Closed-Form Approximate Inversion of the Black-Scholes Formula

M. Li (USA)


BlackScholes Formula, Closedform Inversion, Rational Approximation, Implied Volatility, Dimensional Reduction


The Black-Scholes formula is arguably the most frequently used formula in finance. In practice, it is often used in the backward direction to invert the implied volatility from observed option prices, usually with some solver method. Solver methods, being aesthetically unappealing, are also slower than closed-form approximations. However, closed form approximations in previous works lack accuracy, of ten providing option pricing errors well exceeding the bid ask spreads. We develop a new closed-form method based on the rational approximation. The rational approximation is much faster than typical solver methods and very accu rate for both at-the-money and away-from-the-money op tions.

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