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Recovery of the local volatility function using regularization and a gradient projection method. (English) Zbl 1305.91241
Summary: This paper considers the problem of calibrating the volatility function using regularization technique and the gradient projection method from given option price data. It is an ill-posed problem because of at least one of three well-posed conditions violating. We start with the European option pricing problem. We formulate the problem by obtaining the integral equation from Dupire equation and provide a theory of identifying the local volatility function $$\sigma(y,\tau)$$ when the parameter $$\mu\neq 0$$, and then we apply regularization technique for volatility function retrieval problems. A projected gradient method is developed for recovering the volatility function. Numerical simulations are given to illustrate the feasibility of our method.
MSC:
 91G60 Numerical methods (including Monte Carlo methods) 91G20 Derivative securities (option pricing, hedging, etc.) 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
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