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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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A New Approximation to Standard Normal Distribution Function

Abderrahmane M1* and Kamel B2

1Algeria Mathematical Department, EPST School of Algiers, Algeria

2Mathematical 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.

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Abstract

 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.

Keywords

Cumulative distribution function; Normal distribution; Maximum absolute error

Introduction

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

Equation (1)

Equation (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

Equation (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 Equation.

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].

Improving Polya’s Approximation

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

The original Polya’s approximation given by:

Equation (2.1)

The Maximum absolute Error (MAE)

Equation (2.2)

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

Equation

Equation

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

Equation (2.3)

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

Equation (2.4)

Was the smallest possible using the following algorithm?

Equation

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

Equation (2.5)

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

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

applied-computational-mathematics-comparison-absolute

Figure 1: Comparison of absolute error for Polya, Aludaat and Malki1.

Improving Tocher’s Approximation

The Original Tocher approximation is

Equation

Equation (3.1)

These approximations have the form (3.2)

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

Equation (3.3)

Was the smallest possible using the following algorithm?

Equation

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

Equation (3.5)

Equation (3.6)

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

applied-computational-mathematics-gives-curves

Figure 2: Gives the curves of original absolute error and the new absolute error.

Combined Formula

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

Equation

Hence, we consider the third new formula as

Equation (4.1)

We search the optimum parameter ω that the

Equation

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

The new third approximation is

Equation (4.2)

The adjusted formula is

Equation (4.3)

For, this approximation we have:

Equation (4.4) (Figure 3).

applied-computational-mathematics-absolute-function

Figure 3: Absolute error as function of Z variable for Malki1, Malki2 and Malki3.

Application with Black-Scholes Model

For

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

To calculate a Call European option we compute

Equation (5.2)

And Equation (5.3)

Hence

Equation (5.4)

Using, NMalki3 we have

Equation (5.5)

The absolute error is Equation (5.6)

Conclusion

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.

References

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