  In complete markets, there are risky assets and a riskless asset. It is
assumed that the riskless asset and the risky asset are traded continuously in
time and that the market is frictionless. In this paper, we propose a new
method for hedging derivatives assuming that a hedger should not always rely on
trading existing assets that are used to form a linear portfolio comprised of
the risky asset, the riskless asset, and standard derivatives, but rather
should design a set of specific, most-suited financial instruments for the
hedging problem. We introduce a sequence of new financial instruments best
suited for hedging jump-diffusion and stochastic volatility market models. The
new instruments we introduce are perpetual derivatives. More specifically, they
are options with perpetual maturities. In a financial market where perpetual
derivatives are introduced, there is a new set of partial and partial-integro
differential equations for pricing derivatives. Our analysis demonstrates that
the set of new financial instruments together with a risk measure called the
tail-loss ratio measure defined by the new instrument's return series can be
potentially used as an early warning system for a market crash.
