Applied Modeling Techniques and Data Analysis 2. Группа авторов

      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 I₂,h are I₂,₁ = {(2)}, I₂,₂ = {( 1, 1)}. We have a₁₁,₂(x,y,z) = 0. It follows that equation [2.10] with n = 2 includes only summation over the set I₂,₂ and 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].

2.4. References

      Gatheral, J. (2008). Consistent modelling of SPX and VIX options. The Fifth World Congress of the Bachelier Finance Society, London.

      Latané, H.A. and Rendleman Jr., R.J. (1976). Standard deviations of stock price ratios implied in option prices. J. Finance, 31(2), 369–381.

      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.

      Pagliarani, S. and Pascucci, A. (2012). Analytical approximation of the transition density in a local volatility model. Cent. Eur. J. Math., 10(1), 250–270.

      Pagliarani, S. and Pascucci, A. (2017). The exact Taylor formula of the implied volatility. Finance Stoch., 21(3), 661–718.

      Chapter written by Mohammed ALBUHAYRI, Anatoliy MALYARENKO, Sergei SILVESTROV, Ying NI, Christopher ENGSTRǑM, Finnan TEWOLDE and Jiahui ZHANG.