Stable local volatility function calibration using spline kernel

• Published in: Computational optimization and applications Vol. 55; no. 3; pp. 675 - 702
• Main Authors: Coleman, Thomas F., Li, Yuying, Wang, Cheng
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.07.2013; Springer Nature B.V
• Subjects: Accuracy; Calibration; Computational mathematics; Convex and Discrete Geometry; Engineering research; Management Science; Markets; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Partial differential equations; Securities prices; Splines; Statistics; Stochastic models; Studies; Vectors (mathematics); Volatility; Spline kernel; optimization; Option pricing; Local volatility function; Calibration; L; Trust region method
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-013-9543-x

We propose an optimization formulation using the l 1 norm to ensure accuracy and stability in calibrating a local volatility function for option pricing. Using a regularization parameter, the proposed objective function balances calibration accuracy with model complexity. Motivated by the support vector machine learning, the unknown local volatility function is represented by a spline kernel function and the model complexity is controlled by minimizing the 1-norm of the kernel coefficient vector. In the context of support vector regression for function estimation based on a finite set of observations, this corresponds to minimizing the number of support vectors for predictability. We illustrate the ability of the proposed approach to reconstruct the local volatility function in a synthetic market. In addition, based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is simple and resembles the observed implied volatility surface in shape. Stability is illustrated by calibrating local volatility functions using market option data from different dates.