be rejected in favor of the alternative hypothesis at one percent level of significance. Moreover, we fit ARMA(1,1) to ˆrt using the FitARMA package in the R programming
language written by McLeod and we derive: ˆ
rt= 0.9353 ˆrt−1 + εt − 0.1103εt−1.
6.2.1.1 Illustration of Simple Trading Strategy:
Although the optimal ∆ can be derived using the expected first passage time for the estimatedARM A(p, q) model for the spread series, to illustrate how the algorithm works, we choose ∆ to be the volatility of the spread process, ˆrt. Therefore, ∆ = σr = 0.816.
Figure 6.1 demonstrates the time plot of the spread series ˆrt. In figure 6.1, strategy
(a), each triangle represents the time when the existing paired trade is closed with profit of at least 2 ∆ = $1.632 ( assuming the transition cost is zero) and enter into a new paired trade in the opposite direction of closed position. In figure 6.1, strategy (b), each triangle also represents the time that we decide to open a new paired trade in appropriate direction and square represents the time that we decide to close the only existing position with a profit of at least ∆ = $0.816 (assuming the transition cost is zero). In figure 6.1, profit/loss (c), we also depict how the profit or loss in both strategies, (a) and (b) evolves through the specified time frame. We summarize the trades with respect to two different strategies (a) and (b) in our sample data in table 6.1. As we can see, for this data set, strategy (a) has better performance than strategy (b) in two aspects: first it has higher profit and second it has lower number of trades meaning that it has lower transaction costs. We would like to come up with a rigorous method to select optimal ∆ and optimal closing time for the trades.
Table 6.1: Summary of trades in two different strategies (a) & (b) shown in figure6.1 in the spread process between WTI crude oil and Brent oil daily front contracts prices from April 1994 to January 2005 in cointegration approach.
Strategy (a) Strategy (b)
Number of Profit per Expected Total Number of Profit per Expected Total
trades trade total profit profit trades trade total profit profit
31 $1.632 $48.96 $67.60 48 $0.816 $38.35 $60.22
6.3
Trading Strategy for One-factor Spread Process
This trading strategy assumes that the spread process follows the Vasicek process, pro- posed by Elliottet al.(2005), and explained in detail in section 4.3 of this thesis. Notice
6.3. Trading Strategy for One-factor Spread Process 137 that first, we assume that the linear combination of underlying assets or commodities to form the spread process is already estimated and known. We also estimated the pa- rameters of the model using the sample data (the algorithm was explained in chapter
4). As we explained before, the spot spread process, Xt is assumed to follow the Vasicek
stochastic process as follows:
dXt=a(b−Xt)dt+σdWt (6.1)
where the model assumes that the linear combination Xt evolves as an Ornstein-
Uhlenbeck process with constant coefficients, a > 0 is mean-reversion rate, b is the long-run mean of the spread, σ is the volatility of the process, anddWt is the increment
of a standard Brownian motion.
Let −∆ and ∆ (∆ > 0) be the lower and upper deviations from the equilibrium level b where we decide to open a new position, unwind the existing position, or both. Therefore, the lower barrier is`1 =b−∆ and the upper barrier is`1 =b+∆. We attempt
to determine ∆ so as to maximize the profits of the trades. The profitability of the trades can be measured by the mean-reversion rate a which here is related to the zero-crossing rate and the location of ∆. When the zero-crossing rate is high, the process quickly reverts back to the equilibrium level b meaning the holding time of a paired position is significantly short and frequency of trades is high. Now, we need to define notations that are required here. For the stochastic process Xt, the first passage time τx is defined as
τx = inf{t ≥ 0 : Xt = x | X0 = x0} where if Xt never hits x, τx is assigned a value of
infinity (τx =∞). We also define the first exit time from interval [`1, `2] as follows:
τ = inf{t≥0 : Xt =`1or`2 | X0 =`0 and `0[`1, `2]}
Letτ∆ be the expected first passage time forXtstarting at `1 and ending at`2. SinceXt
is symmetric, the expected passage time for starting at `2 and ending at `1 will be the
same as τ∆. The rateR at which the trades deliver is R= 2∆τ
∆. We select ∆ to maximize
this rate. Assume the optimal ∆ is estimated. Then a simple trading strategy can be as follows:
• When the spread process has diverged ∆ from the long-run mean, b, we place an appropriate position in the two underlying assets or commodities based on the linear combination coefficient,β,
• Wait for the mispricing to be corrected. Depending on the spread series, we can either unwind the position at timeT (T > t), when the spread process reverts back to the long-run meanb with profit ∆−η, ηis the transaction costs of carrying out of the paired trade, or unwind the position at time T (T > t), when the spread process hit the barrier in the opposite direction of previous hit with profit of 2∆−η and enter into a new position in the opposite direction.
6.3. Trading Strategy for One-factor Spread Process 138 Note that we assume that we only hold at most one paired trade at any given time in both strategies, and we do not open new position unless we close the previous trade.
6.3.1
Empirical Demonstration
To illustrate one-factor strategy, we consider the location spread process between WTI crude oil,PC
t and Brent oil,PtBfor daily price of five contracts from April 1994 to January
2005 where the involved future contracts are the 1,3,6,9 and 12 months to maturities. In chapter 4, we explained the model parameters’s estimation methodology. We fit the model for location spreads between WTI and Brent oils. Figure 6.2 depicts the daily front contract future prices of location spread process between WTI and Brent oils. Table
6.2 lists the estimation results for the one factor model using our sample data.
Table 6.2: Estimated parameters for one-factor model using future contracts of 1,3,6,9 and 12 months to maturities (daily location spread between WTI and Brent oils from April 1994 to January 2005).
a σ b λ ζ
value 1.158 0.843 1.649 0.467 0.281 Sd.Err 0.0174 0.0198 0.0498 0.0510 0.0142
6.3.1.1 Illustration of Simple Trading Strategy:
As we mentioned prior in this section, −∆ and ∆ (∆ > 0) is the lower and upper deviations from the equilibrium level b where we decide to open a new position, unwind the existing position or both. Although the optimal ∆ can be derived using the expected first passage time for the estimated one-factor model for the location spread process (Vasicek), to illustrate how the algorithm works, we consider ∆ to be the estimated volatility of the location spread process, ˆσ. Therefore, ∆ = ˆσ = 0.843. Figure6.2 shows the time plot of the location spread process for the front contract. In figure6.2, strategy (a), each triangle represents the time when the existing paired trade is closed with profit of at least 2 ∆ = $1.686 (assuming the transition cost is zero) and enter into a new paired trade in the opposite direction of closed position. In figure 6.2, which depicts strategy (b), each triangle also represents the time that we decide to open a new paired trade in appropriate direction and square represents the time that we decide to close the only existing position with profit of at least ∆ = $0.843 ( assuming the transition cost is zero). In figure6.2, profit/loss (c), we also demonstrate how the profit or loss in both strategies, (a) and (b) evolves through the specified time frame.
6.3. Trading Strategy for One-factor Spread Process 139 Strategy (a) Time Spread ($/barrel) 0 100 300 500 700 900 1100 1300 1500 1700 1900 2100 2300 2500 2700 −1 1 2 3 4 5 6 Strategy (b) Time Spread ($/barrel) 0 100 300 500 700 900 1100 1300 1500 1700 1900 2100 2300 2500 2700 −1 1 2 3 4 5 6 Profit/Lost Diagram (c) Time $P/L 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 −5 5 15 25 35 45
Strategy (a) Open/Close at +−Sigma Strategy (b) Open at +−Sigma close at mu
Figure 6.2: Time plot of the location spread process between WTI crude oil and Brent oil daily front contracts prices from April 1994 to January 2005 in one-factor method: Plot (a) shows the series of trades in strategy (a) in which whenever we hit the lower barrier, `1 = b− ∆ = 1.649 −0.843 or upper barrier, `2 = b + ∆ = 1.649 + 0.843,
we close existing paired trade and open new position in opposite direction. Plot (b) shows the series of trades in strategy (b) in which whenever we hit the lower barrier, `1 =b−∆ = 1.649−0.843 or upper barrier, `2 =b+ ∆ = 1.649 + 0.843, we only open a
new position in proper direction and we wait until to revert to long-run mean, b= 1.649 when we unwind existing paired trade. Plot (c) depict how profit/loss growth is both strategies.
6.4. Cointegration approach and One-factor Method results Comparison: 140