  The third moment variation of a financial asset return process is defined by
the quadratic covariation between the return and square return processes. The
skew and fat tail risk of an underlying asset can be hedged using a third
moment variation swap under which a predetermined fixed leg and the floating
leg of the realized third moment variation are exchanged. The probability
density function of the hedged portfolio with the third moment variation swap
was examined using a partial differential equation approach. An alternating
direction implicit method was used for numerical analysis of the partial
differential equation. Under the stochastic volatility and jump diffusion
stochastic volatility models, the distributions of the hedged portfolio return
are symmetric and have more Gaussian-like thin-tails.
