## 6 Results
### Evaluation Metrics  
The open-source stylefact library implemented by [PERSON] et al. [11] was used and adapted to compare and evaluate the performance of RSQGAN versus QGAN.  
Definitions for SF test metrics used to evaluate and compare the QGAN and RSQGAN performance are shown in Table 2 below.  
The model performance is measured for each SF loss metric. An SF loss metric is the mean absolute deviation between the SF metric for real and synthetic data, which are then mean averaged against overall stocks in the training or validation set. SF loss metrics can be used to compare the model performance for each regime but can also be averaged over all regimes to evaluate the general performance. The results are shown for training and validation stock sets.  
For crisis risk management applications, the focus should be placed on four SF test metrics marked with asterisks in Table 2: [vc_act, leveff, cfvol, loss_cdf_\(\theta\)]. This is because each of these 'crisis-sensitive' SF metrics is some measure of time or path-dependent (rather  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**SF Test Losses** & **Interpretation and Sources** & **Expression** \\ \hline acf & ST unpredictability & \(\sum_{k}|\hat{\gamma}_{R}(k)-\frac{1}{M}\sum_{i=1}^{M}\hat{\gamma}_{G_{i}}(k)|\) \\  & [PERSON] et al. [43] & \(\gamma(k)=\rho(r_{t},r_{t+k}),k\in\mathbb{N}\) \\  & Return density & \\ pdf & [PERSON] et al. [43] & \(\int\|\hat{f}_{R}(r)-\frac{1}{M}\sum_{i=1}^{M}\hat{f}_{G_{i}}(r)\|dr\) \\  & [PERSON] et al. [53] & \\ \(\text{vc\_act}\)* & Volatility clustering & \(\sum_{k}|\hat{\gamma}_{R}(k)-\frac{1}{M}\sum_{i=1}^{M}\hat{\gamma}_{G_{i}}(k)|\) \\  & Cont [42] & \(\
u(k)=\rho(|r_{t}|_{r+k}|),k\in\mathbb{N}\) \\ \(\text{vc\_pdf}\) & Abs. return density & \(\int\|\hat{f}_{R}(r)-\frac{1}{M}\sum_{i=1}^{M}\hat{f}_{G_{i}}(r)\|dr\) \\  & Loss future vol lag & \(\sum_{k}|\hat{L}_{R}(k)-\frac{1}{M}\sum_{i=1}^{M}\hat{L}_{G_{i}}(k)|\) \\ leveff* & [PERSON] et al. [54] & \(L(k)=\mathbb{E}\left[r_{t}|r_{t+k}|^{2}-r_{t}|r^{2}\right]/\mathbb{E}\left[|r _{t}^{2}|\right]^{2}\) \\  & Coarse-fine vol lag & \\ cfvol* & [PERSON] et al. [56] & \(\sum_{k}|\hat{\gamma}_{R}^{x}(k)-\frac{1}{M}\sum_{i=1}^{M}\hat{\rho}_{G_{i}}^{x }(k)|\) \\  & [PERSON] [57] & \(\rho^{x}(k)=\rho(|\sum_{i=1}^{x}r_{t+k-i}|,\sum_{i=1}^{x}|r_{t-i}|)\) \\  & [PERSON] and [PERSON] [58] & \\  & Coarse-fine vol skew & \\ cfvol\_diff & [PERSON] et al. [56] & \(\sum_{k}|\hat{\Delta}_{R}^{x}(k)-\frac{1}{M}\sum_{i=1}^{M}\hat{\Delta}_{G_{i}}^ {x}(k)|\) \\  & [PERSON] [57] & \(\Delta^{x}(k)=\rho^{x}(k)-\rho^{x}(-k)\) \\  & [PERSON] and [PERSON] [58] & \\ gain\_cdf\_\(\theta\) & Gain stopping CDF & \(\sum_{t}|\hat{F}_{T_{\text{sf}}^{\text{max}}}^{t}(t)-\frac{1}{M}\sum_{i=1}^{M} \hat{F}_{T_{\text{sf}}^{\text{max}}}(t)|\) \\  & [PERSON] et al. [59] & \(T_{\text{sf}}^{t}(t;\theta)=\text{inf}_{r}\,r_{t+r}\geq\theta,\theta>0\) \\ loss\_cdf\_\(\theta\)* & Loss stopping CDF & \(\sum_{t}|\hat{F}_{T_{\text{sf}}^{\text{max}}}(t)-\frac{1}{M}\sum_{i=1}^{M} \hat{F}_{T_{\text{sf}}^{\text{max}}}(t)|\) \\  & [PERSON] et al. [59] & \(T_{\text{loss}}^{t}(t;\theta)=\text{inf}_{r}\,r_{t+r}\leq\theta,\theta<0\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Stylised fact (SF) test losses.  
than distributional) behaviour and might be actionable in a real-world crisis scenario. The first three represent the ACF of returns and/or volatility given observations about recent returns and/or volatility. The fourth, loss_cdf_\(\theta\), is the cumulative distribution function of stoptimes needed to reach a cumulative loss of \(-\theta\) for some \(\theta>0\). Knowledge of this distribution could help risk managers determine a time-duration-sensitive portfolio management strategy.