Pair Trading Models

Pair Trading Lab offers pair trading algorithms based on various mathematical models. These are models currently supported in PTL:

Ratio Model

This is one of the standard pair trading models described in literature. It is based in ratio of instrument prices, moving average and standard deviation. In other words, it is based on Bollinger Bands indicator.

Model Support

• in PTL backtester: yes (max Z-score not supported)
• in PTL portfolio backtester: yes
• in PTL Trader: yes

Model Neutrality

We currently support only dollar-neutral version of this model, which means we allocate same amounts of margin to both legs based on current prices at the time of opening the position.

Model Parameters

• entry threshold E_n for Z-score, typical value range is <1.5, 2.5>, 2.0 is used most often
• exit threshold E_x for Z-score, typical value is <-0.5, 0.5>, 0 is used most often...we allow positive values only for now
• max Z-score E_max (optional, to filter out extremes, typical value is >4 if used)
• moving average period P_m (typical range <10, 100>), default = 15
• moving average type T (algorithm), default = exponential
• standard deviation period P_s (typical range <10, 100>), default = 15
• entry mode (simple, uptick, downtick)

Description

• we trade pair of stocks A, B, having price series A(t), B(t)
• we need to calculate ratio time series R(t) = A(t) / B(t)
• let's apply moving average of type T with period P_m on R(t) to get time series M(t)
• let's apply standard deviation with period P_s on R(t) to get time series S(t)
• now we can create Z-score series Z(t) as Z(t) = (R(t) - M(t)) / S(t), this time series can give us z-score to signal trading decision directly
• another common approach (to visualize) is to create bands and put it above the moving average M(t):
  • upper entry band U_n(t) = M(t) + S(t) * E_n
  • lower entry band L_n(t) = M(t) - S(t) * E_n
  • upper exit band U_x(t) = M(t) + S(t) * E_x
  • lower exit band L_x(t) = M(t) - S(t) * E_x
  • these bands are actually the same bands as in Bollinger Bands indicator and we can use crossing of R(t) and bands as trade signals

Entering Position

There are certain possible approaches how to interpret model statistics in order to make trading decisions. For entering position, we used to call them entry modes. This is the list of them and description how they work:

• entry mode = simple:
  • to open short pair position, it is simple enough if the Z-score Z(t) >= E_n (equivalent to R(t) >= U_n(t))
  • to open long pair position, it is simple enough if the Z-score Z(t) <= -E_n (equivalent to R(t) <= L_n(t))
• entry mode = uptick: same as simple, but in addition, previous Z-score must be below the entry band (so we cross the band from inside to outside):
  • to open short pair position, we require Z(t) >= E_n (equivalent to R(t) >= U_n(t)) and Z(t-1) < E_n (same as R(t-1) < U_n(t-1))
  • to open long pair position, we require Z(t) <= -E_n (equivalent to R(t) <= L_n(t)) and Z(t-1) > -E_n (same as R(t-1) > L_n(t-1))
• entry mode = downtick: we wait for the Z-score crossing back the band from outside to inside:
  • to open short pair position, we require Z(t) < E_n and Z(t-1) >= E_n and Z(t) > E_x
  • when using bands, it is the same as having R(t) < U_n(t) and R(t-1) >= U_n(t-1) and R(t) > U_x(t)
  • to open long pair position, we require Z(t) > -E_n and Z(t-1) <= -E_n and Z(t) < -E_x
  • when using bands, it is the same as having R(t) > L_n(t) and R(t-1) <= L_n(t-1) and R(t) < L_x(t)

Why do we have the simple entry mode? In normal situations and backtests, it gives same results as the uptick mode. But the difference comes up while trading multiple pairs in portfolio. The simple mode allows you to jump in the position immediately after a new slot is freed, regardless of the previous Z-scores.

Which entry mode is better? Hard to tell, sometimes the uptick, sometimes the downtick. You have to do your homework and decide, which idea suits your trading style better. In general, uptick/simple mode is more aggressive, as it does not wait for first signs of spread mean reversion.

Exiting Position

For exiting position, we always use only these simple rules:

• we exit short position when Z(t) <= E_x (equivalent to R(t) <= U_x(t))
• we exit long position when Z(t) >= -E_x (equivalent to R(t) >= L_x(t))

Moving Average Types

You can choose from these moving average algorithms:

• Simple Moving Average (SMA)
• Exponential Moving Average (EMA)
• Weighted Moving Average (WMA)
• Double Exponential Moving Average (DEMA)
• Triple Exponential Moving Average (TEMA)
• Triangular Moving Average (TMA)
• Kaufman Adaptive Moving Average (KAMA)
• MESA Adaptive Moving Average (MAMA)
• Triple Exponential T3 Moving Average

Which one is the best? It depends, you have to test for yourself. They mostly differ in the term of memory and how fast they react to changes. Industry standard and the default is EMA. We suggest to try all of them on some sample pair to see how they work.