## 2 Results
### Calculating the Lower Bound
Rearranging to \(b_{1}\) leads to:\[b_{1} = \frac{1-\frac{q_{1}}{M}}{2-\frac{q_{1}}{M}-\frac{m}{q_{1}}}. \tag{10}\]  
Inserting Equation (10) into Equation (8) and Equation (9) leads to:  
\[\frac{OPT(\mathbf{q_{1}})}{ON(\mathbf{q_{1}})}=\frac{OPT(\mathbf{q_{2}})}{ON( \mathbf{q_{2}})}=\frac{2-\frac{q_{1}}{M}-\frac{m}{q_{1}}}{1-\frac{m}{M}}. \tag{11}\]  
To determine the price \(q_{1}=q^{*}\) which leads to the highest ratio of OPT and ON, the first derivative \(\frac{d}{dq^{*}}\left[\frac{2-\frac{q^{*}}{M}-\frac{m}{q^{*}}}{1-\frac{m}{M}}\right]\) of Equation (11) is set equal to zero:  
\[\frac{\frac{M\cdot m}{(q^{*})^{2}}-1}{M-m}=0. \tag{12}\]  
Rearranging Equation (12) to \(q^{*}\) leads to:  
\[\left(q^{*}\right)^{2}=M\cdot m. \tag{13}\]  
Since \(q^{*}>0\) holds true, it follows:  
\[q^{*}=\left(M\cdot m\right)^{\left(\frac{1}{2}\right)}. \tag{14}\]  
For the second derivative holds:  
\[\frac{d^{2}}{d(q^{*})^{2}}\left[\frac{2-\frac{q^{*}}{M}-\frac{m}{q^{*}}}{1- \frac{m}{M}}\right]=\frac{2\cdot M\cdot m}{(m-M)\cdot(q^{*})^{3}}. \tag{15}\]  
Inserting Equation (14) into Equation (15) leads to a negative value. That means, the function of the ratio between OPT and ON has a maximum at this price. Inserting Equation (14) into Equation (11) and rearranging leads to the highest possible error for \(T=2\):  
\[\frac{OPT(\mathbf{q})}{ON(\mathbf{q})} \leq \frac{2}{\left(\frac{m}{M}\right)^{\frac{1}{2}}+1}\quad\forall \mathbf{q}\in\Omega. \tag{16}\]  
Equation (10) leads to an optimal decision. Therefore, it is not possible that an algorithm exists which achieves a smaller competitive ratio than \(\frac{2}{\left(\frac{m}{M}\right)^{\frac{1}{2}}+1}\) for \(T=2\). In the following, the \(EBA\) algorithm for the bi-directional preemptive CP is given (for more details, see [19]). The algorithm is based on Equation (10) and is optimal for \(T=2\):  
**Algorithm EBA:**  
Given: \(M,m,T\)  
Step 1: Determine \(b_{t}\) for \(t=1,\ldots,T-1\) with  
\[E_{1}=\frac{OPT(\mathbf{q_{1}})}{ON(\mathbf{q_{1}})}=\frac{W_{0}\cdot\frac{q_ {1}}{m}}{W_{0}\cdot\left(\frac{q_{1}}{m}+b_{1}\cdot\left(1-\frac{q_{1}}{m} \right)\right)}=\frac{\frac{q_{1}}{m}}{\frac{q_{1}}{m}+b_{1}\cdot\left(1-\frac {q_{1}}{m}\right)}. \tag{8}\]0  
Step 2: Set \(b_{T}=1\). Note that [12] provides the algorithm _OCIP_ to solve the bi-directional preemptive CP with interrelated prices. _OCIP_ is similar to \(EBA\). The CP with a constant price interval can be reformulated as CP with interrelated prices, with the factors \(\theta_{1,t}=\frac{m}{q_{t}}\) and \(\theta_{2,t}=\frac{M}{q_{t}}\forall t=1,\ldots,T\). This problem can be solved by using \(EBA\). In contrast, _OCIP_ can be used if \(\theta_{1,t}=\theta_{1}\) and \(\theta_{2,t}=\theta_{2}\forall t=1,\ldots,T\) hold true. **Theorem 1**. : _There is no deterministic algorithm that achieves a smaller competitive ratio for the bi-directional preemptive conversion problem with given \(M\), \(m\) and \(T\) than \(\left(\frac{2}{\left(\frac{m}{M}\right)^{\frac{1}{2}}+1}\right)^{\left\lfloor \frac{T}{2}\right\rfloor}\)._  
Proof of Theorem 1.: Let \(\mathbf{q}=q_{1},\ldots,q_{T}\) be a price sequence for which it holds that every second price has to be \(m\) or \(M\). On such a sequence, an algorithm which invests all wealth in yen at price \(m\) and all wealth in dollars at price \(M\) makes optimal decisions at every even \(t\). At every odd \(t\), the same two errors can be made. First, investing too much wealth in yen when the next price will be \(M\). Second, investing too little wealth in yen when the next price will be \(m\). On a sequence with \(T\) prices, \(\left\lfloor\frac{T}{2}\right\rfloor\) errors can be made. On each partial sequence consisting of two prices, where \(q_{2}\in\{m,M\}\) holds for the second price, the same maximum ratio between OPT and ON can be achieved. From Lemma 1, it follows that the competitive ratio on a sequence with \(T=2\) cannot be smaller than \(\frac{2}{\left(\frac{m}{M}\right)^{\frac{1}{2}}+1}\).