# Analytical Modeling and Empirical Analysis of Binary Options Strategies
## 2 Materials and Methods
### Investment Strategies  
In this paper, the performance of four easy-to-implement investment strategies for binary options, namely Strategies A, B, C, and D, was introduced and empirically analyzed. These strategies are based on two popular investment principles in practice, _dollar cost averaging_ (_DCA_) [55] and _taking profits_[56]. The _dollar-cost-averaging_ (_DCA_) principle suggests investing regularly over time, regardless of the price of the asset. This principle is represented in our proposed strategies through _proportional investment_, which is reflective of investing consistently and regularly over time. The _taking profits_ principle suggests regularly taking profit as profits are made. This principle is represented in our proposed strategies through _proportional withdrawals_ in strategies C and D.  
The description of each strategy, expression of the strategy as a rule, and the formula for balance \(X_{t+1}\) in each period \(t+1\) are provided. The balance at the beginning of period \(t+1\) is the ending balance at the end of period \(t\) and is a function of the balance \(X_{t}\) at the previous period \(t\), as well as the parameters of the binary options and the decision variables of the investment strategy. **Strategy A:** Proportional Investment of Current Balance  
In this strategy, in each period \(t\), the investor invests a certain proportion \(z\) of the current balance \(X_{t}\). In other words, strategy A is  
\"At each period \(t\), invest \(A_{t}=zX_{t}\)\". For strategy A,  
\[X_{t+1}=\left\{ \begin{matrix}X_{t}+rzX_{t}&\text{ with probability }p\\ X_{t}-zX_{t}&\text{ with probability }1-p\end{matrix} \right. \tag{2}\]  
\[X_{t+1}=\left\{ \begin{matrix}(1+rz)X_{t}&\text{ with probability }p\\ (1-z)X_{t}&\text{ with probability }1-p\end{matrix} \right. \tag{3}\]  
**Strategy B:** Proportional Investment of Initial Capital  
In this strategy, in each period \(t\), the investor invests a certain proportion \(z^{\prime}\) of the initial balance \(X_{0}\). In other words, strategy B is  
\"At each period \(t\), invest \(B_{t}=z^{\prime}X_{0}\)\". For strategy B,  
\[X_{t+1}=\left\{ \begin{matrix}X_{t}+rzX_{0}&\text{ with probability }p\\ X_{t}-z^{\prime}X_{0}&\text{ with probability }1-p\end{matrix} \right. \tag{4}\]  
**Strategy C:** Proportional Withdrawal and Investment from Current Balance  
In this strategy, in each period \(t\), if the balance is higher than a certain \(w\) multiple of the initial balance \(X_{0}\), the investor withdraws a certain percentage \(v\) of the current balance \(X_{t}\) and then invests a certain proportion \(z\) of the remaining balance after the withdrawal. In other words, strategy C is  
\"At each period \(t\), if the balance \(X_{t}\) equals or exceeds \(wX_{0}\) (\(w\geq 1\)), withdraw \(vX_{t}\) at the beginning of \(t\). Then, invest \(C_{t}\) as a portion \(z\) of the remaining available balance\". For strategy C,  
\[C_{t}=\left\{ \begin{matrix}z(1-v)X_{t}&\text{ if }X_{t}\geq wX_{0}\\ zX_{t}&\text{ if }X_{t}<wX_{0}\end{matrix} \right. \tag{5}\]  
\[X_{t+1}=\left\{ \begin{matrix}X_{t}+rz(1-v)X_{t}\text{ if }X_{t}\geq wX_{0}, \text{ with probability }p\\ X_{t}-z(1-v)X_{t}\text{ if }X_{t}\geq wX_{0},\text{ with probability }1-p\\ X_{t}+rzX_{t}\text{ if }X_{t}<wX_{0},\text{ with probability }p\\ X_{t}-zX_{t}\text{ if }X_{t}<wX_{0},\text{ with probability }1-p\end{matrix} \right. \tag{6}\]  
\[X_{t+1}=\left\{ \begin{matrix}(1+rz(1-v))X_{t}\text{ if }X_{t}\geq wX_{0}, \text{ with probability }p\\ (1-z(1-v))X_{t}\text{ if }X_{t}\geq wX_{0},\text{ with probability }1-p\\ (1+rz)X_{t}\text{ if }X_{t}<wX_{0},\text{ with probability }p\\ (1-z)X_{t}\text{ if }X_{t}<wX_{0},\text{ with probability }1-p\end{matrix} \right.