This paper proposes a new hedging scheme of European derivatives
under uncertain volatility environments, in which a weighted variance
swap called the polynomial variance swap is added to the Black-Scholes
delta hedging for managing exposure to volatility risk. In general, under
these environments one cannot hedge the derivatives completely by using
dynamic trading of only an underlying asset owing to volatility risk. Then,
for hedging uncertain volatility risk, we design the polynomial variance,
which can be dependent on the level of the underlying asset price. It is
shown that the polynomial variance swap is not perfect, but more efficient
as a hedging tool for the volatility exposure than the standard variance
swap. In addition, our hedging scheme has a preferable property that
any information on the volatility process of the underlying asset price is
unnecessary. To demonstrate robustness of our scheme, we implement
Monte Carlo simulation tests with three different settings, and compare
the hedging performance of our scheme with that of standard dynamic
hedging schemes such as the minimum-variance hedging. As a result, it is
found that our scheme outperforms the others in all test cases. Moreover,
it is noteworthy that the scheme proposed in this paper continues to be
robust against model risks.
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