A Question About Negative Volatility in SABR Derivation
Summary
The document questions a step in the appendix derivation of the SABR implied-volatility formula in Hagan and coauthors’ paper. The derivation is described as finding a joint density for the underlying and its volatility, integrating an option payoff against that density, and matching the resulting price to a Black or Bachelier price. The question is why the volatility integral extends over the entire real line when volatility is ordinarily positive, and whether that range is connected to terms being discarded in the derivation.
A brief response suggests that the original formula may have a flaw and points to a later correction by Obloj, based on work deriving implied volatilities from the Dupire equation. The respondent explicitly does not verify whether the issue raised is the same as the cited error. As presented, the exchange is a prompt for checking the mathematical derivation and its subsequent literature, not a complete explanation of the integration domain or a proof that the original argument is invalid. Readers should consult the cited research to assess the claim.
Key ideas
- The question concerns why a SABR derivation integrates volatility over negative and positive values.
- The described pricing approach integrates an option payoff against the joint density of the underlying and volatility.
- A response points to a later paper by Obloj as a correction to a possible problem in the original formula.
- The response does not establish that the cited correction addresses the specific integration concern.
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# Question about derivation of SABR volatility formula in original paper 'Managing Smile Risk' by Hagan et al # Question about derivation of SABR volatility formula in original paper 'Managing Smile Risk' by Hagan et al I have a question regarding the starting point of the derivation of SABR volatilities formulas in the appendix of the famous paper 'Managing Smile Risk' by Hagan et al. To derive SABR volatility formulas one need to : - solve a differential equation for the joint probability density on the values of the underlying and underlying's volatility - compute the price of a european option integrating the payoff times the probability density found at point 1 - equate the price found at point 2 with the price of the same option computed assuming Black (or Bachelier) dynamics and solving for the constant volatility of the latter. My question concerns the integration mentioned at point 2 above: In the original paper Hagan et al. perform such integration from $K$ to $\infty$ in the underlying domain (and this is fine) but integrate from $-\infty$ to $\infty$ in the volatility domain and this seems quite odd to me. Moreover they use this fact to let some terms of the integral go to $0$ (see after equation A6). Volatility is a positive number by definition so shouldn't integration of a probability density over negative values of volatility be forbidden? Did anyone have the same doubt in reading Hagan's et al. paper? ## Answer by Mr_3_7 (score 1) https://quant.stackexchange.com/a/51115 Apparently Hagan et al original formula has a problem, and strictly speaking is incorrect. This was later corrected by Obloj 2008 using the BBF paper giving the implied volatilities directly from the Dupire PDE: https://arxiv.org/pdf/0708.0998.pdf I did not check your claims against the original paper but it is a possibility that you have precisely bumped into that error.
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