Deep quantile regression is a neural network-based method for estimating conditional quantiles, used in the context of xVA computation to approximate initial margin requirements. Initial margin is defined as a risk measure (typically Value at Risk) of adverse portfolio value movements over the margin period of risk (MPR), conditioned on the current state. Since the conditional VaR is not Markovian in the state variable alone in a discrete-time setting, the neural network takes both the current state X_n and the current clean portfolio value V_n as inputs.
The method exploits the classical equivalence between quantile estimation and minimisation of the check function (pinball loss): VaR_alpha(L | F) = argmin_{q} E[kappa^alpha(q; L) | F], where kappa^alpha(q; x) = max(alpha * (x - q), (alpha - 1) * (x - q)). This formulation, due to Koenker and Bassett (1978), converts quantile estimation into a regression problem amenable to neural network approximation. For each time step n, a separate fully connected network R^d x R^P → R^2 is trained to output both the upper (alpha-level) and lower ((1-alpha)-level) quantiles simultaneously, from which IMFC (initial margin from counterparty) and IMTC (initial margin to counterparty) are obtained.
In the multi-layer deep BSDE solver, deep quantile regression constitutes Layer 2, sitting between the clean value computation (Layer 1) and the xVA BSDEs (Layers 3-4). The IM estimates feed into the CVA, DVA, MVA, and FVA BSDEs through both the terminal conditions (close-out amounts at default depend on IM) and the BSDE drivers (MVA involves the funding cost of posting IM). The networks use 3 hidden layers of 16 neurons each with ReLU activations, trained on 2^12 samples with 80/20 train/validation split and early stopping with patience of 100 epochs.
Key Details
- Risk measure: VaR at level alpha = 0.99 for IMFC, at level 1 - alpha for IMTC
- MPR is set to min(MPR, T - t), with MPR = 8 time steps (approximately 14.6 days if T is in years)
- Input includes both state X_n and portfolio value V_n to recover the Markov property in discrete time
- Check function loss: kappa^alpha(q; x) = max(alpha(x-q), (alpha-1)(x-q))
- IM is not rehypothecable: IMFC and IMTC are posted simultaneously in segregated accounts and cannot be netted
- The mean-squared error of the IM approximation appears explicitly in the error bounds of Theorem 7.3 for the xVA layers