Bayesian Adaptive Trading with Daily Cycle Mr Chee Tji Hun Ms Loh Chuan Xiang Mr Tie JianWang Algernon

Abstract The Bayesian Adaptive Trading with Daily Cycle (BATDC) paper presents the idea of an optimal trading schedule to minimise the total expected cost of trading. The key idea in BATDC paper is that there exists a value α which represents the true drift generated by Institutional Traders. Institutional traders, start with a daily target size and by observing price evolution throughout the day, one can adjust the α estimate and uncover the order’s nature (Buy/Sell) and tailor an optimal trading schedule using the adjusted α estimate to finally reduce trading cost. As part of the BATDC paper, they have uncovered a proprietary trading strategy but have ignored it for the purpose of their paper. The purpose of this paper is to test the proprietary trading strategy and implement certain variants to the strategy.

Assumptions Prior belief that α is normally distributed α ~ N( ᾱ, v 2 ) Underlying asset price’s distribution, conditional on α, is also normally distributed S t = S 0 + αt + σB t for t ≥ 0 S t ~ N( ᾱ t, (σ 2 + v 2 t)t) Using Bayesian update the optimal value for α α * |S t ~ N( ( ᾱ σ 2 + v 2 (S t – S 0 ))/(σ 2 + v 2 t), σ 2 v 2 /(σ 2 + v 2 t) )

Methodology and Trading Strategy 1 We seek to optimise 2 parameters 1. The size of the rolling window 2. The size of a trading cycle Prop Strategy 1. Start each m-period with 0 position 2. Use a n-period rolling window (past data) & m-period trading cycle S t – S 0 = αt + σB t where t ε [1, 2, …, m] 3. Estimate α by a regression without intercept; 4. Estimate of σ is the standard error of the regression model; and 5. Estimate of v, the prior’s standard deviation, is the standard deviation of the estimate of α 6. With the information calculate α * at time t 7. If α * > 0 long the asset else short the asset 8. If position at period m-1 not equal to 0 then sell at period m end day price so that position is back to 0 at the end of period m.

Methodology and Trading Strategy 2 After taking a sparse sample from the range of days for close out and rolling window. We notice that the strategies performed better within the 4-44 day ranges. Optimization was then applied at a finer grain within this range. We propose 2 different strategies: 1. A pure proprietary strategy as suggested in the paper 2. A modified strategy with a liquidation component that slowly negates the position to 0 over the m- periods

Pure Proprietary Strategy 1 The pure proprietary strategy is explained in slide 4 and exactly replicates the strategy explained in the BATDC paper. Performance data for the strategy is collated and the graphical representation is attached in the following slides.

Pure Proprietary Strategy 2 P&L for Pure Prop Strategy

Pure Proprietary Strategy 3 Sharpe Ratio of Pure Prop Strategy

Pure Proprietary Strategy 4 Omega of Pure Prop Strategy

Pure Proprietary Strategy 5 Observing the data, we notice that using P&L the plateau is achieved between 30, 31 & 32 close out period and 42, 43 & 44 rolling window period. Using Sharpe Ratio and Omega, there does not seem to be a stand out optimal close out and rolling window period.

Modified Proprietary Strategy 1 We now consider an additional liquidating strategy to complement the proprietary strategy. The idea is also in line with the ideas in the BATDC paper. For each n-period rolling window & m-period trading cycle, we apply the proprietary strategy as explained in slide 4 and also consider the following: 1. If time t ε [1, 2, …, m] and position is not 0 2. We choose to sell (position > 0) or buy (position < 0) Position / (m-t) units of the underlying asset in addition to the proprietary strategy We consider this additional liquidation strategy because one cannot reasonably expect to get the price at the end of the m-period if the accumulated position is huge.

Modified Proprietary Strategy 2 P&L for Modified Prop Strategy

Modified Proprietary Strategy 3 Sharpe Ratio for Modified Prop Strategy

Modified Proprietary Strategy 4 Omega for Modified Prop Strategy

Modified Proprietary Strategy 5 Using P&L, we observe from the data that stability is achieved in the period trading cycle and period rolling window. Sharpe Ratio is much improved under this strategy and is stable in the period trading cycle and period rolling window On average, Omega under this strategy is larger than under the pure prop strategy. However, similar to the pure prop strategy, it is stable over the entire range of trading cycle and rolling window periods combinations considered

Conclusions and Comparisons Although the pure Prop Strategy out performs the modified Prop Strategy in P&L, the volatility of the P&L for the modified Prop Strategy is less than that of the pure Prop Strategy. The modified Prop Strategy out performs the pure Prop Strategy in Sharpe Ratio and Omega. However, this difference is not significant. The data seems to suggests a that a 35 period trading cycle and a 18 period rolling window are the optimal parameters for the modified Prop strategy based on Sharpe Ratio and P&L. However, one can only use P&L to decide the optimal parameters for the pure Prop Strategy.