Given Heston model parameters calibrated from vanilla put/call options it is possible to imply a volatility surface by pricing calls or puts for different strikes and maturities and solving the inverse BS equation to find the corresponding black volatilities. Clearly, by construction, using this volatility in the BS formula for puts/calls will yield the same price as pricing the same option directly with the Heston model. To what extent does this hold for other options? My intuition tell me that for path dependent options like barrier options this would not be the same as the black vol and Heston parameters only imply the same distribution of the stock price at maturity, but not the same distributions along the path there. Would it however be the same thing for non-path dependent instruments, such as a binary call option? Meaning that you would retrieve the same price for the binary call option by using the black-vol from your Heston-implied vol surface in a BS model as you would by pricing the binary call option directly with the Hesotn model?
A different way to view this I suppose is: Would you retrieve the same volatility surface by pricing options with the Heston model and solving for the corresponding black vol regardless of the type of option, or would you get a different surface for e.g. puts/call and binary options? What about path dependent options like American or barriers?