Pair_Trading_with_Box_Tiao_Decomposition

Apply Box&Tiao to generate stationary price spread series in steel industry commodity futures market for pair trading

Abstract

In 1977, Box and Tiao provided a method of Canonical Decomposition, which can be used in assessing the predictability of Multidimensional Time Series. With Box-Tiao decomposition, we can get a vector of weights allows the linear combination of the input time series to have the worst forecasting capability. In this case, the output combination is close to a white noise series where we can apply pair-trading strategy due to its mean-reverting property.

In this project, we are going to study the cointegration phenomena among 6 steel and coal assets in Chinese commodity futures market. Box-Tiao decomposition would be used to generate the stationary price spread series among those cointegrated assets for further pair-trading strategy developing.

1. Box & Tiao Canonical Decomposition

In the paper of Box and Tiao, the portfolio $\Pi_t $ is assumed to be satisfied with an VAR(1) model: $\Pi_t=\Pi_{t-1}A+Z $, where $\Pi_{t-1}A $ is an one-step forward prediction for $\Pi_t $ at time $t-1 $. So, we have their volatility had the following relationship: $$Var(\Pi_t)=\sigma_t^2\quad|\quad Var(\Pi_{t-1}A)=\sigma_{t-1}^2\quad|\quad Var(Z_t)=\Sigma$$ $$\sigma_t^2=\sigma_{t-1}^2+\Sigma$$ $$1=\frac{\sigma_{t-1}^2}{\sigma_{t}^2}+\frac{\Sigma}{\sigma_{t}^2}=v+\frac{\Sigma}{\sigma_{t}^2}$$ $v=\frac{\sigma_{t-1}^2}{\sigma_{t}^2}$ is used to describe the predictability of the Var(1) model. So, while minimizing $v $, the portfolio $\Pi_t $ would be close to a white noise series, which is the stationary price spread we want.

In this project, we have the price of commodity futures as $S_t $, the weight of asset $x $, and the linear combination with the worst forecasting capability as $\Pi_t $ with the minimum $v=\frac{\sigma_{t-1}^2}{\sigma_{t}^2} $. So, we have the following formula: $$\Pi_t=S_tx$$ $$S_tx=S_{t-1}Ax+Z_tx$$ $$v=\frac{x^TA^T\Gamma Ax}{x^T\Gamma x}$$ $$\Gamma=\frac{1}{m}S_t^TS_t\quad(S_t=Demean(S_t))$$ The process of minimizing $v $ is equal to have the solution for eigenvalue $\lambda $ satisfied with the following equation: $$det(\lambda\Gamma-A^T\Gamma A)=0$$ So, we have $x=\Gamma^{-\frac{1}{2}}z $, where $z $ is the eigenvector with the minimum eigenvalue of the following matrix: $$\Gamma^{-\frac{1}{2}}A^T\Gamma A\Gamma^{-\frac{1}{2}}$$ Since $A $ is the slope parameter of the Linear Regression for $S_t $ to $S_{t-1} $ and $\Gamma $ is the covariance matric of $S_t $, we have: $$\Gamma^{-\frac{1}{2}}A^T\Gamma A\Gamma^{-\frac{1}{2}}$$ $$=(S_t^TS_t)^{-\frac{1}{2}}A^T(S_{t-1}^TS_{t-1})A(S_t^TS_t)^{-\frac{1}{2}}$$ $$=(S_t^TS_t)^{-\frac{1}{2}}(S_{t-1}A)^T(S_{t-1}A)(S_t^TS_t)^{-\frac{1}{2}}$$ $$=(S_t^TS_t)^{-\frac{1}{2}}(\hat{S_{t}^T}\hat{S_{t}})(S_t^TS_t)^{-\frac{1}{2}}$$

2. Pick up Trading Pairs

Box-Tiao decomposition helps us generate the stationary price spread series for a given set of commodity futures, so the question remains is to identify a set of commodity futures possibly have cointegration phenomena among them. It is obvious that those commodity categories who are in the same supply and demand chain would probably have a close trend, so we pick up 6 categories from the steel and coal industries: rebar(rb), iron ore(i), hot rolled coil(hc), manganese silicon(SM), coking coal(jm), coke(j).

After applying Box-Tiao decomposition on every 3 pairs out of these 6 categories ($C_6^3=20$ pairs in total), we filter those pairs didn't pass the ADF stationary test or have a half-life longer than 21 trading days. Finally, we have the following 5 pairs suitable for pair trading:

No.	Code	Half-Life
1	rb+i+SM	20
2	rb+j+jm	19
3	i+SM+jm	17
4	i+j+jm	20
5	hc+j+jm	21

For the first pair, all these 3 categories are from steel industry, so we pick up this pair for the further pair trading strategy developing.

After applying Box-Tiao decomposition on the first pair, we have the weight vector for these 3 categories as follow:

Code	Weight	Contract Multiplier	Volume
rb	1	10	100
i	-0.75038229	100	8
SM	-0.12836106	5	26

3. Mean-Reverting Breakthrough Signal

After we generated the stationary price spread series, with its mean-reverting property, we can open a position when it is moving away from the mean value and cover the position at its returning. A simple method to catch these arbitrage chances is to build up a channel which is $n\sigma$ away from the mean value $\mu\pm n\times\sigma$, and then trade every time when the series breakthroughs the channel.
For a white Gaussian noise, theoretically, the frequency of series moving within the channel should be equal to the following possibility: $$\frac{\sum_i{I_i(\mu-n\sigma\leq P_t\leq \mu+n\sigma)}}{T}=\phi(\mu+n\sigma)-\phi(\mu-n\sigma)$$ We hope to have around half of the trading days in-sample within the channel, so we set $n=0.75$.

n Times	In-Sample Trading Frequency	Theoretical Frequency
0.75	0.5467	0.561

However, with this simple method, we can only make money when the series passes the channel, whose final profit is close to $\frac{N_{pass}}{2}\times 0.75\sigma$. By introducing the Bollinger Band Breakthrough signal, it is possible to open a position earlier than the passing time, which would bring us more profit. For example, when the series is far above the upper band but it sends out a signal of breaking down through the lower Bollinger band as a sign that the series is in its process of reverting, opening a short position earlier can earn more profit. We use the half-life 20 and $0.75\sigma$ to build the Bollinger Band.

4. Back-Test

item	details
Futures Category	rb, i, SM
Contract Type	continuous contract
Trading Frequency	daily
Trade Executing Time	next day
Price for Signal Construction	back-adjusted closing price
Trading Rules	When the series is above the upper band and a signal of down breaking through the lower Bollinger band or upper band, sell 100 contracts of rb, buy 8 contracts of i and 26 contracts of SM. When the series is under the lower band and a signal of up breaking through the upper Bollinger band or lower band, buy 100 contracts of rb, sell 8 contracts of i and 26 contracts of SM. When the series touches the mean value, cover all the positions.
Executing Price Simulation	average of open price and close price next day
Executing Cost	0.03%/RMB for rb and i; 3.00RMB/contract for SM
Price for Profit Calculation	average of open price and close price

Back-Test	in-sample
Period	03/19/2017 - 03/19/2021
Capital	1000000.00RMB
Trading Volume	rb: 100, i: 8, SM: 26
Trading Fee	rb: 0.03%/RMB, i: 0.03%/RMB, SM: 3.00RM/contract

Statistics	Value
Annual Return	7.5612%
Max Drawdown	66616.00RMB
Sharpe Ratio	1.8118
Calmar Ratio	1.1350*10^-6
Win Rate	84.6154%
Profit-Loss Ratio	2.2122
Average Position Holding Period	59.3077 days

Back-Test	out-of-sample
Period	03/19/2021 - 03/18/2022
Capital	1000000.00RMB
Trading Volume	rb: 100, i: 8, SM: 26
Trading Fee	rb: 0.03%/RMB, i: 0.03%/RMB, SM: 3.00RM/contract

Statistics	Value
Annual Return	1.3788%
Max Drawdown	127679.00RMB
Sharpe Ratio	0.1643
Calmar Ratio	1.0799*10^-7
Win Rate	50.00%
Profit-Loss Ratio	1.1247
Average Position Holding Period	66.25 days