Recovery of the local volatility function using regularization and a gradient projection method.

*(English)*Zbl 1305.91241Summary: 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 |

##### Keywords:

volatility function; linearization; regularization; projective gradient methods; European option pricing problem
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\textit{Q. Ma} et al., J. Ind. Manag. Optim. 11, No. 2, 421--437 (2015; Zbl 1305.91241)

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