_t$ , if $o_t = +1$ , is the highest value of a high from the beginning of the uptrend. Similarly, if market $o_t = -1$ , it is the lowest value of low from the beginning of the downtrend. Formally we can defined it as follows:

$EP_t = \begin{cases} max(h_1, h_2) & t = 2 \wedge O_2 = +1 \\ min(l_1, l_2) & t = 2 \wedge O_2 = -1 \\ max_{c < i \leq t} h_i & O_t = +1 \wedge t \geq 3 \\ min_{c < i \leq t} l_i & O_t = -1 \wedge t \geq 3 \end{cases}$

where $c < t$ is the biggest value for which $O_{c-1} \neq O_{c}$ . If there is no such $c$ (there was no trend change from the time when the strategy starts) then we should set $c$ to 1. To sum up: the value of $EP_t$ is the highest high or the lowest low of the current trend.

A variable $AF$ is a kind of a counter which is set to Acceleration Coefficient when the strategy’s opinion about market trend has changed, it is usually equal to 0.02. Moreover it is increased by Acceleration Coefficient when the new extreme value of EP is found (either high or low depending on the strategy’s opinion about market trend). It is worth to mention that its value cannot exceed Acceleration Limit, by default 0.2. Lets define it formally:

$AF_t = \begin{cases} Acceleration~coefficient & O_t \neq O_{t-1} \vee t = 2 \\ min(Acceleration~limit, AF_{t-1} + Acceleration~coefficient) & EP_{t-1} \neq EP_{t} \\ min(Acceleration~limit, AF_{t-1}) & otherwise \end{cases}$

The $SAR$ coefficient is defined as a function of its previous value, AF and EP. Formally we can defined it as follows:

$SAR_t = \begin{cases} h_t & O_{t} = -1 \wedge t = 2 \\ l_t & O_{t} = +1 \wedge t = 2 \\ SAR_{t - 1} + AF_{t-1} \times (EP_{t-1} - SAR_{t-1}) & O_{t} = O_{t-1} \wedge t \geq 3 \\ min(l_t, EP_{t-1}) & O_{t} \neq O_{t-1} \wedge O_{t} = +1 \wedge t \geq 3 \\ max(h_t, EP_{t-1}) & O_{t} \neq O_{t-1} \wedge O_{t} = -1 \wedge t \geq 3 \end{cases}$

Additionally, when SAR indicator is rising it cannot stay above last two intervals’ low price. When the indicator is falling – it cannot stay below maximum of last two intervals’ high price, either. So, if $O_t = 1$ and calculated $SAR$ value is bigger than last two periods’ low price ( $l_{t-1}$ , $l_{t-2}$ ), then $SAR$ value is set to $min(l_{t-1}, l_{t-2})$ . Analogous, if $O_t = -1$ and calculated $SAR$ value is smaller than last two periods’ high price ( $h_{t-1}$ , h_{t-2}), then $SAR$ value is set to $max(h_{t-1}, h_{t-2})$ .

Finally we can give a formal definition of the $O_t$ . Its definition is the following:

$O_t = \begin{cases} \frac{h_2 - h_1}{|h_2 - h_1|} & t = 2 \\ +1 & SAR_t < l_t\wedge t \geq 3 \\ -1 & SAR_t > h_t \wedge t \geq 3 \\ -O_{t-1} & otherwise \end{cases}” /></p> </div> <p>As you can see the decision about a actual trend is normally made on the basis of the comparision between the SAR coefficient and the local high (low). At the beginning (<img decoding=$ ) we have to determine whether we have an uptrend or downtrend. In above formula fleeting high prices have been used, but this automatic method can be replaced by a parameter Initial Market Trend basing on the user’s opinion.

Market Data

Last trade
Historical market data

Parameters

PARAMETER NAME	DESCRIPTION	ESSENTIAL
Acceleration Coefficient	Coefficient of acceleration factor, usually set to 0.02 by default	Yes
Acceleration Limit	Upper limit of acceleration factor, usually set to 0.2 by default	Yes
Initial Market Trend	Initial market trend set arbitrary by an user	No

Conditions

Open position

Side Buy or Sell
Amount One asset per order
Price Last market price
Type Price limit

Consider day $t \geq 3$ . If strategy does not have an open position and:

$O_{t} = +1 \wedge O_{t-1} = -1 \rightarrow$ algorithm opens long position
$O_{t} = -1 \wedge O_{t-1} = +1 \rightarrow$ algorithm opens short position

Close position

Consider day $t \geq 3$ . If strategy does have an open position and:

$O_{t} = +1 \wedge O_{t-1} = -1 \rightarrow$ algorithm closes short position
$O_{t} = -1 \wedge O_{t-1} = +1 \rightarrow$ algorithm closes long position

Termination

There is no strategy termination condition.

Time frame

Strategy is designed for long-term trading, usually for whole weeks.

Further information & source

A complete guide to Algorithmic Trading

Parabolic SAR Strategy

Description

Market Data

Parameters

Conditions

Open position

Close position

Termination

Time frame

Further information & source

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