Pricing Barrier Options under Local Volatility

We study pricing under the local volatility. Our research is mainly intended for pedagogical purposes. In the first part of our work we study the local volatility modeling. We derive the local volatility formula in terms of the European call prices and in terms of the market implied volatilities. We propose and calibrate to the DAX option data a functional form for the implied volatility which simplifies pricing under the local volatility. In the second part of our work we analyze pricing of vanilla and double barrier options under the local volatility. To carry out our analysis of the pricing problem, we code three finite-difference solvers to compute vanilla and double barrier option prices using the local volatility function. At first, we verify that the local volatility solver produces vanilla prices which are exactly compatible with the Black-Scholes prices. Then we compare prices of double barrier options which are computed using market implied volatility and the corresponding local volatility. We find that using the local volatility yields prices and deltas of double barriers which are considerably higher than those computed using the constant volatility.