New approximations for standard normal distribution function

Eidous, Omar; Abu-Shareefa, Rima

doi:10.1080/03610926.2018.1563166

Cited by 17 publications

(11 citation statements)

References 28 publications

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“…As previously mentioned, there is a large number of approximations for the standard normal distribution or, equivalently, for the error function. In Table 8, we compare our approximations with Bowling et al (2009) [which is equivalent to Page (1977)], and Eidous and Abu-Shareefa (2020). As shown, the approximations are very close and has the smallest RMSE.…”

Section: Call When S = Xmentioning

confidence: 90%

“…Alternatively N can be approximated with suitable functions, generally of rational form. In this framework, there is a variety of contributions, see for example Bowling et al (2009), Choudhury (2014) and Eidous and Abu-Shareefa (2020) who focus on the choice of the best coefficients. Hofstetter and Selby (2001) obtained approximations by replacing the cndf with the logistic distribution, Li (2008) developed an heuristic closed-form method based on the rational functions which requires one or two steps of Newton-Raphson algorithm to improve the accuracy.…”

Section: N (X)mentioning

confidence: 99%

“…Hence, in order to approximate the call function, we have to approximate the function N (x) or equivalently erf(x). There are a lot of approximated formulae for N (x) as well as for erf(x), see Choudhury (2014), Eidous and Abu-Shareefa (2020) and the references therein. However, not all of these formulas are useful for our purpose, since the inverse is not easy to compute; (e.g.…”

Section: Approximating the Call Function When S = Xmentioning

confidence: 99%

See 2 more Smart Citations

Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

Mininni

Orlando

Taglialatela

2020

Decisions Econ Finan

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In this paper, we introduce the concept of standardized call function and we obtain a new approximating formula for the Black and Scholes call function through the hyperbolic tangent. Differently from other solutions proposed in the literature, this formula is invertible; hence, it is useful for pricing and risk management as well as for extracting the implied volatility from quoted options. The latter is of particular importance since it indicates the risk of the underlying and it is the main component of the option’s price. That is what trading desks focus on. Further we estimate numerically the approximating error of the suggested solution and, by comparing our results in computing the implied volatility with the most common methods available in the literature, we discuss the challenges of this approach.

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Section: Call When S = Xmentioning

confidence: 90%

Section: N (X)mentioning

confidence: 99%

Section: Approximating the Call Function When S = Xmentioning

confidence: 99%

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Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

Mininni

Orlando

Taglialatela

2020

Decisions Econ Finan

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“…Due to the relationship between Φ(𝑥) and the above functions along with the Mills' ratio (Fan, 2013), it makes sense that the results would find application in the broader fields such as statistical computations, applied statistics, mathematical models in biology, mathematical physics and diffusion theory (see Bercu, 2020 andLipoth et al, 2022). Eidous and Abu-Shareefa (2020) have reviewed 45 approximations for Φ(𝑥) reported in literature from 1945 to 2019. They also provided nine accurate approximations for Φ(𝑥).…”

Section: Introductionmentioning

confidence: 99%

Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions

Eidous

2022

Preprint

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Although there is extensive literature on the upper bound for cumulative standard normal distribution Φ(χ), there are relatively not sharp for all values of the interested argument χ. The aim of this paper is to establish a sharp upper bound for Φ(χ), in the sense that its maximum absolute difference from Φ(χ) is less than 5.785 x10-5 for all values of χ ≥ 0. The established bound improves the well-known Polya upper bound and it can be used as an approximation for Φ(χ) itself with very satisfactory accuracy. Numerical comparisons between the proposed upper bound and some other existing upper bounds have been achieved, which shows that the proposed bound is tighter than alternative bounds found in the literature.

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“…While the above approximations used one term in to approximate ( ), there are many others approximations -in the literature-for ( ) with more than one term and give more accurate. See for example Eidous and Abu-Shareefa [7] at which most of these approximations have been collected.…”

Section: Introductionmentioning

confidence: 99%

A Simple Approximation for Normal Distribution Function

Edous¹,

Eidous²

2018

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This paper proposes an approximation to the standard normal distribution function. The introduced approximation formula is very simple and it has a very acceptable accurate. By comparing the proposed approximation with other existing approximations, it can be observed that the proposed one has a simple, easily computable formula and it gives a good accurate with maximum absolute error equals 0.000444.

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New approximations for standard normal distribution function

Cited by 17 publications

References 28 publications

Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions

A Simple Approximation for Normal Distribution Function

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