Definition (D)

Time Ratio ($\theta$)

$$\theta := \frac{d_{DTM}}{365} \text{ (Eq. D-1)}$$

Price

$$p_{ft} := \frac{1}{1+r\theta} \text{ (Eq. D-2)}$$

$$p_{xt} = p_{underlying} - \varepsilon p_{ft} = 1 - \frac{\varepsilon}{1+r\theta} \\ \because \varepsilon p_{ft} + p_{xt} := p_{underlying} = 1 \text{ (Eq. D-3)}$$

$$P := \frac{\^R_{y} \text{ (virtual XT reserve)}}{\^R_{x} \text{ (virtual FT reserve)}} = \frac{p_{ft}}{p_{xt}} = \frac{\frac{1}{1+r\theta}}{1 - \frac{\varepsilon}{1+r\theta}} = \frac{1}{1 + r \theta - \varepsilon} \text{ (Eq. D-4)}$$

APR (Annual Percentage Rate)

$$r=\frac{\frac{1}{P}+\varepsilon-1}{\theta}=\frac{\frac{\^R_x}{\^R_y}+\varepsilon-1}{\theta} \text{ (Eq. D-5)}$$

$$\begin{align*} \frac{N_r}{D_r} & = \frac{\frac{1}{P}+\frac{N_{\varepsilon}}{D_{\varepsilon}}-1}{\frac{d_{DTM}}{365}} \\ & = \frac{\frac{\^R_x}{\^R_y}+\frac{N_{\varepsilon}}{D_{\varepsilon}}-1}{\frac{d_{DTM}}{365}} \\ & = \frac{365 (\^R_x+\frac{N_{\varepsilon}}{D_{\varepsilon}} \^R_y-\^R_y)}{d_{DTM} \times \^R_y} \\ & = \frac{365 (D_{\varepsilon} \^R_x + N_{\varepsilon} \^R_y - D_{\varepsilon}\^R_y)}{D_{\varepsilon} d_{DTM} \^R_y} \\ \Rightarrow N_r & = \frac{365 D_r (D_{\varepsilon} \^R_x + N_{\varepsilon} \^R_y) - 365 D_r D_{\varepsilon}\^R_y}{D_{\varepsilon} d_{DTM} \^R_y} \text{ (Eq.D-6)} \end{align*}$$