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In a realistic market with transaction costs, the option pricing problem is known to lead to solving nonlinear partial differential equations even in the simplest model. The nonlinear term in these partial differential equations (PDE) reflects the presence of transaction costs. In this article we consider an underlying general stochastic volatility model. In this case the market is incomplete and the option price is not unique. Under a particular market completion assumption where we use a traded proxy for the volatility, we obtain a non-linear PDE whose solution provides the option price in the presence of transaction costs. This PDE is studied and under suitable regularity conditions, we prove the existence of strong solutions of the problem.