q-fin.PMDec 8, 2024
SaveMean--Variance Portfolio Selection by Continuous-Time Reinforcement Learning: Algorithms, Regret Analysis, and Empirical Study
Organizations: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027, USA. · Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, N.T. · Department of Industrial Engineering and Operations Research & Data Science Institute, Columbia University, New York, NY, 10027, USA.
Abstract
We study continuous-time mean--variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes, yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL approach that learns the pre-committed investment strategy directly without attempting to learn or estimate the market coefficients. For multi-stock Black--Scholes markets without factors, we further devise an algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of the Sharpe ratio. We then carry out an extensive empirical study implementing this algorithm to compare its performance and trading characteristics, evaluated under a host of common metrics, with a large number of widely employed portfolio allocation strategies on S&P 500 constituents. The results demonstrate that the proposed continuous-time RL strategy is consistently among the best, especially in a volatile bear market, and decisively outperforms the model-based continuous-time counterparts by significant margins.
Figures & tables
Figure 1: Wealth trajectories under the proposed CTRL algorithm and 13 alternative methods over 100 independent experiments, each with 10 randomly selected stocks (except the S&P 500 index), from 2000 to 2020. The solid line shows the median across experiments at each time, and the shaded region is the interquartile range.
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| S&P500 | 4.077% (0.000%) | 0.189 (0.000) | 0.216 (0.000) | 0.341 (0.000) | 0.072 (0.000) | 0.568 (0.000) | 1021 (0) |
| ew | 10.343% (0.185%) | 0.218 (0.002) | 0.483 (0.011) | 0.789 (0.018) | 0.182 (0.005) | 0.585 (0.008) | 544 (31) |
| mv | 4.226% (0.274%) | 0.148 (0.002) | 0.304 (0.020) | 0.491 (0.032) | 0.113 (0.008) | 0.443 (0.013) | 1349 (77) |
| js | 7.493% (0.537%) | 0.240 (0.019) | 0.397 (0.023) | 0.644 (0.038) | 0.156 (0.011) | 0.577 (0.016) | 1124 (71) |
| bl | 5.180% (1.962%) | 0.618 (0.092) | 0.195 (0.027) | 0.350 (0.041) | 0.079 (0.020) | 0.851 (0.013) | 1180 (70) |
| lw | 3.797% (0.226%) | 0.154 (0.002) | 0.261 (0.016) | 0.421 (0.026) | 0.097 (0.007) | 0.456 (0.013) | 1452 (80) |
Table 1: Comparison of out-of-sample performance of different allocation methods from 2000 to 2020. We report annualized return, volatility, Sharpe ratio, Sortino ratio, and Calmar ratio, together with maximum drawdown (MDD) and recovery time (RT) in days, over 100 independent experiments. Each experiment is based on 10 randomly selected stocks (except for S&P 500 index). For each cell, the upper number is the average (over the 100 experiments) while the lower one with parentheses is the standard error.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| ew | 0% | 5.798% | 10.000% | 0.000% | 100.000% | 100.000% |
| mv | 0% | 16.437% | 113.971% | 61.028% | 113.558% | 260.610% |
| js | 1% | 15.345% | 171.726% | 106.333% | 140.192% | 542.149% |
| bl | 16% | 15.900% | 271.773% | 150.030% | 149.242% | 1000.542% |
| lw | 0% | 18.745% | 116.278% | 82.500% | 117.245% | 389.493% |
| ff | 0% | 8.841% | 116.479% | 53.619% | 96.796% | 274.978% |
Table 2: Trading and exposure statistics over the period 2000–2020. BKR is the bankruptcy rate across experiments, TO is the average monthly turnover, MaxL and MaxS are the maximum single-asset long and short positions, respectively, GE is the average gross exposure, and GEMax is the maximum gross exposure observed across all experiments.
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| S&P500 | 4.077% (0.000%) | 0.189 (0.000) | 0.216 (0.000) | 0.341 (0.000) | 0.072 (0.000) | 0.568 (0.000) | 1021 (0) |
| ew | 9.959% (0.185%) | 0.218 (0.002) | 0.465 (0.011) | 0.760 (0.018) | 0.174 (0.005) | 0.588 (0.008) | 568 (31) |
| mv | 3.203% (0.279%) | 0.148 (0.002) | 0.234 (0.019) | 0.381 (0.031) | 0.085 (0.007) | 0.462 (0.013) | 1490 (72) |
| js | 6.507% (0.548%) | 0.240 (0.019) | 0.352 (0.023) | 0.572 (0.037) | 0.135 (0.010) | 0.587 (0.016) | 1199 (64) |
| bl | 4.186% (1.958%) | 0.618 (0.092) | 0.172 (0.027) | 0.312 (0.041) | 0.066 (0.020) | 0.854 (0.013) | 1378 (88) |
| lw | 2.635% (0.230%) | 0.154 (0.002) | 0.185 (0.016) | 0.300 (0.026) | 0.067 (0.006) | 0.478 (0.013) | 2027 (120) |
Table 3: Net-of-cost performance over the full sample period (2000–2020) under 50 bps proportional transaction costs. We report annualized return, volatility, Sharpe ratio, Sortino ratio, and Calmar ratio, together with maximum drawdown (MDD) and recovery time (RT) in days, over 100 independent experiments. Each experiment is based on 10 randomly selected stocks (except for S&P 500 index). Numbers in parentheses are the standard errors across 100 experiments.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| ew | 0% | 5.798% | 10.000% | 0.000% | 100.000% | 100.000% |
| mv | 0% | 16.432% | 113.971% | 61.028% | 113.558% | 260.610% |
| js | 1% | 15.341% | 171.726% | 106.333% | 140.192% | 542.149% |
| bl | 16% | 15.912% | 271.773% | 150.030% | 149.242% | 1000.542% |
| lw | 0% | 18.741% | 116.278% | 82.500% | 117.245% | 389.493% |
| ff | 0% | 8.840% | 116.479% | 53.619% | 96.796% | 274.978% |
Table 4: Trading and exposure characteristics under 50 bps proportional transaction costs over the full sample period (2000–2020). BKR denotes the bankruptcy rate across experiments. TO is the average monthly turnover. MaxL and MaxS report the maximum single-asset long and short positions, respectively. GE is the average gross exposure, and GEMax is the maximum gross exposure observed across all experiments.
Appendix figures & tables21 assets
Supplementary material from the paper’s appendix.
Appendix
| Strategy | Symbol | Reference / Description |
| Buy-and-hold S&P 500 index | S&P 500 | Capitalization‑weighted U.S. market index |
| Equally weighted allocation | ew | DeMiguel et al., 2009b () |
| Mean–variance (single‑period) | mv | Markowitz, (1952) |
| Minimum variance | min_v | Markowitz, (1952) |
| James–Stein shrinkage (mean) | js | Jorion, (1986) |
| Ledoit–Wolf shrinkage (covariance) | lw | Ledoit and Wolf, (2003) |
Table 5: Alternative asset allocation strategies compared in this study.
Figure 2: Error of parameter The solid curves and the upper and lower boundaries of the shaded regions represent the average, 2.5% and 97.5% percentile of the error over 1000 independent simulation runs, respectively. The slope for by least squares regression is -1.09. The vertical and horizontal axes are on natural log-scale.
Figure 3: Error of parameter The solid curves and the upper and lower boundaries of the shaded regions represent the average, 2.5% and 97.5% percentile of the error over 1000 independent simulation runs, respectively. The slope for by least squares regression is -0.91. The vertical and horizontal axes are on natural log-scale.
Figure 4: Error of parameter The solid curves and the upper and lower boundaries of the shaded regions represent the average, 2.5% and 97.5% percentile of the error over 1000 independent simulation runs, respectively. The slope for by least squares regression is -0.97. The vertical and horizontal axes are on natural log-scale.
Figure 5: Cumulative regret rate in number of episodes. The solid blue curve and the upper and lower boundary of the shaded region represent the mean, 2.5% and 97.5% percentile of the regret over 1000 independent simulation runs, respectively. The red dashed line is the fitted value by linearly regressing the log average regret against the logarithm of the number of episodes starting from the 100th episode. The fitted slope by least squares regression is . The vertical and horizontal axes are on natural log-scale.
Figure 6: Log–log plot of versus , showing how the exploration level affects the stability of the update for . The shaded area represents the 95% confidence interval.
Figure 7: Median wealth trajectories with interquartile ranges for CTRL, mv, ctmv, and ew over the full sample period.
Figure 8: Box plots of Sharpe ratios over the full sample period across 100 experiments. Extremely negative values are truncated at .
| S&P500 | ew | mv | js | bl | lw | ff | rp | min_v | pmv | ctmv | ddpg | ppo | CTRL | |
| S&P500 | 1 | 1 | 1 | 1 | 0.953 | 1 | 1 | 1 | 0.997 | 0.143 | 0.826 | 0.999 | 1 | |
| ew | 2.01e-18 | 3.83e-16 | 8.1e-06 | 8.86e-16 | 4.01e-18 | 1.12e-07 | 0.274 | 0.578 | 1.34e-14 | 2.97e-16 | 6.71e-13 | 2.99e-12 | 0.999 | |
| mv | 0.000184 | 1 | 1 | 0.941 | 9.82e-05 | 1 | 1 | 1 | 0.248 | 0.000963 | 0.0824 | 0.561 | 1 | |
| js | 1.98e-12 | 1 | 1.79e-16 | 1.1e-06 | 8.77e-15 | 0.974 | 0.999 | 1 | 2.69e-09 | 2.28e-11 | 2.61e-06 | 0.000232 | 1 | |
| bl | 0.000381 | 1 | 0.0589 | 1 | 0.00166 | 1 | 1 | 1 | 0.0317 | 0.000191 | 0.0463 | 0.373 | 1 | |
| lw | 0.047 | 1 | 1 | 1 | 0.998 | 1 | 1 | 1 | 0.954 | 0.0527 | 0.462 | 0.958 | 1 |
Table 6: One-sided paired Wilcoxon p-values on per-seed Sharpe. Entry (i,j) tests .
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| S&P500 | 4.077% (0.000%) | 0.189 (0.000) | 0.216 (0.000) | 0.341 (0.000) | 0.072 (0.000) | 0.568 (0.000) | 1021 (0) |
| ew | 10.247% (0.185%) | 0.218 (0.002) | 0.478 (0.011) | 0.782 (0.018) | 0.180 (0.005) | 0.586 (0.008) | 551 (31) |
| mv | 3.970% (0.275%) | 0.148 (0.002) | 0.287 (0.020) | 0.463 (0.031) | 0.106 (0.008) | 0.447 (0.013) | 1359 (70) |
| js | 7.246% (0.540%) | 0.240 (0.019) | 0.386 (0.023) | 0.626 (0.037) | 0.151 (0.011) | 0.579 (0.016) | 1126 (65) |
| bl | 4.932% (1.960%) | 0.618 (0.092) | 0.189 (0.027) | 0.340 (0.041) | 0.076 (0.020) | 0.851 (0.013) | 1250 (79) |
| lw | 3.506% (0.227%) | 0.154 (0.002) | 0.242 (0.016) | 0.391 (0.026) | 0.090 (0.006) | 0.462 (0.013) | 1487 (78) |
Table 7: Net-of-cost performance over the full sample period (2000–2020) under 12.5 bps proportional transaction costs. We report annualized return, volatility, Sharpe ratio, Sortino ratio, and Calmar ratio, together with maximum drawdown (MDD) and recovery time (RT) in days, over 100 independent experiments. Each experiment is based on 10 randomly selected stocks (except for S&P 500 index). Numbers in parentheses are the standard errors across 100 experiments.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| ew | 0% | 5.798% | 10.000% | 0.000% | 100.000% | 100.000% |
| mv | 0% | 16.435% | 113.971% | 61.028% | 113.558% | 260.610% |
| js | 1% | 15.344% | 171.726% | 106.333% | 140.192% | 542.149% |
| bl | 16% | 15.903% | 271.773% | 150.030% | 149.242% | 1000.542% |
| lw | 0% | 18.744% | 116.278% | 82.500% | 117.245% | 389.493% |
| ff | 0% | 8.840% | 116.479% | 53.619% | 96.796% | 274.978% |
Table 8: Trading and exposure characteristics under 12.5 bps proportional transaction costs over the full sample period (2000–2020). BKR denotes the bankruptcy rate across experiments. TO is the average monthly turnover. MaxL and MaxS report the maximum single-asset long and short positions, respectively. GE is the average gross exposure, and GEMax is the maximum gross exposure observed across all experiments.
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| S&P500 | -2.633% (0.000%) | 0.222 (0.000) | -0.118 (0.000) | -0.190 (0.000) | -0.046 (0.000) | 0.568 (0.000) | N/A (N/A) |
| ew | 7.467% (0.286%) | 0.252 (0.003) | 0.301 (0.012) | 0.497 (0.020) | 0.133 (0.006) | 0.585 (0.008) | 259 (3) |
| mv | -0.166% (0.358%) | 0.169 (0.004) | 0.013 (0.019) | 0.027 (0.030) | 0.011 (0.008) | 0.431 (0.012) | 1488 (17) |
| js | 0.242% (0.636%) | 0.282 (0.026) | 0.064 (0.020) | 0.105 (0.031) | 0.027 (0.010) | 0.570 (0.016) | 476 (5) |
| bl | -3.817% (1.830%) | 0.731 (0.123) | -0.030 (0.020) | -0.021 (0.032) | -0.030 (0.018) | 0.850 (0.014) | 223 (2) |
| lw | 0.411% (0.338%) | 0.178 (0.003) | 0.041 (0.019) | 0.071 (0.030) | 0.024 (0.008) | 0.440 (0.012) | 1790 (43) |
Table 9: Comparison of out-of-sample performance of different allocation methods in the bear period (2000-2010) We report annualized return, volatility, Sharpe ratio, Sortino ratio, and Calmar ratio, together with maximum drawdown (MDD) and recovery time (RT) in days, over 100 independent experiments. Each experiment is based on 10 randomly selected stocks (except for S&P 500 index). For each cell, the upper number is the average (over the 100 experiments) while the lower one with parentheses is the standard error.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| ew | 0% | 6.559% | 10.000% | 0.000% | 100.000% | 100.000% |
| mv | 0% | 13.854% | 81.364% | 61.028% | 94.358% | 260.610% |
| js | 1% | 13.370% | 136.498% | 106.333% | 114.192% | 461.708% |
| bl | 16% | 18.404% | 271.773% | 150.030% | 130.610% | 1000.542% |
| lw | 0% | 16.854% | 100.599% | 82.500% | 103.348% | 389.493% |
| ff | 0% | 10.359% | 93.438% | 53.619% | 94.511% | 274.978% |
Table 10: Trading and exposure statistics over the bear period 2000–2010. BKR is the bankruptcy rate across experiments, TO is the average monthly turnover, MaxL and MaxS are the maximum single-asset long and short positions, respectively, GE is the average gross exposure, and GEMax is the maximum gross exposure observed across all experiments.
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| S&P500 | 11.070% (0.000%) | 0.148 (0.000) | 0.750 (0.000) | 1.173 (0.000) | 0.560 (0.000) | 0.198 (0.000) | 81 (0) |
| ew | 13.335% (0.248%) | 0.177 (0.002) | 0.772 (0.020) | 1.249 (0.032) | 0.526 (0.018) | 0.270 (0.006) | 195 (11) |
| mv | 8.873% (0.375%) | 0.123 (0.001) | 0.736 (0.031) | 1.224 (0.052) | 0.515 (0.031) | 0.207 (0.008) | 303 (25) |
| js | 15.379% (0.602%) | 0.177 (0.006) | 0.941 (0.031) | 1.564 (0.053) | 0.761 (0.034) | 0.240 (0.011) | 177 (17) |
| bl | 15.348% (2.260%) | 0.419 (0.049) | 0.647 (0.047) | 1.074 (0.071) | 0.477 (0.038) | 0.494 (0.025) | 242 (29) |
| lw | 7.359% (0.330%) | 0.123 (0.002) | 0.608 (0.027) | 1.004 (0.045) | 0.382 (0.023) | 0.225 (0.008) | 430 (37) |
Table 11: Comparison of out-of-sample performance of different allocation methods in the bull period (2010-2020). We report annualized return, volatility, Sharpe ratio, Sortino ratio, and Calmar ratio, together with maximum drawdown (MDD) and recovery time (RT) in days, over 100 independent experiments. Each experiment is based on 10 randomly selected stocks (except for S&P 500 index). For each cell, the upper number is the average (over the 100 experiments) while the lower one with parentheses is the standard error.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| ew | 0% | 5.036% | 10.000% | 0.000% | 100.000% | 100.000% |
| mv | 0% | 19.019% | 113.971% | 58.252% | 132.757% | 260.003% |
| js | 1% | 17.320% | 171.726% | 82.796% | 166.192% | 542.149% |
| bl | 16% | 13.396% | 168.732% | 85.625% | 167.874% | 574.338% |
| lw | 0% | 20.636% | 116.278% | 62.322% | 131.142% | 295.426% |
| ff | 0% | 7.323% | 116.479% | 38.422% | 99.081% | 225.771% |
Table 12: Trading and exposure statistics over the bull period 2010–2020. BKR is the bankruptcy rate across experiments, TO is the average monthly turnover, MaxL and MaxS are the maximum single-asset long and short positions, respectively, GE is the average gross exposure, and GEMax is the maximum gross exposure observed across all experiments.
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| CTRL-baseline | 12.211% (0.219%) | 0.232 (0.003) | 0.531 (0.010) | 0.865 (0.017) | 0.200 (0.005) | 0.621 (0.009) | 521 (30) |
| CTRL-lr3/2 | 12.202% (0.220%) | 0.232 (0.003) | 0.530 (0.010) | 0.864 (0.017) | 0.200 (0.005) | 0.621 (0.009) | 523 (30) |
| CTRL-lr2/1 | 12.192% (0.221%) | 0.232 (0.003) | 0.530 (0.010) | 0.863 (0.017) | 0.200 (0.005) | 0.621 (0.009) | 529 (30) |
| CTRL-lr2/3 | 12.217% (0.219%) | 0.233 (0.003) | 0.531 (0.010) | 0.865 (0.017) | 0.201 (0.005) | 0.620 (0.009) | 519 (30) |
| CTRL-lr1/2 | 12.220% (0.219%) | 0.233 (0.003) | 0.531 (0.010) | 0.866 (0.017) | 0.201 (0.005) | 0.620 (0.009) | 513 (30) |
| CTRL-gamma3/2 | 12.211% (0.219%) | 0.232 (0.003) | 0.531 (0.010) | 0.865 (0.017) | 0.200 (0.005) | 0.621 (0.009) | 521 (30) |
Table 13: Performance summary: No transaction cost.
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| CTRL-baseline | 12.115% (0.219%) | 0.232 (0.003) | 0.527 (0.010) | 0.858 (0.017) | 0.199 (0.005) | 0.621 (0.009) | 523 (30) |
| CTRL-lr3/2 | 12.106% (0.220%) | 0.232 (0.003) | 0.526 (0.010) | 0.857 (0.017) | 0.198 (0.005) | 0.622 (0.009) | 528 (30) |
| CTRL-lr2/1 | 12.097% (0.220%) | 0.232 (0.003) | 0.526 (0.010) | 0.856 (0.017) | 0.198 (0.005) | 0.622 (0.009) | 537 (30) |
| CTRL-lr2/3 | 12.121% (0.218%) | 0.233 (0.003) | 0.527 (0.010) | 0.859 (0.017) | 0.199 (0.005) | 0.621 (0.009) | 522 (30) |
| CTRL-lr1/2 | 12.124% (0.218%) | 0.233 (0.003) | 0.527 (0.010) | 0.859 (0.017) | 0.199 (0.005) | 0.621 (0.009) | 521 (30) |
| CTRL-gamma3/2 | 12.115% (0.219%) | 0.232 (0.003) | 0.527 (0.010) | 0.858 (0.017) | 0.199 (0.005) | 0.621 (0.009) | 523 (30) |
Table 14: Performance summary: Transaction cost 12.5 bps.
| Return | Volatility | Sharpe | Sortino | Calmar | MDD | RT | |
| CTRL-baseline | 11.828% (0.218%) | 0.233 (0.003) | 0.514 (0.010) | 0.837 (0.017) | 0.193 (0.005) | 0.623 (0.009) | 538 (31) |
| CTRL-lr3/2 | 11.820% (0.219%) | 0.232 (0.003) | 0.514 (0.010) | 0.837 (0.017) | 0.193 (0.005) | 0.624 (0.009) | 548 (31) |
| CTRL-lr2/1 | 11.812% (0.219%) | 0.232 (0.003) | 0.514 (0.010) | 0.836 (0.017) | 0.193 (0.005) | 0.624 (0.009) | 556 (31) |
| CTRL-lr2/3 | 11.834% (0.217%) | 0.233 (0.003) | 0.515 (0.010) | 0.838 (0.017) | 0.194 (0.005) | 0.623 (0.009) | 537 (31) |
| CTRL-lr1/2 | 11.836% (0.217%) | 0.233 (0.003) | 0.515 (0.010) | 0.838 (0.017) | 0.194 (0.005) | 0.623 (0.009) | 536 (31) |
| CTRL-gamma3/2 | 11.828% (0.218%) | 0.232 (0.003) | 0.514 (0.010) | 0.837 (0.017) | 0.193 (0.005) | 0.623 (0.009) | 538 (31) |
Table 15: Performance summary: Transaction cost 50 bps.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| CTRL-baseline | 0% | 5.679% | 18.778% | 1.903% | 102.301% | 124.034% |
| CTRL-lr3/2 | 0% | 5.658% | 18.903% | 1.930% | 102.479% | 125.026% |
| CTRL-lr2/1 | 0% | 5.637% | 18.901% | 1.957% | 102.657% | 126.016% |
| CTRL-lr2/3 | 0% | 5.693% | 18.653% | 1.885% | 102.179% | 123.366% |
| CTRL-lr1/2 | 0% | 5.700% | 18.590% | 1.876% | 102.118% | 122.977% |
| CTRL-gamma3/2 | 0% | 5.679% | 18.780% | 1.903% | 102.297% | 124.026% |
Table 16: Trading statistics: No transaction cost.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| CTRL-baseline | 0% | 5.679% | 18.778% | 1.903% | 102.301% | 124.034% |
| CTRL-lr3/2 | 0% | 5.658% | 18.903% | 1.930% | 102.479% | 125.026% |
| CTRL-lr2/1 | 0% | 5.638% | 18.901% | 1.957% | 102.657% | 126.016% |
| CTRL-lr2/3 | 0% | 5.693% | 18.653% | 1.885% | 102.179% | 123.366% |
| CTRL-lr1/2 | 0% | 5.701% | 18.590% | 1.876% | 102.118% | 122.977% |
| CTRL-gamma3/2 | 0% | 5.679% | 18.780% | 1.903% | 102.297% | 124.026% |
Table 17: Trading statistics: Transaction cost 12.5 bps.
| BKR | TO | MaxL | MaxS | GE | GEMax | |
| CTRL-baseline | 0% | 5.680% | 18.778% | 1.903% | 102.301% | 124.034% |
| CTRL-lr3/2 | 0% | 5.659% | 18.903% | 1.930% | 102.479% | 125.026% |
| CTRL-lr2/1 | 0% | 5.638% | 18.901% | 1.957% | 102.657% | 126.016% |
| CTRL-lr2/3 | 0% | 5.694% | 18.653% | 1.885% | 102.179% | 123.366% |
| CTRL-lr1/2 | 0% | 5.701% | 18.590% | 1.876% | 102.118% | 122.977% |
| CTRL-gamma3/2 | 0% | 5.680% | 18.780% | 1.903% | 102.297% | 124.026% |
Table 18: Trading statistics: Transaction cost 50 bps.
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