Approximation to portfolio liquidation value with calculation of its skewness

A boundary problem for a homogeneous multidimensional diffusion process is considered under the assumption of small perturbations. Approximations to the mean, second and third central moments of the process at the moment of the first crossing a given plane in phase space are presented as a solution of ordinary differential equations with an additional transformation (“projection onto the boundary”). The quantile of a linear combination of coordinates is estimated by the second order expansion in powers of a small parameter determining the magnitude of perturbations. In the first approximation, this expansion corresponds to the Gaussian distribution, the next term contains skewness. The result is extended to a process with multiple boundaries, upon reaching each of them the equation of the process changes. Such a model describes the liquidation of a portfolio of financial instruments in which the closing rate of each of the positions is a random process. The result is illustrated by two examples. In the first example a portfolio consists of linear instruments (such as stocks, futures), prices are correlated Geometric Brownian Motions with zero drift. The closing rates are constant, but with random noise due to daily fluctuations in trading volume. In this particular case approximations for the mean, variance, skewness, VaR and CVaR of the financial result of portfolio liquidation are given explicitly. In the second example, the liquidation of an exchange-traded option position is considered under the assumption that the closing rate depends on the ratio of the underlying price to the strike of the option. Numerical calculations demonstrate that taking into account skewness significantly increases the accuracy of estimates.