Applied Modeling Techniques and Data Analysis 2. Группа авторов

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^BS and 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 a₁₁(z) = -a₁(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.

      From Lorig et al. (2017), Lemma 3.4, we have

[2.11]

      where

      and where

      is the mth Hermite polynomial.

      We must still calculate the expression in the third line of equation [2.6] for h ≥ 2 (it is equal to 1 when h = 1). From Lorig et al. (2017), Proposition 3.5, we have

      [2.12]

      where the coefficients ch,h−2q are defined recursively by

Using equations [2.7] and [2.10], we explicitly calculate:

      and

      where the dots denote the terms containing and ∙ The functions take the form

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