Option prices encode the market's collective outlook through implied density and implied volatility. An explicit link between implied density and implied volatility translates the risk-neutrality of the former into conditions on the latter to rule out static arbitrage. Despite earlier recognition of their parity, the two had been studied in isolation for decades until the recent demand in implied volatility modeling rejuvenated such parity. This paper provides a systematic approach to build neural representations of option implied information. As a preliminary, we first revisit the explicit link between implied density and implied volatility through an alternative and minimalist lens, where implied volatility is viewed not as volatility but as a pointwise corrector mapping the Black-Scholes quasi-density into the implied risk-neutral density. Building on this perspective, we propose the neural representation that incorporates arbitrage constraints through the differentiable corrector. With an additive logistic model as the synthetic benchmark, extensive experiments reveal that deeper or wider network structures do not necessarily improve the model performance due to the nonlinearity of both arbitrage constraints and neural derivatives. By contrast, a shallow feedforward network with a single hidden layer and a specific activation effectively approximates implied density and implied volatility.

Theconventional closed-set DAmethods generally assume that the source and target domains share the same label space. However, this assumption is often not realistic in practice.

The proof of infeasibility condition (Theorem 3.2) is provided in Section B. Explanations on conditions derived in Theorem 3.2 are included in Section C. The proof of properties of the proposed model (r)LogSpecT (Proposition 3.4 The truncated Hausdorff distance based proof details of Theorem 4.1 and Corollary 4.4 are Details of L-ADMM and its convergence analysis are in Section F. Additional experiments and discussions on synthetic data are included in Section G. ( m 1) Again, from Farkas' lemma, this implies that the following linear system does not have a solution: Example 3.1 we know δ = 2|h Since the constraint set S is a cone, it follows that for all γ > 0, γ S = S . Opt(C, α) = α Opt(C, 1), which completes the proof. The proof will be conducted by constructing a feasible solution for rLogSpecT. Since the LogSpecT is a convex problem and Slater's condition holds, the KKT conditions We show that it is feasible for rLogSpecT. R, its epigraph is defined as epi f: = {( x, y) | y f ( x) }. Before presenting the proof, we first introduce the following lemma.

First, note that we shall use the same MDPs defined in Appendix D.1 as follows

This appendix presents the technical details of efficiently implementing Algorithm 2. B.1 Computing Intermediate Quantities We argue that in the setting of neural networks, Algorithm 2 can obtain the intermediate quantities ζ Algorithm 3 gives a subroutine for computing the necessary scalars used in the efficient squared norm function of the embedding layer.Algorithm 3 Computing the Nonzero V alues of n In the former case, it is straightforward to see that we incur a compute (resp. F .1 Effect of Batch Size on Fully-Connected Layers Figure 4 presents numerical results for the same set of experiments as in Subsection 5.1 but for different batch sizes |B | instead of the output dimension q . Similar to Subsection 5.1, the results in Figure 4 are more favorable towards Adjoint compared to GhostClip.

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