Applied Modeling Techniques and Data Analysis 2

and Theorem 2.1 follows from [2.2] and [2.5].

PROOF OF THEOREM 2.2.- Let n ≥ 1, and let h be an integer with 1 ≤ h ≤ n . The Bell polynomials are defined by Pagliarani and Pascucci (2017) in Equation E.5

where the sum is taken over all sequences { ji : 1 ≤ i ≤ n – h + 1 } of non-negative integers such that

Let u BS( σ; τ,x,k ) be the Black-Scholes price [2.1]. Pagliarani and Pascucci (2017, Equation 3.15) has the form

[2.6]

For the sake of simplicity, we have omitted the last three arguments of the function u BSand all arguments of the functions and .

To define , consider the differential operator

where

[2.7]

and

are the terms of the Taylor expansions of the functions aij( z ) and ai( z ) around the point .

Following Pagliarani and Pascucci (2017), define the vector by

the matrix by

and the operator by

[2.8]

Define the set In,h by

and the operator as the differential operator acting on the z-variable and defined by (Pagliarani and Pascucci 2017, Equation D.2) as

The function in equation [2.6]is defined by (Pagliarani and Pascucci 2017, Equation D.1)

[2.9]

Here, we wrote all the arguments of the function to show that it

does not depend on y and z .

Note that a 11( z) = -a 1( z). It follows that

and equation [2.9]can be written as

where the operator is given by (Lorig et al . 2017, Equation 3.14) as

[2.10]

It is well-known that

The first term on the right-hand side of equation [2.6]takes the form of (Lorig et al . 2017, Equation 3.13)

It follows that there exist functions such that

(see Lorig et al . (2017), Equation 3.15). This is because the function does not depend on y and z .