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Using an Attention-Based Permutation-Invariant VAE to Model Volatility Surfaces

Sjögren, Ludvig LU (2026) In Master's Theses in Mathematical Sciences FMSM01 20261
Mathematical Statistics
Abstract (Swedish)
This thesis investigates whether a variational autoencoder (VAE) can be trained directly on raw, ir-
regularly sampled option quotes rather than on pre-interpolated volatility surfaces. A Raw Set VAE
is proposed, combining a Set Transformer encoder with a pointwise decoder to process unordered,
variable-length sets of SPX option contracts without requiring prior grid construction. To isolate the
contribution of the attention-based encoder architecture, two grid-based models are implemented and
compared against each other on fixed interpolated surfaces: an MLP-VAE replicating prior work and
a Grid Set VAE using the Set Transformer encoder. All models are evaluated on their ability to
reconstruct full volatility surfaces from a small... (More)
This thesis investigates whether a variational autoencoder (VAE) can be trained directly on raw, ir-
regularly sampled option quotes rather than on pre-interpolated volatility surfaces. A Raw Set VAE
is proposed, combining a Set Transformer encoder with a pointwise decoder to process unordered,
variable-length sets of SPX option contracts without requiring prior grid construction. To isolate the
contribution of the attention-based encoder architecture, two grid-based models are implemented and
compared against each other on fixed interpolated surfaces: an MLP-VAE replicating prior work and
a Grid Set VAE using the Set Transformer encoder. All models are evaluated on their ability to
reconstruct full volatility surfaces from a small number of observed contracts. When evaluated head-
to-head on identical raw market data, the Raw Set VAE outperforms the MLP-VAE as the number
of observed contracts increases, achieving a relative MAE improvement of approximately 10% at 40
observed points. The two grid-based models perform nearly identically across all settings, suggesting
that attention-based aggregation provides limited benefit when the input structure is fixed and regular,
and that permutation invariance is simply not a relevant property for structured grid data. Arbitrage
compliance is achieved through soft penalty regularization, whereby the constrained model produces
e!ectively arbitrage-free surfaces across the full validation period. Latent space analysis reveals that
the model organizes market information in a structured and interpretable manner, with the dominant
latent dimensions capturing volatility level and term structure. The results demonstrate that it is
possible to train and deploy a generative model for implied volatility surfaces without constructing
pre-interpolated inputs, eliminating a significant preprocessing dependency (Less)
Popular Abstract (Swedish)
Volatilitetsytor är centrala verktyg inom kvantitativ finans och används dagligen för prissättning av optioner och riskhantering. Existerande metoder bygger på att kompletta, interpolerade ytor konstrueras som träningsdata innan ett neuralt nätverk tränas, ett steg som introducerar antaganden och begränsar flexibiliteten. I detta examensarbete föreslås en modell som tränas direkt på rå, oregelbundet samplade optionskontrakt från S&P 500-indexet utan något förbearbetningssteg. Modellen kombinerar en permutationsinvariant Set Transformer-encoder med en punktvis decoder i ett variational autoencoder-ramverk och kan rekonstruera hela volatilitetsytor från så få som fem observerade kontrakt. Arbitragefrihet uppnås genom strafftermer i... (More)
Volatilitetsytor är centrala verktyg inom kvantitativ finans och används dagligen för prissättning av optioner och riskhantering. Existerande metoder bygger på att kompletta, interpolerade ytor konstrueras som träningsdata innan ett neuralt nätverk tränas, ett steg som introducerar antaganden och begränsar flexibiliteten. I detta examensarbete föreslås en modell som tränas direkt på rå, oregelbundet samplade optionskontrakt från S&P 500-indexet utan något förbearbetningssteg. Modellen kombinerar en permutationsinvariant Set Transformer-encoder med en punktvis decoder i ett variational autoencoder-ramverk och kan rekonstruera hela volatilitetsytor från så få som fem observerade kontrakt. Arbitragefrihet uppnås genom strafftermer i träningsfunktionen och modellen producerar i princip helt arbitragefria ytor över hela valideringsperioden. En analys av det latenta rummet visar att modellen, utan explicit styrning, organiserat så att olika dimensioner kontrollerar distinkta egenskaper hos ytorna. Resultaten visar att det är möjligt att eliminera beroendet av förinterpolerade träningsytor utan att förlora i prestanda. (Less)
Please use this url to cite or link to this publication:
author
Sjögren, Ludvig LU
supervisor
organization
course
FMSM01 20261
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Implied volatility surface, variational autoencoder, VAE, Set Transformer, permutation invariance, raw option data, surface reconstruction, arbitrage-free, SPX options, latent space, attention mechanism
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3625-2026
ISSN
1404-6342
other publication id
2026:E43
language
English
id
9233863
date added to LUP
2026-06-11 13:24:24
date last changed
2026-06-11 13:24:24
@misc{9233863,
  abstract     = {{This thesis investigates whether a variational autoencoder (VAE) can be trained directly on raw, ir-
regularly sampled option quotes rather than on pre-interpolated volatility surfaces. A Raw Set VAE
is proposed, combining a Set Transformer encoder with a pointwise decoder to process unordered,
variable-length sets of SPX option contracts without requiring prior grid construction. To isolate the
contribution of the attention-based encoder architecture, two grid-based models are implemented and
compared against each other on fixed interpolated surfaces: an MLP-VAE replicating prior work and
a Grid Set VAE using the Set Transformer encoder. All models are evaluated on their ability to
reconstruct full volatility surfaces from a small number of observed contracts. When evaluated head-
to-head on identical raw market data, the Raw Set VAE outperforms the MLP-VAE as the number
of observed contracts increases, achieving a relative MAE improvement of approximately 10% at 40
observed points. The two grid-based models perform nearly identically across all settings, suggesting
that attention-based aggregation provides limited benefit when the input structure is fixed and regular,
and that permutation invariance is simply not a relevant property for structured grid data. Arbitrage
compliance is achieved through soft penalty regularization, whereby the constrained model produces
e!ectively arbitrage-free surfaces across the full validation period. Latent space analysis reveals that
the model organizes market information in a structured and interpretable manner, with the dominant
latent dimensions capturing volatility level and term structure. The results demonstrate that it is
possible to train and deploy a generative model for implied volatility surfaces without constructing
pre-interpolated inputs, eliminating a significant preprocessing dependency}},
  author       = {{Sjögren, Ludvig}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's Theses in Mathematical Sciences}},
  title        = {{Using an Attention-Based Permutation-Invariant VAE to Model Volatility Surfaces}},
  year         = {{2026}},
}