Direct Fit for SVI Implied Volatilities

Abstract

The stochastic volatility inspired (SVI) formula is one of the mainstream models for fitting the option implied volatility smile. Herein I fully linearize the SVI equation by rewriting it into the algebraic form of a conic section, limited to the geometric shape of a hyperbola. This step reduces the complexity of the otherwise non-linear optimization problem significantly. Based on the conic representation, I introduce the direct least-squares for SVI, allowing us to fit the model in a computationally efficient and non-iterative manner. The performance of the proposed method is evaluated upon empirical data of seven different asset classes. It turns out to deliver a very good fit, and is about 25 times faster than the existing 'quasi-explicit' benchmark algorithm. Following the outstanding computational speed combined with the high accuracy, the direct SVI fit qualifies as a robust method for calibrating implied volatilities in real-time, and for applications to big option datasets.

Keywords: implied volatility, volatility smile, calibration, SVI, hyperbola, high-frequency

JEL Classification: C58, G12, C61