In this study, we develop a calibration method of the temporally varying volatility and interest rate functions using the Black–Scholes (BS) partial differential equation and the observed market option prices with different strikes and expiries. The proposed method uses the piecewise linear interpolations between data points which are defined at the middle points of maturity dates. When we construct the volatility and interest rate, we use the exponential function so that the interpolated values are always positive. Numerical experiments with synthetic and real market data demonstrate the superior performance of the proposed method.

中文翻译：

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校准随时间变化的波动率和利率函数

在这项研究中，我们使用 Black-Scholes (BS) 偏微分方程和观察到的具有不同行使价和到期日的市场期权价格开发了一种随时间变化的波动率和利率函数的校准方法。所提出的方法使用在到期日中间点定义的数据点之间的分段线性插值。当我们构造波动率和利率时，我们使用指数函数，因此插值始终为正。合成和真实市场数据的数值实验证明了所提出方法的优越性能。