An Adjoint Equation-based Method For Determining The Implied Volatility

Black-Scholes (Abbr, B-S) option pricing model is one of the most widely used models in finance. Its appearance is nothing short of a revolution in finance. Since then, many economists have done a great deal of fruitful research on the basis of the B-S model. Especially in recent years some new types of options have come into being, which show some characteristics absent in the general options and bring new challenges for option pricing methods. Therefore, to study their numerical methods has important practical significance. The most commonly used numerical methods include the lattice method, the Monte-Carlo method, the finite difference method, etc.An important parameter in the B-S model is the asset volatility, using historical volatility to replace it has a serious defect so that we need to solve the implied volatility. Under the framework of the generalized B-S model, the present thesis discusses the inverse problem:how to reconstruct the implied volatility provided that the price of the option is known. To determine the implied volatility is a typical partial differential equation (Abbr, PDE) inverse problem. Based on the Tikhonov regularization model and the total variation (Abbr, TV) regularization model for determining the implied volatility, we propose a new TV regularization model for solving the implied volatility. By deriving the corresponded adjoint equation and under the discretization of space and time, we present an adjoint equation-based method for solving the implied volatility, and determine the solution by combining the BFGS inverse Newton method. Numerical experiments show that the proposed method can generate more accurate numerical results.Option pricing is a very important and challenging task, so far, there are still lots of problems unsolved so as to deserve our further research.