$$ X_t^{R}=1-\frac{R_t}{C_t},\qquad X_t^{R,\prime}=\frac{X_t^{R}-\alpha^{\mathrm{tr}}}{\beta^{\mathrm{tr}}} $$

$$ Y_t=g_{\mathrm{link}}(X_t^{R,\prime})\in\mathbb{R} $$

$$ C_t(1-\alpha^{\mathrm{tr}}-\beta^{\mathrm{tr}})<R_t<C_t(1-\alpha^{\mathrm{tr}}) $$

$$ Y_t=\log\!\left[-\log\!\left(\frac{1-R_t/C_t-\alpha^{\mathrm{tr}}}{\beta^{\mathrm{tr}}}\right)\right] $$

$$ \bar{\sigma}_t^2=b_0+b_1\cos\!\left(\frac{2\pi t}{365}\right)+b_2\sin\!\left(\frac{2\pi t}{365}\right) $$

$$ \bar{Y}_t=a_0+a_1\cos\!\left(\frac{2\pi t}{365}\right)+a_2\sin\!\left(\frac{2\pi t}{365}\right) $$

$$ \sigma_t^2=\omega+\sum_{k=1}^{r}\alpha_k\tilde{\varepsilon}_{t-k}^{2}+\sum_{k=1}^{s}\beta_k\sigma_{t-k}^2, \quad \tilde{\varepsilon}_t=\varepsilon_t/\bar{\sigma}_t $$

$$ \tilde{Y}_t=\sum_{k=1}^{p}\phi_k^{\mathrm{AR}}\tilde{Y}_{t-k}+\sum_{k=1}^{q}\theta_k^{\mathrm{MA}}\varepsilon_{t-k}+\varepsilon_t, \quad \tilde{Y}_t=Y_t-\bar{Y}_t $$

$$ u_t\mid B_t=b\sim\mathcal{N}\!\left(\mu_t^{(b)},\sigma_t^{2(b)}\right),\quad b\in\{0,1\} $$

$$ \Gamma_{\ell}^{\mathrm{put}}(T,K;R)=\zeta^{R}\left(K-R_{\ell,T}\right)^{+} $$

$$ \mathrm{P\&L}_{\mathrm{hedged}}=\mathrm{exposure}+q\, (\mathrm{payoff}-\mathrm{price}) $$

$$ F_R(K;t,T)=\mathbb{P}(R_T\leq K\mid\mathcal{F}_t) =\mathbb{P}(\Gamma_T>0\mid\mathcal{F}_t) $$

$$ \mathbb{E}\{\Gamma_T\mid\mathcal{F}_t\}=\zeta^R\int_{-\infty}^{K}(K-r)f_R(r;t,T)\,dr $$

$$ R_{t_m}^{\text{spp}(b)} = \text{BNI}_{t_m} \cos(\text{AOI}_{t_m}^{\text{spp}(b)}) + \text{DHI}_{t_m} \left(\frac{1- \cos(\theta_{\text{tilt}}^{\text{spp}(b)})}{2}\right) + \text{alb}^{\text{spp}(b)} \cdot \text{GHI}_{t_m} \left(\frac{1+ \cos(\theta_{\text{tilt}}^{\text{spp}(b)})}{2}\right) $$

$$ X_t^{E}=\log(E_t) $$

$$ dX_t^{E}=\kappa^E\!\left(\mu_X-\frac{\sigma_X^2}{2\kappa^E}-X_t^{E}\right)dt+\sigma_X\,dW_t $$

$$ dY_t=d\bar{Y}_t+\theta(\bar{Y}_t-Y_t)\,dt+\mu_Y(t,B_t)\,dt+\sigma_Y(t,B_t)\,dM_t $$

$$ Y_t=g_{\mathrm{link}}(X_t^{R,\prime}) $$

$$ dM_t=B_t\,dW_{1,t}+(1-B_t)\,dW_{0,t} $$

$$ \sigma_Y(t,b)=\bar{\sigma}_t\sigma_{\mathrm{m}(t)}^{(b)} $$

$$ \mu_Y(t,b)=\bar{\sigma}_t\mu_{\mathrm{m}(t)}^{(b)} $$

$$ \Gamma^{\mathrm{SoRad}}=\zeta^{R}\left(K-R_T\right)^{+} $$

$$ \Gamma^{\mathrm{SoREd}}=\zeta^{E}E_T\left(K-R_T\right)^{+} $$

$$ f_{\mathbf{Y}}(\mathbf{y};t,t+1)=\sum_{\mathbf{b}\in\{0,1\}^{2}}\pi_{\mathbf{B}_{t+1}}^{(\mathbf{b})}\,\phi_2\!\left(\mathbf{y};\mathbf{M}_{\mathbf{Y}}^{(\mathbf{b})},\boldsymbol{\Sigma}_{\mathbf{Y}}^{(\mathbf{b})}\right) $$

$$ f_{\mathbf{R}}(\mathbf{r};t,T)=f_{\mathbf{Y}}\!\left(\mathbf{y}_{T}(\mathbf{r});t,T\right)\prod_{\ell=1}^{2}\frac{\left|g_{\mathrm{link}}^{\prime}(\eta_{\ell,T}(r_{\ell}))\right|}{\beta_{\ell}^{\mathrm{tr}}C_{\ell,T}} $$

$$ F_{\Pi_a}(p;t,T)=\Pr(\Pi_{a,T}\le p\mid\mathcal F_t)=\int_{\mathbb R^d}\!\mathbf 1_{\{\mathbf w_a^\top\mathbf r_T(\mathbf y)\le p\}} \times f_{\mathbf Y}(\mathbf y;t,T)\,\mathrm d\mathbf y $$

$$ \Pi_{a,T}=\mathbf w_a^\top\mathbf R_T $$

$$ \operatorname{VaR}_{0.05\mid t}=\inf\{p:F_{\Pi_a}(p;t,T)\ge0.05\} $$

$$ \Pi_{a,T}=\mathbf w_a^\top\mathbf R_{a,T},\qquad\widetilde\Pi_{a,T}=\Pi_{a,T}+\mathbf q_a^\top(\boldsymbol\Gamma_T-A\mathbf V) $$

$$ \max_{\mathbf q_a}\;\mathbb E_t\{\widetilde\Pi_{a,T}\}-\frac{\nu_a}{2}\mathbb{V}_t\{\widetilde\Pi_{a,T}\},\qquad\nu_a>0 $$

$$ \mathbf q_a(\mathbf V)=\boldsymbol\Sigma_{\Gamma}^{-1}\!\left[\frac{\mathbf M_{\Gamma}-A\mathbf V}{\nu_a}-\mathbf c_a\right] $$

$$ \mathcal R_\nu=\sum_{a=1}^{N_A}\nu_a^{-1},\qquad\mathbf c=\sum_{a=1}^{N_A}\mathbf c_a,\qquad\sum_{a=1}^{N_A}\mathbf q_a(\mathbf V)=\mathbf q_t^{\mathrm{ext}} $$

$$ \begin{aligned} {} & \mathbf{M}_{\Gamma}=\mathbb{E}_t\{\boldsymbol\Gamma_T\}, \\ & \boldsymbol\Sigma_{\Gamma}=\mathbb{C}v_t\{\boldsymbol\Gamma_T,\boldsymbol\Gamma_T\} \\ & c_a=\mathbb{C}v_t\{\boldsymbol\Gamma_T,\Pi_{a,T}\} \end{aligned} $$

$$ \mathbf V^{\mathrm{mv}}=D\!\left[\mathbf M_{\Gamma}-\frac{\mathbf c+\boldsymbol\Sigma_{\Gamma}\mathbf q_t^{\mathrm{ext}}}{\mathcal R_\nu}\right] $$