An Innovation for the Local Volatility Surface

This paper proposes a new innovative method for the calibration of local volatility under Dupire's model. First, our proposed method approximates the Arrow-Debreu(AD) prices on the finite difference nodes as a low-dimensional function based on implied distribution estimation. This makes the function values from the continuous state space and the point values of the finite difference method coincide at the nodes. As a result, the instability caused by the discretization is structurally eliminated. Second, we derive three different Kolmogorov forward equations that can be used to generate the local volatility surface, and for Dupire's PDE, we can change the problem of finding the local volatility to the sole calculation of a simple linear equation. This gives us the remarkable results that are very robust, include no significant reconstruction error, and do not have any of the calculation time issues that the existing methods have. We show our researching process in detail, including the analysis of the existing articles, a detailed description of the algorithm used, and test results with figures