Why Dupire Local Variance Can Turn Negative on an SVI Surface

Summary

The document discusses negative values arising when Dupire local variance is computed from an implied total variance surface fitted with SVI. The questioner estimates the time derivative by finite differences and finds a negative numerator or denominator, which prevents taking a real square root for local volatility. They believe the surface is arbitrage-free and ask whether regularization could fix the issue.

The response emphasizes that local volatility requires an arbitrage-free surface and cautions that this is difficult to ensure in practice, particularly for equity options with wide bid–ask spreads. It explains that fitting SVI separately by maturity can control strike arbitrage while still allowing calendar arbitrage across maturities. Quasi-SVI or adjustments across slices are mentioned as alternatives, but the response warns that slice adjustments can produce unrealistic surfaces. These are practical cautions rather than a complete derivation or robust calibration recipe; no data, diagnostics, or validation results are supplied.

Key ideas

• Dupire local variance calculations can fail when the input implied volatility surface is not sufficiently arbitrage-free.
• Separate SVI fits by maturity may control strike arbitrage without ensuring calendar-arbitrage consistency.
• Wide bid–ask spreads in equity options can make arbitrage-free surface calibration difficult.
• Quasi-SVI and cross-maturity adjustments are possible approaches, though adjustments may yield unrealistic surfaces.

Tags

# Negative Dupire Variance


# Negative Dupire Variance












I want to compute Dupire Local volatility using the identity that links Dupire local variance to BS implied total variance. I calibrated an SVI on options data to get the implied total variance surface and to compute the derivative of the implied total variance wrt time, i do a finite difference. The problem is i always stumble on a negative nominator or denominator, hence I cannot apply a square root to get dupire local volatility.

Is there a solution to the problem ? some kind of regularization to do ? I am 99% sure that my surface is free of arbitrage

## Answer by THATS MY QUANT MY QUANTITATIVE (score 1)

https://quant.stackexchange.com/a/79432

What data are you using? Local volatility requires an arbitrage-free implied volatility surface. In general, equities rarely satisfy these conditions on their options because their bid-ask spreads are too large. I have had more success calibrating local volatility surfaces with FX options because of this very reason.

Additionally, SVI is generally used for each tenor slice and so you will satisfy strike arbitrage, but will generally fail calendar arbitrage. There is another method called quasi-SVI, which has worked for me, or you can calibrate each slice and then attempt to adjust the slices - but I often find this will lead to unrealistic results for the surface.

But in general, there isn't a robust method to calibrate equity options across a vast amount of strikes and maturities.

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