Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs

We present a novel penalty approach to the Hamilton-Jacobi-Bellman (HJB) equation arising from the valuation of European options with proportional transaction costs. We first approximate the HJB equation by a quasilinear 2nd-order partial differential equation containing two linear penalty terms with penalty parameters λ1 and λ2 respectively. Then, we show that there exists a unique viscosity solution to the penalized equation. Finally, we prove that, when both λ1 and λ2 approach infinity, the viscosity solution to the penalized equation converges to that of the corresponding original HJB equation.