Financial Modelling in Python

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c04 JWBK378-Fletcher May 23, 2009 4:21 Printer: Yet to come

Basic Mathematical Tools 49

if biga <> 0: bigb = q/biga

r1 = (biga+bigb)-a/3.0

if r1 >= xl and r1 <= xh:

roots.append(r1)

return roots

else:

return quadratic

roots(b, c, d, xl, xh)

Finally, note that in the actual implementations we only return the real roots, quadratic or

cubic, if they lie in the range [x

l

, x

h

]. Moreover we always sort the real roots. The reason for

doing this will become clear in the next section when we apply the above algorithms in the

context of integrating a polynomial.

4.8 INTEGRATION

In finance we often need to calculate the expectation of some function f : R

n

→ R of a

number of random variables X :  → R

n

. Throughout this section we will only consider

financial payoffs that can be written in terms of a single random variable X :  → R and

belong to the space C

3

(R), that is the space of continuous three-times differentiable functions

on R. But the following can be extended to higher dimensions with more effort.

4.8.1 Piecewise Constant Polynomial Fitting

Let X denote a random variable on R which we sample on a uniform lattice {x

1

, x

2

, ..., x

N

}

with spacing . The corresponding values of the function f on the uniform lattice are denoted

by {f

1

, f

2

, ..., f

N

}. Similarly the first, second and third derivatives of f at any node of the

lattice x

i

are denoted by f

i



, f

i



and f

i



respectively. Our aim is to fit the function ...