Implied volatility expansions allow calibration of sophisticated volatility models. They provide an accurate fit and parametrization of implied volatility surfaces that is consistent with empirical observations. Fine-grained higher order expansions offer a better fit but pose the challenge of finding a robust, stable, and computationally tractable calibration procedure due to a large number of market parameters and nonlinearities. We propose calibration schemes for second order expansions that take advantage of the model's structure via exact parameter reductions and recoveries, reuse and scaling between expansion orders where permitted by the model asymptotic regime, and numerical iteration over bounded significant parameters. We perform a numerical analysis over 12 years of real S&P 500 index options data for both multiscale stochastic and general local-stochastic volatility models. Our methods are validated empirically by obtaining stable market parameters that meet the qualitative and numerical constraints imposed by their functional forms and model asymptotic assumptions.


  1. implied volatility expansions
  2. numerical calibration
  3. parameter reductions
  4. nonlinear least squares

MSC codes

  1. 91G20
  2. 91G60
  3. 90C30
  4. 60-08

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Information & Authors


Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 917 - 946
ISSN (online): 1945-497X


Submitted: 17 August 2015
Accepted: 20 September 2016
Published online: 1 December 2016


  1. implied volatility expansions
  2. numerical calibration
  3. parameter reductions
  4. nonlinear least squares

MSC codes

  1. 91G20
  2. 91G60
  3. 90C30
  4. 60-08



Funding Information

Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 : DTA

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