Applied Modeling Techniques and Data Analysis 2. Группа авторов
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where
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 uBS (σ; τ,x,k) be the Black-Scholes price [2.1]. Pagliarani and Pascucci (2017, Equation 3.15) has the form
For the sake of simplicity, we have omitted the last three arguments of the function uBS and all arguments of the functions
To define
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
the matrix
and the operator
[2.8]
Define the set In,h by
and the operator
The function
[2.9]
Here, we wrote all the arguments of the function
does not depend on y and z.
Note that a11(z) = -a1(z). It follows that
and equation [2.9] can be written as
where the operator
[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
(see Lorig et al. (2017), Equation 3.15). This is because the function
From Lorig et al. (2017), Lemma 3.4, we have
[2.11]
where
and where
is the mth Hermite polynomial.