# Analytical Modeling and Empirical Analysis of Binary Options Strategies
## 3 Results
### All Strategies  
Figure 4 illustrates the _roi_ of all of the strategies in relation to prediction accuracy \(p\) and payout percentage \(r\). The sizes of the circles denote the _roi_, and the color denotes profits/losses (orange for profits, _roi_\(\geq 1\), and gray for losses, _roi_\(<1\)). A change in color from gray to orange denotes the start of profits (_roi_\(\geq 1\)). The growth of the circle sizes denotes the growth in profits. Missing circles denote either infeasibility (lower-left triangle) or _roi_\(>10\). For example, for \(r=0.65\), Strategy B has _roi_\(>10\) after \(p>0.68\), and hence, there are no more circles. \(p=0.64\) (orange-colored bubble), whereas A, B, and D start to profit at \(p=0.67,\ 0.65\), and \(0.65\), respectively. In contrast, profits in other strategies grow faster than those in Strategy C. For example, for \(r=0.65\), profits for all strategies grow for higher values of \(p\), yet the circles for C grow the slowest. For Strategy A, profits start the latest (for highest values of \(p\)) but also grow the fastest. For Strategy B, profits start after C but before A. Therefore, B seems to exhibit a balance of benefits, with profits starting earlier than A yet growing faster than C.  
It is to be noted that although Strategies A and B yield a much higher _roi_ for high values of \((p,r)\), these take place at the end of the trading horizon \(T=260\). The models in this study assume that the same values hold for \((p,r)\) throughout the trading horizon, which most likely does not hold true in the real world, where prediction accuracy is never a constant or granted on average. In contrast, Strategies C and D start withdrawing profits earlier, and thus they are more robust against possible changes in \((p,r)\) over the trading horizon. The line charts in the following figures show changes with increasing values of \(p\) and \(r\), revealing many inflection points, after which the behavior changes significantly. Figure 5 displays the change in the (average of median) _roi_ of the strategies in relation to prediction accuracy \(p\) on the \(x\)-axis over all considered values of payout percentage \(r\). The figure suggests that Strategies A and B yield the highest _roi_ as p increases despite their lower initial values. For Strategy B, _roiB_ exhibits a higher yield until _roiA_ for Strategy A eventually catches up at \(p\cong 0.605\) and starts to exceed the yields of Strategy B because the prediction accuracy \(p\) is higher. Figure 5 also suggests that as the prediction accuracy increases, the _roi_ increases for all strategies but unevenly. The figure also suggests that _roiB_ for Strategy B increases at a faster rate as prediction accuracy \(p\) increases, whereas _roi_ for Strategies C and D increases only slightly. An analysis of skewness can be found in Supplementary File S9. Figure 6 exhibits the change in (the medians of) the standard deviation of _roi_ for different strategies when plotted against prediction accuracy \(p\) on the \(x\)-axis. Figure 6 indicates that the _roiA StdDev_ for Strategy A is substantially higher than the standard deviations of _roi_ for other strategies for higher prediction accuracy values when \(p>0.63\). Strategy A _roiA_ can be termed the most volatile (less clustered around the mean); hence, Strategy A could be considered the riskiest, particularly for high \(p\) values. Figure 6 suggests that standard deviations of _roi_ for all strategies increase, although unevenly, with higher values of prediction accuracy \(p\). Figure 7 displays the change in (average of median) the _roi_ of the strategies in relation to payout percentage \(r\) on the \(x\)-axis over all considered values of prediction accuracy \(p\). Strategies A and B have the highest _roi_ among all strategies, and _roiA_ for Strategy A yields more than the _roiB_ for Strategy B when the payout percentage parameter increases to beyond \(r>0.645\). Similar to Figure 5, this phenomenon occurs despite their lower initial values.