An Adaptive Successive Over-Relaxation Method for Computing the Black-Scholes Implied Volatility

Abstract

A new successive over-relaxation method to compute the Black-Scholes implied volatility is introduced. Properties of the new method are fully analyzed, including the well-definedness, and local and global convergence patterns. Quadratic order of convergence is achieved by either a transformation of sequence technique or dynamic relaxation. The method is further enhanced by introducing a rational approximation on initial values. Numerical implementation shows that uniformly in a very large approximation domain, the new method converges to the true implied volatility with very few iterations. Overall, the new method achieves a very good combination of efficiency, accuracy and robustness.

Keywords: Successive over-relaxation, Black-Scholes formula, Implied volatility, Convergence acceleration, Rational approximation

JEL Classification: C02, G13