Applied Modeling Techniques and Data Analysis 2

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

[2.11]

where

and where

is the m th 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

Equation [2.6]gives

Then,

and

As T → t and k → x, the second and third terms disappear. Calculating the derivative with respect to k , we obtain

and Theorem 2.2 follows.

PROOF OF THEOREM 2.3.-

Equation [2.6]takes the form

[2.13]

The sets I 2, h are I 2, 1= {(2)}, I 2, 2= {( 1, 1)}. We have a 11, 2(x,y,z) = 0. It follows that equation [2.10]with n = 2 includes only summation over the set I 2, 2and takes the form

While calculating the operator using equation [2.8], we need to calculate only the coefficients of the three partial derivatives with respect to the variable x . We obtain

The following integrals are important for calculations:

The operator takes the form

Calculation of the first term on the right-hand side of equation [2.13]using equation [2.11]may be left to the reader.

Next, we calculate the left-hand side of equation [2.12]for h = 2. Using the Hermite polynomials H0(ζ) = 1, H1 (ζ) = 2ζ and H2(ζ) = 4ζ2 - 2, we obtain

Combining everything together, we obtain the formula for

[2.14]

where the ellipsis denotes the terms satisfying the following condition: the limits of the term, its first partial derivative with respect to T and its first two partial derivatives with respect to k as ( T,k ) approaches ( t,x ) within are all equal to 0.

On the right-hand side of equation [2.14], the first term, the partial derivatives with respect to T of the second, fourth and sixth terms, the first partial derivative with respect to k of the third term, and the second partial derivative with respect to k of the fifth term give nonzero contributions to the right-hand side of the asymptotic expansion [2.4].

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Lorig, M., Pagliarani, S., Pascucci, A. (2017). Explicit implied volatilities for multifactor local-stochastic volatility models. Math. Finance , 27(3), 926–960.

Orlando, G. and Taglialatela, G. (2017). A review on implied volatility calculation. J. Comput. Appl. Math ., 320, 202–220.