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How to calculate natural log?in option pricing model - black

Anoop kumar (Article) (570 Points)

24 October 2012

How to calculate natural Log?in option pricing model - Black Scholes Model

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Tehsinkhan Pathan (CA CMA DISA(ICAI) B.COM) (3956 Points)
Replied 24 October 2012

There are Three methods of calculation of value of Natural Log :-
(a) Using Natural log table
(b) Using ex/e-x table
(c) Using ordinary log tables

I Method: Using Natural log tables

Example 1 : Find the natural log of (a) 0.75 (b) 1.24 (c) 1.0379 (d) 1.2667
Answer
(a) ln(0.75) = -0.2877

(b) ln(1.24) = 0.2151

(c) ln(1.0379) =?
ln(1.03) = 0.0296
ln(1.04) = 0.0392
For LHS diff. of 0.01, RHS diff. is 0.0096
For LHS diff. of 1, RHS diff. is 0.96
For LHS diff. of 0.0079, RHS diff. is:
(0.96)x 0.0079 i.e. 0.0076
ln (1.0379) = 0.0296 +0.0076 = 0.03702

(d) ln(1.2667) =?
ln(1.26) = 0.2311
ln(1.27) = 0.2390
For LHS diff. of 0.01, RHS diff. is 0.0079
For LHS diff. of 1, RHS diff. is 0.79
For LHS diff. of 0.0067, RHS diff. is :
(0.79)x 0.0067 i.e. 0.005293
ln (1.2667) = 0.2311 + 0.005293 = 0.236393

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Tehsinkhan Pathan (CA CMA DISA(ICAI) B.COM) (3956 Points)
Replied 24 October 2012

II Method: Using ex/e-x table

Let ex = y
then, ln(y) =x

Example 2 :
Find natural log of (i) 0.2567(ii) 0.3570 (iii)0.2466 (iv) 0.1979 and (v) 4.0960
Answers
(i) Consulting the table of values of ex/ e-x, we find :
0.2567 = e-1.36
Hence, natural log of 0.2567 = -1.36

(ii) Consulting the table of values of ex/e-x, we find :
0.3570 = e-1.03
Hence, natural log of 0.3570 = -1.03

(iii) Consulting the table of values of ex/e-x, we find :
0.2466 = e-1.40
Hence, natural log of 0.2466 = -1.40

(iv) Consulting the table of values of ex/e-x, we find :
0.1979 = e-1.62
Hence, natural log of 0.1979 = -1.62

(v) Consulting the table of values of ex/e-x, we find :
4.0960 = e1.41
Hence, natural log of 4.0960 = 1.41

Example 3 : Find the natural log of (a) 0.75 (b) 1.24 using ex/e-x table
Answer
(a) ln(0.75) = ?
e-0.28 = 0.7558
e-0.29 = 0.7483
0.7558 = e-0.28
0.7483 = e-0.29
For 0.0075 ↓ in LHS, e’s power ↓ by 0.01
For 1↓ in LHS, e’s power ↓ by 0.01/0.0075
For 0.0058 ↓ in LHS, e’s power ↓ by (0.01/0.0075)x0.0058 i.e. by 0.0077
= 0.7558 – 0.0058 = e-0.28-0.0077
0.75 = e-0.287686
ln 0.75 = -0.287686

(b) ln(1.24) = ?
e0.21 = 1.2337
e0.22 = 1.2461
1.2337 = e0.21
1.2461 = e0.22
For 0.0124 ↑ in LHS, e’s power ↑ by 0.01
For 1 ↑ in RHS, e’s power ↑ by 0.01/0.0124
For 0.0063 ↑ in RHS, e’s power ↑ by
(0.01/0.0124)x0.0063 i.e. 0.0050806
1.2337 +0.0063 = e0.21+0.0050806
= 1.24 = e0.215097
ln 1.24 = 0.215097