# Analytical Modeling and Empirical Analysis of Binary Options Strategies
## 3 Results
### Strategy B  
Figure 9 analyzes the relation between the binary option parameters \((p,r)\) and investment proportion \(z^{\prime}\) in Strategy B as factors versus (average of the median) _roiB_ as the response. The \(y\)-axis of the matrix denotes prediction accuracy \(p\), the \(x\)-axis of the matrix denotes payout percentage \(r\), the \(x\)-axis of each plot denotes investment proportion \(z^{\prime}\), and the \(y\)-axis of each plot denotes roiB for Strategy B. In the visualization, the yellow color denotes lower roiB values, and the orange to red color tones denote higher roiB values. The matrix suggests that roiB increases as prediction accuracy--payout percentage and investment proportion--increases.  
Figure 9 shows that the relation between \(z^{\prime}\) and _roiB_ can be nonlinear or (almost) linear, depending on the \((p,r)\) values. The best value to select for \(z^{\prime}\) also changes with different \((p,r)\) value pairs. For example, for \((p,r)=(0.65,\ 0.75)\), _roiB_ first increases as \(z^{\prime}\) increases. Then, after a certain point (\(z^{\prime}\cong 0.55\)), _roiB_ decreases as \(z^{\prime}\) increases. Therefore, the investor should use the \(z^{\prime}\) value that (nearly) maximizes _roiB_, which is empirically found to be \(z^{\prime*}=0.55\) (also displayed in Figure 10).  
Figure 9 also plots for each \((p,r)\) the best-fit nonlinear regression model for which _roiB_ is a function of investment proportion \(z^{\prime}\). In the regression analysis for Strategy B, various functional forms are tested. Eventually, polynomial functions were found to be most robust across different \((p,r)\) value pairs with respect to fit despite simplicity and consistent statistical significance. Thus, to derive the nonlinear regression equations, a polynomial trend model of Degree 3 was computed. The model response was (average of median) _roiB_ as a function of the factor investment proportion \(z^{\prime}\) for each \((p,r)\) value pair. The regression model results are presented visually in Figure 9 and provided in full as regression equations in Supplementary File S7.  
During the development of the regression functions for Strategy B, two further research questions were identified, as follows: \"Can (near-)optimal values \(z^{\prime*}\) for the decision variable investment proportion \(z^{\prime}\) be derived? Are there any patterns in these (near-)optimal values?\" Because the data generated in the experiments are for discrete points in continuous space, the derived values could only be near-optimal rather than optimal. To this end, for each \((p,r)\) value pair, database queries were used to search within the experimental results for the value of \(z^{\prime}\) that yields the highest _roiB_ for that pair.  
Figure 10 displays the near-optimal investment proportion values (\(z^{\prime*}\)) for Strategy B for each \((p,r)\) value pair. Darker colors denote higher near-optimal investment proportion values \(z^{\prime*}\). The visualization suggests that \(z^{\prime*}\) also increases (higher value captions, darker-colored circles) as payout percentage \(p\) increases. In other words, a higher percentage of the initial balance can be invested in each period as the prediction accuracy increases; that is, one can stake more as the instrument becomes safer.