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Unbalanced and Pairwise Optimal Transport for Implied Volatility Surfaces

Article Quant Q&A · Author: Kevin
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Summary

The document proposes two possible adaptations of optimal transport methods for implied volatility surfaces. One concerns arbitrage-free projection methods that assume each maturity’s distribution integrates to one. When observed strike coverage is incomplete at the tails, the author asks whether unbalanced optimal transport could relax that constraint by allowing mass to be created or removed at a controlled cost, avoiding the need to extrapolate the surface first.

The second idea addresses the computational burden of coupling distributions across many maturities. The author suggests replacing a high-dimensional multi-marginal problem with successive pairwise couplings along the maturity sequence. These are presented as research hypotheses, not validated methods: the document provides no implementation, formal analysis, or empirical comparison. It also notes a practical tradeoff, since unbalanced transport may not produce implied volatility values at every strike, while tail extrapolation can fill those locations. Whether estimates at illiquid strikes are useful remains open.

Key ideas

  • Unbalanced optimal transport could relax exact probability-mass constraints when strike tails are missing.
  • Pairwise couplings along maturities may reduce the dimensionality of a multi-marginal transport problem.
  • Both proposals are exploratory and the document gives no mathematical or empirical validation.
  • Relaxing marginal constraints may leave some strikes without implied volatility estimates.

Tags

OptionsVolatilityDerivatives pricingStatistics
Full text
# The emerging use of Optimal Transport theory to Implied Volatility surfaces


# The emerging use of Optimal Transport theory to Implied Volatility surfaces












I've been reading up on the emerging application of OT to several areas of implied volatility surfaces, and I was looking for feedback on some of my ideas that I think are unique (haven't been explored yet):

The paper 'https://arxiv.org/abs/2501.12195' explores OT projection on IV surfaces to ensure no arbitrage. I noticed that before they set up the OT problem, they assume each maturity's marginals is a probability measure, and hence they integrate to 1. To implement this method, they first extrapolate at the tails to ensure this assumption holds. In real world IV surfaces, there might be missing strikes at the tails in which case the maturity's distribution doesn't integrate exactly to 1. So could you use unbalanced OT to relax the hard marginal constraints, and allow mass creation and deletion at a controlled cost where needed? This way, you don't have to first process the IV surface to meet the hard 'must integrate to 1' constraint.

From the same paper, they formulate the problem as an N marginal coupling which becomes extremely computationally intensive, leaving it hard to use for front office work. Could you solve a series of pairwise OT couplings along a maturity chain, linking each pair? It would leverage the fact that maturities form a natural time like chain (a tree structure). In theory, this would change a high dimensional coupling into a sequence of lower dimensional couplings

Any critique is welcomed, no matter how harsh. These could all be meaningless

EDIT: With the unbalanced OT approach, I noticed that we might not get an implied volatility value for every strike, whereas with extrapolation we could. I guess it comes down to if having quotes at very illiquid spots is even beneficial

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.