# Optimal Hedging of Options - asymmetry between long and short vol positions

Going over Zakamouline's Approximation method for optimal delta hedging of options, it is claimed that the result remains valid for both buying options (long vol positions) or selling options (short vol positions). Furthermore, when the final delta bands are plotted for these two different cases, they are clearly different charts (see image):

However, mathematically looking at the result, I can't see why the bands will be different. My reasoning is that the main difference between both positions are the sign of the Gamma, since all the other inputs are the same. However, the results of Zakamouline, seem to only depend on the absolute value of the Gamma:

All images were used from Euan Sinclair's book. The Zamakouline papers cited in the book are:

Optimal Hedging of Options with Transaction Costs

Efficient Analytic Approximation of the Optimal Hedging Strategy for a European Call Option with Transaction Costs

European Option Pricing and Hedging with both Fixed and Proportional Transaction Costs

My question is: why are the delta bands different for long/short option positions, when looking at the mathematical equations above? Thanks in advance!

$$\sigma_m^2 = \sigma^2(1+K)$$
$$\sigma_m^2 = \sigma^2(1-K)$$