HyperIV: Real-time Implied Volatility Smoothing

Thank you for your insightful comments.

• Data allocation between 1-day and 1-minute intervals

To clarify the data usage: while the 1-day dataset covers more assets (8 vs 2), the 1-minute data constitutes the majority (about 87%) of surfaces analyzed (~130,000 out of ~150,000 total, see Table 2). We included the diverse 1-day assets partly because end-of-day data is more accessible, aiding reproducibility, and also to provide a testbed for the cross-asset generalization study.

• Performance comparison on 1-minute SPX data

It is true that GNO achieved a slightly lower MAE on 1-minute SPX data (0.0140 vs 0.0167, Table 4). However, this should be considered alongside computational costs (Table 3): HyperIV uses about 1/4 the memory and is nearly 4x faster at inference. Furthermore, HyperIV demonstrates superior robustness, performing consistently well across all assets, whereas GNO's performance degrades significantly on VIX and MXEF. For context, if we scaled up HyperIV to use half of GNO’s resources (memory/runtime), its MAE could be reduced to 0.0128, easily surpassing GNO. We presented the lightweight version as this marginal accuracy difference did not outweigh the significant efficiency advantage.

• Using more market features

Yes, it is possible to extend HyperIV (and potentially the baselines) to include additional features beyond moneyness and maturity. This could be an interesting direction for future work.

• Density function in Equation (11) and the second derivative

The term $p(k,t)$ in Eq. (11) represents the risk-neutral probability density function of the terminal log-moneyness, $\log(S_t/F)$, for maturity $t$.

The second derivative of the undiscounted call price $C(K)$ with respect to the strike price $K$ yields the risk-neutral probability density function $Q(K)$ of the terminal asset price $S_T$, evaluated at $K$:

The undiscounted call price is:

$$C(K) = \mathbb{E}^Q[\max(S_T - K, 0)] = \int_{K}^{\infty} (S_T - K) Q(S_T) dS_T$$

Taking the first derivative w.r.t. $K$:

$$\frac{\partial C(K)}{\partial K} = - \int_{K}^{\infty} Q(S_T) dS_T$$

Taking the second derivative w.r.t. $K$:

$$\frac{\partial^2 C(K)}{\partial K^2} = Q(K)$$

Therefore, the second derivative yields the density of the underlying asset price under the risk-neutral measure $Q$.

This derivation is model-agnostic. Our $p(k,t)$ is directly related to this density $Q(K)$ through a change of variable from strike price $K$ to log-moneyness $k=\log(K/F)$.

Thank you for your insightful comments.

• Dividend modelling

The method itself does not rely on specific dividend assumptions like proportional dividends. It uses log forward moneyness ($k = \log(K/F)$), where the forward price $F$ (taken from data vendors in our study) already incorporates the impact of rates and dividends. We will clarify this in the revised manuscript's Preliminaries section.

• Literature on illiquid options

Thank you for recommending these papers. We agree they are relevant and will add them to the literature review in the revised version.

• More experiments on 2024 data

The original work used data available up to late 2023, as the 2024 data snapshot was not yet released by the vendor at the time of experimentation. We have run preliminary experiments on available 2024 futures options data. The results support our original findings:

Average MAE on 2024 Data (%)

90th Percentile MAE on 2024 Data (%) (representing the tail/worst cases)

We will add these results to the revised paper.

Thank you for your insightful comments.

• Justification of real use cases.

The implied volatility surface is a starting point for option trading and hedging. HyperIV's ability to generate an arbitrage-free surface in ~2 ms from sparse data (9 contracts) is useful for intra-day option traders. Specifically, it enables:

1. Updating the market view based on the latest transitions (e.g., the past minute).

2. Providing timely option quotes.

3. Calculating real-time Option Greeks for dynamic hedging (e.g., delta hedging).

4. Using the latest surface for anomaly detection in subsequent quotes, potentially identifying trading opportunities.

• Connection to Yang & Hospedales (2023) [ECML PKDD]

Both papers use HyperNetworks, but their work focuses on accelerating calibration for models like rough Bergomi, still requiring iterative optimization (~5 seconds). Our method is calibration-free at inference time, constructing a surface in ~2 ms via a single forward pass. This fundamental difference in approach and speed is why their method wasn't selected as a direct baseline for our specific calibration-free, real-time, sparse-data setting.

• Performance with more data points and computation power.

HyperIV is specifically designed for the challenging scenario of sparse data and high-frequency updates. For general cases like fitting end-of-day surfaces with thousands of options, where time sensitivity is low (once per day), directly training a standard network or calibrating a model might suffice, possibly without needing a hypernetwork. However, the sparse-data setting is a realistic reflection of high-frequency trading conditions where only a few contracts have reliable quotes at any instant, making HyperIV's speed and data efficiency valuable.