OPTION HEDGING AND IMPLIED VOLATILITIES IN A STOCHASTIC VOLATILITY MODEL1

Eric Renault and Nizar Touzi

Mathematical Finance, 1996, vol. 6, issue 3, 279-302

Abstract: 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 variation m 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.

Date: 1996
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