  In the present paper, a decomposition formula for the call price due to
Al\`{o}s is transformed into a Taylor type formula containing an infinite
series with stochastic terms. The new decomposition may be considered as an
alternative to the decomposition of the call price found in a recent paper of
Al\`{o}s, Gatheral and Radoi\v{c}i\'{c}. We use the new decomposition to obtain
various approximations to the call price in the Heston model with sharper
estimates of the error term than in the previously known approximations. One of
the formulas obtained in the present paper has five significant terms and an
error estimate of the form $O(\nu^{3}(\left|\rho\right|+\nu))$, where $\nu$ is
the vol-vol parameter, and $\rho$ is the correlation coefficient between the
price and the volatility in the Heston model. Another approximation formula
contains seven more terms and the error estimate is of the form
$O(\nu^4(1+|\rho|)$. For the uncorrelated Hestom model ($\rho=0$), we obtain a
formula with four significant terms and an error estimate $O(\nu^6)$. Numerical
experiments show that the new approximations to the call price perform
especially well in the high volatility mode.
