Never have looked before at the pricing details. In my understanding stochastic volatility should imply stochastic BS pricing. In other words regardless of a particular distribution of the [$]\sigma ( * , \omega )[$] any path of the sigma leads to unique BS price of the option. Hence I thought that option price under a SV model should be a statistics of
[$]C ( t , S ( t ) ; T , K ; \sigma ( * , \omega )[$] (1)
ie it should be mean of (1) or something like adjustment of this mean. Note that in this case we should arrive at explicit market risk implied by the SV model. Such risk will represent overvalued / undervalued option price with respect to real world realization of the [$]\sigma ( * , \omega )[$]
I have looked at some papers and discovered somewhat different approaches. I will thankful for comments or a reference with primary idea of option valuation under SV.
In particular I looked through 1 ch Gatheral book and found his derivation of the 'BSE' with reference on (1998) Wilmott paper. The derivation somewhat confused me. a) we assume that market is underlying with SV, option contract and bond. There is no other assets. In derivation it was used undefined asset [$] V_1 [$] for hedged portfolio. It was shown that option pricing eq does not depend on this auxiliary asset, ie it's existence does not actually needed for pricing but it is used to eliminate stock risk term dS and volatility risk term dv in derivation of the 'BSE'. How can one use undefined function to eliminate some terms? It might be I missed here something?