Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration

Jean Pierre Fouque, Matthew Lorig, Ronnie Sircar

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


Multiscale stochastic volatility models have been developed as an efficient way to capture the principal effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book by Fouque et al. (Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives, 2011) analyzes models in which the volatility of the underlying is driven by two diffusions – one fast mean-reverting and one slowly varying – and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.

Original languageEnglish (US)
Pages (from-to)543-588
Number of pages46
JournalFinance and Stochastics
Issue number3
StatePublished - Jul 1 2016

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Finance
  • Statistics, Probability and Uncertainty


  • Calibration
  • Implied volatility
  • Multiscale asymptotics
  • Stochastic volatility


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