박판 스플라인 함수를 이용한 내재변동성 곡면 생성 방법 연구

In this article, we work on the construction of an implied volatility surface by using radial basis functions (RBFs) based on implied volatility data in equity index markets including KOSPI200, HSCEI, Eurostoxx50. Construction of the implied volatility surface is the ‘missing data problem’ in Statistics. The choice of a 2 dimensional interpolation function for surface construction and the data filtering are important. To solve such problems in Statistics, RBF is used popularly. We first explain multi regression method known for practitioner’s implied volatility surface, and suggest an estimation method using RBF. In particular, we review merits and demerits of the Gaussian RBF, multiquadric RBF, and thin plate spline(TPS) RBF, and study concrete methods for the construction of implied volatility surfaces using these RBFs. After finding volatility data which are seriously different from their neighbors, we remove them. To fix an incomplete shape of the surface caused by missing data, we provide an adapted fitting procedure using the leave one out cross validation (LOOCV) method. Finally, using those we provide an optimal approximation algorithm to the implied volatility surface. In the construction of the implied volatility surface, the most important issue is the inference of missing data. In this work, we show that it is most efficient to estimate the implied volatility from the variance surface obtain by using TPS approximation function. Another important issue in the implied volatility construction is the satisfaction of the no-arbitrage condition (NAC) by volatility data. By applying the quadratic programming with NAC to the TPS-smoothed implied volatility, we solve this problem.