Efficiency and Convergence of Bisection, Secant, and Newton Raphson Methods in Estimating Implied Volatility

This study aims to estimate volatility prices based on the black-Scholes model (BSM) function with research data taken during the COVID-19 pandemic. The estimates of the volatility values are obtained by using three numerical methods, namely the bisection, secant, and Newton Raphson methods. The numerical processes that produce some iteration results in the three methods are then analyzed and the best convergence is sought. As a result, Newton Raphson method produces the smallest number of iterations, which stops at the 3rd iteration and gets a volatility value of 0.500451 with an absolute error value of 0.000388. However, the method requires an initial approximation which lies only in two intervals on the axis σ which are close to the true root. Meanwhile, for the other two methods, namely Bisection and Secant, this limitation does not apply, as long as there is an interval that guarantees the existence of roots. In this case, bisection method requires11 iterations to converge with volatility value of 0.500342 and error value of 0.000878. Whereas secant method requires 4 iterations to converge with a volatility value of 0.500449 and error value of 1.68938E-05. This suggests, that in some cases the use of Newton method, should be initialized with the use of bisection or secant method, to ensure successful iteration and accelerate the rate of convergence.