  This paper investigates the pricing and hedging of variance swaps under a
$3/2$ volatility model. Explicit pricing and hedging formulas of variance swaps
are obtained under the benchmark approach, which only requires the existence of
the num\'{e}raire portfolio. The growth optimal portfolio is the num\'{e}raire
portfolio and used as num\'{e}raire together with the real world probability
measure as pricing measure. This pricing concept provides minimal prices for
variance swaps even when an equivalent risk neutral probability measure does
not exist.
