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Numerical Computation of Implied Volatility Using Root-Finding Methods for Black-Scholes Option Pricing

Author(s)

Xinzi Li

Affiliation(s)

Xi'an Jiaotong-Liverpool University, School of Mathematics and Physics, Suzhou, Jiangsu, China

Abstract

Implied volatility is a crucial parameter in option pricing,which reflects the predicted trend in future asset market and includes more effective information compared with historical volatility(Jiang and Tian, 2005, cited in Shao, Zhou and Gong, 2025). With simplifying assumptions, Black-Scholes(B-S) method(Black and Scholes, 1973)are widely used to model implied volatility indirectly(Zhou and Gong, 2025). This article investigates three numerical root-finding methods—bisection, Newton's, and secant—for directly calculating implied volatility from the B-S model given market option prices. Through a case study of a European call option (S=K=100, T=1 year, r=5%, C_market=10), we rigorously compare the bisection, Newton's, and secant methods in MATLAB, evaluating convergence speed (iterations), computational time, and accuracy.Our results demonstrate that while Newton's method offers the fastest convergence (4 iterations), the bisection method provides the most robust solution, with all methods achieving high accuracy(|f(σ)| < 10^-6). These findings provide practical guidance for financial analysts implementing volatility estimation in trading systems and risk management applications.

Keywords

Implied Volatility; Black-Scholes Model; Root-Finding Methods; Numerical Analysis; Option Pricing

References

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