# American Institute of Mathematical Sciences

April  2015, 11(2): 421-437. doi: 10.3934/jimo.2015.11.421

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

 1 College of Applied Arts and Science of Beijing Union University, Beijing 100191, China 2 Renmin University of China, Beijing 100872, China 3 Hebei Normal University, Shijiazhuang 050024, China

Received  February 2013 Revised  April 2014 Published  September 2014

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.
Citation: Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
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