Let's Be Rational

Abstract

Improving on [Jäc06] and [Vog07], we show how Black's volatility can be implied from option prices with as little as two iterations to maximum attainable precision on standard (64-bit floating point) hardware for all possible inputs. The method is based on four rational function branches for the initial guess adapted to the log-moneyness x, two of which are combined with nonlinear transformations of the input price, and the use of the convergence order four Householder method which comprises a rational function of the residual. Despite sounding difficult, the method is simple in practice, and a reference implementation is provided in [Jäc13]. As was perhaps previously underestimated, of crucial importance for the precision of the implied volatility is a highly accurate Black function that minimizes round-off errors and numerical truncations in the various parameter limits. We implement the Black call option price with the aid of Cody's [Cod69, Cod90] rational approximation for the complementary error function erfc(.) and its little known cousin, the scaled complementary error function erfcx(.). The source code of the reference implementation is available at www.jaeckel.org/LetsBeRational.7z. Keywordsimplied volatility, rational functions, normalized Black function Introd uctionIn [Jäc06], we provided a robust and comparatively efficient method to imply the volatility ‚ from the undiscounted price p of an option via the Black formula( 1.1) where =1 for call options and = -1 for put options. The starting point was gaining a fundamental understanding of the difficulties involved, which lie largely in the fact that the Black function permits no Taylor expansion around ‚ = 0 when F ≠ K. Based on the asymptotics of the Black function for small and large values of ‚, the key components of the method published in 'By Implication' were:• Separate the input price domain into a lower and an upper half at the point of inflexion of B over ‚. • In the upper half, estimate an initial guess on a functional form that is essentially a linear rescaling of the asymptotics for large ‚. • In the lower half, estimate an initial guess based on a geometric interpolation between the initial guess function for the upper half and a functional form that, in a certain sense, dominates the asymptotics for small ‚.• Define an objective function based on the price error in the upper half and based on the reciprocal of the logarithm of the price in the lower half.• Invoke an iteration procedure known as Halley's method of convergence order three on the respective objective function.It was clear in [Jäc06] that this can, for very low input prices p, still lead to the requirement for a significant number of iterations.1 It has since been pointed out [Vog07] that this can be improved upon with the aid of a different functional form for the asymptotics for low ‚. In this article, we will review and refine some of the choices made in [Jäc06] to arrive at an industrial solution that for standard IEEE 754 (53-bit mantissa) floating point hardwar...

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“…As the volatility is unknown, it is found using the Black and Scholes (1973) formula; hence, it is the option‐implied volatility, the volatility implied by the Black and Scholes (1973) formula. The Jäckel (2015) algorithm has been implemented to find the implied volatility.…”

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“…Therefore, efficient volatility inversion methods has been a research topic of interest in computational finance. For the progress made in implied volatility computation under the BS model, see Li (2008); Jäckel (2015); Stefanica and Radoičić (2017); Pötz (2019).…”

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