Medical, Pharma, Engineering, Science, Technology and Business

^{1}Algeria Mathematical Department, EPST School of Algiers, Algeria

^{2}Mathematical Faculty Algiers, Algeria

- *Corresponding Author:
- Malki Abderrahmane

Algeria Mathematical Department

EPST School of Algiers, Algeria

**Tel:**+21321621841

**E-mail:**[email protected]

**Received Date:** December 26, 2016; **Accepted Date:** June 22, 2017; **Published Date:** June 30, 2017

**Citation: **Abderrahmane M, Kamel B (2017) A New Approximation to Standard
Normal Distribution Function. J Appl Computat Math 6: 351. doi: 10.4172/2168-9679.1000351

**Copyright:** © 2017 Abderrahmane M, et al. This is an open-access article
distributed under the terms of the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided
the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

This paper, presents three news-improved approximations to the Cumulative Distribution Function (C.D.F.). The first approximation improves the accuracy of approximation given by Polya (1945). In this first new approximation, we reduce the maximum absolute error (MAE) from0.000314 to 0.00103. For this first new approximation, Aludaat and Alodat were reduce the (MAE) from 0.000314 to 0.001972. The second new approximation improve Tocher’s approximation, we reduce the (MAE) from, 0.166 to 0.00577. For the third new approximation, we combined the two previous approximations. Hence, this combined approximation is more accurate and its inverse is hard to calculate. This third approximation reduces the (MAE) to be less than 2.232e-004. The two improved previous approximations are less accurate, but his inverse is easy to calculate. Finally, we give an application to the third approximation for pricing a European Call using Black-Scholes Model.

Cumulative distribution function; Normal distribution; Maximum absolute error

The cumulative distribution function (CDF) plays an important role in financial mathematics and especially in pricing options with Black-Scholes Model. The European option pricing call given by Black- Scholes Model is

(1)

(2)

S, the current price, K the exercise price, r interest rate, T time option and σ volatility [1-8]. The cumulative distribution function (CDF) is

(3)

The (CDF) has not a closed form. His evaluation is an expensive task. For evaluate the (CDF) at a point z we need compute the integral under the probability density function (PDF) given by .

In much research, we find approximations, with closed forms, for the area under the standard normal curve. Otherwise, we need consulting Tables of cumulative standard normal probabilities. Hence, in the literature, we find several approximations to this function from Polya [7] to Yerukala et al. [9]. For this raison, we use some approximations to this CDF. (Polya’s approximation and Tocher’s approximation) [10-11].

We consider the case of *z* ≥ 0. (For *z* ≥ 0, *N* (*z*) =1− *N* (−*z*)).

The original Polya’s approximation given by:

(2.1)

The Maximum absolute Error (MAE)

(2.2)

*Aludaat K.M and Alodat* M.T [1] proposed the same formula with

In this paper, we write the formula (2.1) and (2.2) as

(2.3)

Hence, we search the parameters a,b and c that

(2.4)

Was the smallest possible using the following algorithm?

Repeat 3) to 6) until convergence

Using our algorithm, we find the best parameters

a^{*}=0.50103;b^{*}=0.49794;c^{*}=0.62632 (2.4)

Hence the best formula is

(2.5)

Note that the absolute error as function of z variable noted by

shows the graph of Absolute Error for Polya, Aludaat and Malki as
function of -5 ≤ z ≤ -5, (**Figure 1**)

The Original Tocher approximation is

(3.1)

These approximations have the form (3.2)

Hence, we search the parameters a, b and c that

(3.3)

Was the smallest possible using the following algorithm?

Repeat (3) to( 6) until convergence

Using our algorithm, we find the best parameters

a^{*}=0.97186; b^{*}=0.97186; c^{*}=1.69075 (3.4)

Hence the best-improved formula for Tocher’s approximation is

(3.5)

(3.6)

Comparison of Absolute Error for Original Tocher, Modified
Tocher and Malki 2 as function of z variable (-5 ≤ z ≤ 5) (**Figure 2**).

As the third new approximation formula, we consider the two previous formula

Hence, we consider the third new formula as

(4.1)

We search the optimum parameter *ω* that the

Was the smallest possible. We find optimum parameter *ω*^{*} = 0.16

The new third approximation is

(4.2)

The adjusted formula is

(4.3)

For, this approximation we have:

(4.4) (**Figure 3**).

For

*S* = 35;*K* = 30;*r* = 0.065;*T* =1.2;σ = 0.35; (5.1)

To calculate a Call European option we compute

(5.2)

And (5.3)

Hence

(5.4)

Using, *N _{Malki3}* we have

(5.5)

The absolute error is (5.6)

We have proposed three approximations to the cumulative distribution function of the standard normal distribution. The first approximation improves the Polya’s formula in accuracy. The second new approximation improve the accuracy of Tocher’s formula. The third formula is a combination of the two previous formula. The MAE for the first approximation is 0.00103. The MAE for the second approximation is 0.00577. For the third approximation the MAE is less than 2.232e−004. Finally, we insert an application to option pricing of a Call European option based on Black-Scholes formula.

- Aludaat KM, Alodat MT (2008) A Note on Approximating the Normal Distribution Function. App Math Sci 2: 425-429
- Choudhury A, Ray S, Sarkar P (2007) Approximating the Cumulative Distribution Function of the Normal Distribution. J Sta Res 41: 59-67.
- Choudhury A (2014) A Simple Approximation to the Area under Standard Normal Curve Math Stat 2: 147-149.
- Hammakar HC (1978) Approximating the Cumulative Normal Distribution and its Inverse. App Stat 27:76-77
- Hart RG (1957) A Formula for the Approximation of Definite Integrals of the Normal Distribution Function Math Tables Aids Com 11
- Lin JT (1990) A Simpler Logistic Approximation to the Normal Tail Probability and its Inverse. Appl Statist 39: 255-257
- Polya G (1945) Remarks on computing the probability integral in one and two dimensions. Proce first Berk Symp Math Stat Prob, pp: 63-78
- Waissi GR, Rossin DF (1996) A Sigmoid Approximation of the Standard Normal Integral. App Math Compu 77: 91-95
- Yerukala R, Boiroju NK, Reddy KM (2015) Approximations to Standard Normal Distribution Function Intern. J Scie Eng Res 2
- Lin JT (1990) A Simpler Logistic Approximation to the Normal Tail Probability and its Inverse. Appl Statist 39: 255-257
- Tocher KD (1963) The Art of Simulation. Eng Univ, pp: 184

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