Forecasting implied volatility surface with generative diffusion models

Introduction to the show: ident: Quantitative Finance. The day in quantitative finance: models of markets, risk and prices.

Ruth: I'm Ruth.

Warren: And I'm Warren, here with Iris.

Ruth: We're looking at the paper: "Forecasting implied volatility surface with generative diffusion models".

Warren: Here's the gist of the paper. We introduce a conditional Denoising Diffusion Probabilistic Model (DDPM) for generating one-day ahead arbitrage-free implied volatility surfaces The paper presents a conditional Denoising Diffusion Probabilistic Model (DDPM) designed to forecast one-day ahead arbitrage-free implied volatility surfaces, addressing the challenge of capturing path-dependent dynamics while suppressing arbitrage violations through a dynamically weighted loss function.

Ruth: This work is significant because it develops a novel method to enforce financial validity within the diffusion framework by incorporating an explicit financial arbitrage penalty that adjusts dynamically during training.

Warren: Modeling Volatility Dynamics and Constraints The model is developed to capture the dynamics of the vol-surface rather than just fitting a static surface, moving beyond classical local volatility models which imply unrealistic future dynamics.

Ruth: The task is framed as a conditional generation problem where the model learns to sample the distribution of one-day-ahead vol-surface conditioned on path-dependent market variables such as exponentially weighted moving averages (EWMAs) of historical vol-surfaces, asset returns, and scalar risk indicators.

Paper summary: Ruth: So we're looking at this paper titled "Forecasting implied volatility surface with generative diffusion models" by Chen Jin, Ankush Agarwal, on page one of that work. The core idea here is they introduce a conditional Denoising Diffusion Probabilistic Model called a DDPM for generating one-day ahead arbitrage-free implied volatility surfaces.

Warren: Arbitrage-free implies they're trying to build something that respects the basic rules of finance, which is hard when you're dealing with complex, path-dependent things like volatility.

Iris: It seems the paper is tackling two big problems at once: capturing the path-dependent nature of volatility dynamics and making sure those generated surfaces don't have arbitrage opportunities.

Ruth: Exactly. To capture that path dependence, they condition the model on a set of market variables, which includes exponential weighted moving averages of historical vol-surfaces, along with returns and squared returns of the underlying asset, plus some scalar risk indicators associated with it.

Warren: That conditioning set sounds pretty comprehensive for capturing the dynamics you need for a forecast. But they immediately hit a snag in their abstract about arbitrage opportunities in the earlier training data, particularly at the start of the sample.

Iris: That’s where things get tricky, because if you train on data that already has arbitrage violations, it makes generating something free of them really challenging.

Ruth: They address that specific conflict by using a parameter-free weighting scheme based on the signal-to-noise ratio or SNR to bake an explicit financial arbitrage penalty right into the loss function.

Warren: So they’re not just trying to fit a surface; they’re actively penalizing things that look like arbitrage while the model is learning how volatility moves over time during diffusion.

Iris: The paper mentions two structural constraints for a valid surface, which are calendar arbitrage penalty phi tau sigma based on the non-decreasing total implied variance with respect to tau, and butterfly arbitrage constraint phi b sigma ensuring convexity across moneyness m.

Ruth: Those constraints sound like they're trying to enforce basic financial validity in the output surface itself before you even look at the forecast quality.

Warren: And how do they actually build that training signal? The summary mentions a composite loss function that augments the standard score-matching objective with an explicit financial arbitrage penalty.

Paper summary: Iris: The key mechanism for enforcing those constraints is this parameter-free weighting scheme based on SNR, defined in Equation fourteen as Ltotal(theta) equals Lsimple(theta) plus lambda times Et,x(k+one)0,epsilon h min of SNR at k plus one gamma times phi sigma hat of k plus one.

Ruth: They cap the SNR at a predefined maximum threshold gamma to ensure training stability, which is smart because you don't want the penalty term just to overwhelm everything during diffusion.

Warren: So they’re essentially using the SNR to dynamically adjust how much they care about those arbitrage violations as they move through each step of the diffusion process.

Iris: That dynamic adjustment is interesting because it suggests that early in training, when noise is high, maybe the model needs a weaker penalty than when it's closer to generating a final surface.

Ruth: It implies that the financial validity constraint isn't static; it evolves with the generation process itself.

Warren: Where do we see if this actually works on real data? The summary mentions numerical experiments using market data on S andP five hundred options, which they use to compare their model against existing modern and classical approaches <ref:2511.07571#pg2>.

Iris: They show superior performance compared to existing benchmarks, specifically noting a lower Mean Absolute Error of zero point seven zero six two percent for their model versus the benchmark’s error of zero point nine one two zero percent on volatility forecasting.

Ruth: That difference in error suggests that incorporating this arbitrage penalty actually helps improve the predictive accuracy of the volatility surface compared to methods like VolGAN or other GAN-based approaches they tested.

Warren: It’s also interesting because they provide uncertainty estimates, showing that ninety percent confidence intervals exhibit breach rates close to the theoretical ten percent target for ATM and ITM points.

Iris: So it's not just about getting a lower error number; it’s about having well-calibrated uncertainty estimates, which is crucial when you're using this for risk management purposes.

Ruth: And they found that including the arbitrage penalty term actually reduces all four statistics of the arbitrage penalties without compromising predictive accuracy when measured by MAPE.

Warren: That suggests it cleans up the surface structure while still keeping the forecast reliable across different metrics, which is a good sign for practical use in trading strategies.

Iris: The simulation study using synthetic data from the Heston model serves as a sanity check, validating that the model can learn known stochastic dynamics in an arbitrage-free environment first.

Paper summary: Ruth: But then they test it on real market data, and when they do, the overall Mean Absolute Percentage Error MAPE is three point zero zero two six percent, which is lower than the benchmark’s three point eight four one eight percent <ref:2511.07571#pg3>.

Warren: The Diebold-Mariano test confirms that this diffusion model reduces MAPE by approximately zero point eight four percentage points on average relative to VolGAN, showing a statistically robust improvement in forecasting error over that competitor.

Iris: The text also notes that the SNR-weighted regularization term successfully minimizes arbitrage violations across the majority of the time series, shifting the penalty distribution toward zero during training.

Ruth: So what does this mean for us? It means we have a new way to enforce financial rules—the arbitrage constraints—directly into a powerful generative modeling framework like diffusion models.

Warren: It suggests that instead of fixing arbitrage after generating a surface, you can bake the avoidance of those issues into the generation process itself using dynamic weighting.

Iris: The limitation they state is that while this method works well for forecasting surfaces for risk management, it still struggles to capture those complex, idiosyncratic shapes observed in equity vol-surfaces.

Ruth: So it’s not a perfect solution for modeling every single nuance of the market structure, but it's a much better tool than what they were compared against.

Warren: And if you look at the architecture, they use a conditional U-Net with an encoder-decoder design and skip connections to reintegrate information from the input tensor X(k) which is in R four times Nm times N tau <ref:2511.07571#pg3>.

Iris: They integrate time t and the conditioning vector c of k into this structure by creating a joint context embedding e(k)ctx using a Multi-Layer Perceptron applied to sinusoidal embeddings of t and the conditioning vector c of k.

Ruth: That embedding is then integrated into the convolutional blocks via Feature-wise Linear Modulation or FiLM to dynamically shape the final output, which is pretty sophisticated engineering work there.

Warren: The overall performance metrics are impressive, with a lower error than GAN approaches like VolGAN and better calibration on uncertainty estimates compared to those benchmarks.

Iris: So, for someone who only listens to this show, what does this mean? It means we have a new generative tool that can produce volatility forecasts that are both more accurate and better understood in terms of risk.

Conclusion: Ruth: So, we're wrapping up this look at "Forecasting implied volatility surface with generative diffusion models." This paper introduces a conditional Denoising Diffusion Probabilistic Model, or DDPM, that tries to generate one-day ahead implied volatility surfaces that are free of arbitrage.

Warren: It’s about taking those complex path-dependent dynamics—how things move over time—and trying to capture them using this diffusion framework, which is a generative model.

Iris: The authors are essentially tackling the problem of making sure these generated surfaces actually make sense in a financial world, not just mathematically plausible.

Ruth: They do this by conditioning the model on historical market variables like exponential weighted moving averages and asset returns to guide its generation process.

Warren: But the real meat here is how they handle those arbitrage opportunities that sneak into the training data—the paper uses a dynamic weighting scheme based on signal-to-noise ratio to penalize violations during training.

Iris: So, it's not just a static constraint; the penalty strength changes as the model learns, which they call Ltotal(theta) incorporating min of SNR and gamma.

Ruth: That’s a clever way to enforce financial validity without completely ruining the training stability by capping that signal-to-noise ratio at some threshold.

Warren: The results show that this model beats existing benchmarks, like VolGAN, in terms of Mean Absolute Error, hitting zero point seven zero six two percent compared to their benchmark’s zero point nine one two zero percent on volatility forecasting.

Iris: And they showed that the uncertainty estimates are well-calibrated; the breach rates for ATM and ITM points are very close to the theoretical ten percent target.

Ruth: It seems like this work suggests that combining a generative diffusion model with an explicit, dynamically weighted arbitrage penalty can lead to a more accurate and reliable volatility forecast.

Warren: It moves the goal from just generating *a* surface to generating one that is both financially plausible and predictive for short-term movements.

Iris: That’s what we need to think about—how this method changes the way we model risk when dealing with these complex, time-dependent surfaces.

Ruth: It opens up a new avenue for trying to build generative models that respect fundamental financial constraints during the learning process.