Financial Modelling in Python

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46 Financial Modelling in Python

>>> x = numpy.zeros([ndata, 1])

>>> a = 0.25

>>> b = -0.1

>>> for i in range(ndata):

... v = generator.gauss(0, 1.0)

... x[i, 0] = v

... y[i] = a*v+b*v*v

>>> fit

fos = []

>>> fit

fos.append(linear fo())

>>> fit

fos.append(quadratic fo())

>>> coeffs = generalised

least squares fit(y, x, sig, fit fos)

>>> coeffs

array([ 0.25, -0.1 ])

4.7 QUADRATIC AND CUBIC ROOTS

It is not uncommon in finance to want to find the real roots of either a quadratic or cubic

equation. The module ppf.math.quadratic

roots provides an implementation for

finding the real roots of a quadratic. The real roots of the quadratic equation

ax

2

+bx + c = 0, a, b, c ∈ R (4.14)

exist provided b

2

− 4ac ≥ 0 and are given by the expression

r

±

=−

b

2a

±

√

b

2

−4ac

2a

. (4.15)

For numerical stability reasons one shouldn’t use the above equation to determine the real

roots. Instead it is better to calculate the roots via the relations below

q =−

1

2



b +sgn(b)



b

2

−4ac



(4.16)

r

+

=

q

a

(4.17)

r

−

=

c

q

(4.18)

and this is precisely the way the real roots are calculated in the ppf.math.quadratic

roots module, as can been seen below.

import math

def quadratic

roots(a, b, c, xl, xh):

# find roots

roots = []

d = b*b-4*a*c

if d > 0:

r1 = 0

r2 = 0

if a <>0:

sgn = 1

if b < 0: sgn = -1

q = -0.5*(b+sgn*math.sqrt(d))

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Basic Mathematical Tools ...