Using algorithmic trading to analyze short term profitability of Bitcoin

Reservation price algorithms

Reservation price algorithm calculates a threshold price and generates a buy signal when the offered exchange rate is less than or equal to the threshold. A sell signal is generated when the price is at least threshold (Iqbal, Ahmad & Schmidt, 2012; Iqbal, Ahmad & Shah, 2019; Kao & Tate, 1999). A number of reservation price algorithms are presented in the literature (Mohr, Ahmad & Schmidt, 2014; Kao & Tate, 1999; Iqbal, Ahmad & Shah, 2019; El-Yaniv et al., 2001). In the following we present the selected set of reservation price algorithms considered for our study.

El-Yaniv et al. (2001) assumed a priori information about the lower (minimum possible price m) and upper (maximum possible price M) bound of prices, and presented a reservation price algorithm. Let e_t be the current exchange price. Algorithm 1 provides formal description for El-Yaniv reservation price algorithm for generating buy and sell signals respectively.

Iqbal, Ahmad & Shah (2019) presented a modified version of the reservation price policy of El-Yaniv et al. (2001). The authors critiqued the assumption of fixed values of m and M and argued that inter-day price fluctuation is not arbitrary but is instead governed by inter-day price fluctuation function as shown in Eq (1). (1)${\displaystyle \left(1-\gamma \right)e_{t-1}\le e_t\le \left(1+\gamma \right)e_{t-1}}$ Note that e_t is the exchange rate offered on day t, and γ ∈ 0, 1. The formal description of algorithm is given in Algorithm 2. For detailed working of the algorithm, the reader is referred to Iqbal, Ahmad & Shah (2019).

Kao & Tate (1999) presented a reservation price algorithm based on the perceived rank of the offered exchange rate. The perceived exchange rank is calculated based on the current rank x_t of the offered exchange rate e_t in all the exchange prices observed so far. The formal algorithm is presented in Algorithm 3.

T represents the number of days in a trading period, ${\mathcal{L}}_T\left(t\right)$ and ${\mathcal{H}}_T\left(t\right)$ are the thresholds for buy and sell signals respectively and are computed as shown in Eqs. (2) and (4) respectively; (2)${\displaystyle {\mathcal{L}}_T\left(t\right)=\left\{\begin{array}{cc}0\hfill & \hfill :t=T\\ ⌊\frac{t+1}{T+1}\left(R_T\left(t+1\right)-P_T\left(T+1\right)\right)⌋\hfill & \hfill :t<T\end{array}\right.}$ Note that P_T(t) is the expected difference between the buy and sell prices if the optimal strategy is followed at t, and is calculated as shown in Eq. (3). (3)${\displaystyle P_T\left(t\right)=\left\{\begin{array}{cc}0\hfill & \hfill :t=T\\ P_T\left(t+1\right)+\frac{{\mathcal{L}}_T\left(t\right)}{t}\left(R_T\left(t+1\right)-P_T\left(t+1\right)-\frac{T+1}{t+1}\frac{{\mathcal{L}}_T\left(t\right)+1}{2}\right)\hfill & \hfill :t<T\end{array}\right.}$ (4)${\displaystyle {\mathcal{H}}_T\left(t\right)=⌈\frac{t+1}{T+1}{R}_T\left(t+1\right)⌉}$ Note that R_T is the expected final rank of e_t for selling, if an optimal strategy is followed starting from time t, and is calculated as given in Eq. (5). (5)${\displaystyle R_T\left(t\right)=\frac{{\mathcal{H}}_T\left(t\right)-1}{t}\left(R_t\left(t+1\right)-\frac{T+1}{2\left(t+1\right)}{\mathcal{H}}_T\left(t\right)\right)+\frac{T+1}{2}.}$

Moving average based rules

Moving average (MA) rule is the simplest and popular technical analysis trading rule. The basic idea of MA based rules is to generate buy and sell signals based on the short vs long-term moving averages. More specifically, a buy signal is generated when the short term moving average cuts the long term moving average from below. On the contrary, a sell signal is generated when the short-term moving average cuts the long-term moving average from above. However, in a market the crossing between short-term and long-term moving averages can occur on multiple instances in a short period, resulting in a large number of buy and sell signals (Zhu et al., 2015). The resulting large number of signals are hardly profitable and can force a large transaction fee as well. To avoid this, a minimum threshold called band is introduced. The band introduces a specific percentage difference between the short and long term moving averages in order to generate buy and sell signals.

In the literature two variants of the moving averages, called Variable Length Moving Average (VLMA) and Fixed Length Moving Average (FLMA) are used (Brock, Lakonishok & LeBaron, 1992; Gunasekarage & Power, 2001; Zhu et al., 2015). Let ${\mathcal{A}}_{\mathcal{S}}$ be the short term moving average, ${\mathcal{A}}_{\mathcal{L}}$ be the long term moving average, and $\mathcal{B}$ the band value. Algorithm 4 describes variable length moving average strategy for buy and sell signals.

In VLMA a buy signal is generated when the short term moving average cuts the long term moving average (taking into account the band value) from below, i.e., ${\mathcal{A}}_{\mathcal{S}}>\left(1+\mathcal{B}\right){\mathcal{A}}_{\mathcal{L}}$. Likewise, VLMA generates a sell signal when ${\mathcal{A}}_{\mathcal{S}}<\left(1-\mathcal{B}\right){\mathcal{A}}_{\mathcal{L}}$. A rule is represented by the combination of three values $\mathcal{S}$ (length of short-term moving average), $\mathcal{L}$ (length of long-term moving average), $\mathcal{B}$ (band). For instance, VLMA(5, 30, 0.02) represents a rule where short term average is taken over a period of 5 days, long term over a period of 30 days, and the band value is 2%. FLMA works on the same principle as stated in Algorithm 4. However, FLMA differs from the VLMA by introducing a holding period, i.e., once a signal is generated then the position must be held for a fixed number of days. Any signal generated during the holding period is ignored.

Buy and hold strategy

Buy and hold (BH) is widely used in the literature as a benchmark strategy (Mohr, Ahmad & Schmidt, 2014; Chang, Jong & Wang, 2017; Baur et al., 2018), and is therefore used in our study as well. In BH an investor executes the buy transaction on the first day of the investment period and holds the position until the last day T. On the last day, a sell transaction is executed to complete the trading. The formal description of buy and sell signals of BH algorithm is given in Algorithm 5.

We test the profitability of the algorithms for various parameters and for various durations. We consider short term moving averages over 5 and 10 days, long term moving averages over 15, and 30 days, and band values 0.01 and 0.02 (Brock, Lakonishok & LeBaron, 1992; Fang, Jacobsen & Qin, 2014). Thus we produce a total of 8 trading rules, 4 each for VLMA and FLMA. Likewise, we consider 15 and 30 days durations for Algorithms 1, 2, 3 and 5. Thus we have a total of 16 variants of algorithms to evaluate. Table 2 presents a summary of the selected algorithms and their variants.