Option hedging and implied volatilities in a stochastic volatility model

Eric Renault, Nizar Touzi

Research output: Contribution to journalArticlepeer-review


In the stochastic volatility framework of Hull and White (1987), we characterize the so-called Black and Scholes implied volatility as a function of two arguments: the ratio of the strike to the underlying asset price and the instantaneous value of the volatility. By studying the variations in the first argument, we show that the usual hedging methods, through the Black and Scholes model, lead to an underhedged (resp. overhedged) position for in-the-money (resp. out-of-the-money) options, and a perfect partial hedged position for at-the-money options. These results are shown to be closely related to the smile effect, which is proved to be a natural consequence of the stochastic volatility feature. The deterministic dependence of the implied volatility on the underlying volatility process suggests the use of implied volatility data for the estimation of the parameters of interest. A statistical procedure of filtering (of the latent volatility process) and estimation (of its parameters) is shown to be strongly consistent and asymptotically normal.

Original languageEnglish (US)
Pages (from-to)279-302
Number of pages24
JournalMathematical Finance
Issue number3
StatePublished - Jul 1996


  • Black and Scholes implied volatility
  • EM algorithm
  • Options
  • Stochastic volatility models

ASJC Scopus subject areas

  • Accounting
  • Finance
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Applied Mathematics


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