I am learning how to price SVJ options and am reading some stuff on no-arbitrage pricing for SVJ model using the typical approach you would use (like in BSM option pricing) of creating a risk free portfolio. I understand the part on why you CAN'T use the typical approach and just create a portfolio that consists of a long option and short underlying as you still have the Poisson process that isn't hedged out.
However, why can't you create a portfolio of long option, short underlying, and short a "magic" basket of options that just happens to match the gamma and delta of your long option. When deriving the pricing equation for Heston's stochastic volatility model, we do the exact same thing by selling options to hedge the stochastic volatility process.
Why can't we do the same when trying to price options using SVJ model?