Hi everyone,I have seen in several papers the following computation for the delta of an option O when the volatility is stochastic :with O^BS the Black Scholes price of the option (when the vol. is constant).What's the justification of this formula ? I don't understand why we can't write : ??

This paper may interest you Delta hedging the vega riskIn this paper, the guy considers two deltas: the implied delta and the local deltaThe implied delta (or BS delta)=d(BS(t,S,Implied_vol(T,K)))/dS where BS is the price computed with the BS formula for your implied vol.The local delta is d(Model(t,S,local_vol(T,K)))/dS where Model is the price computed by your model with the local vol calibratedwith the implied volatility surfaceSince the two prices must be equal by definition BS(t,S,Implied_vol(T,K))=Model(t,S,local_vol(T,K))you will have Local_delta=BS_delta+BS_vega*d(Implied_vol)/dSThink about this formula it's logic: your local delta=variation of the option price/dS=(BS_delta*dS+BS_vega*dsigma)/dS because a change in dS implied a change in volatility.The hard point is to estimate d(Implied_vol)/dS For example an approximation is to replace d(Implied_vol)/dS by d(Implied_vol)/dK (the skew of your implied vol surface).The book of Gatheral and the papers of Derman are must read in this domain. I agree that's a very tricky concept.

QuoteOriginally posted by: frenchXSince the two prices must be equal by definition BS(t,S,Implied_vol(T,K))=Model(t,S,local_vol(T,K))small detail: i think you mean / should write BS(t,S,K,Implied_vol(T,K))=Model(t,S,K,local_vol(S,t))local vol doesn't depend on strike.