We review several problems on the quantile hedging given by [FL99].
We assume that the discounted price process of stock is given by a semimartingale on a probability space
. Under self-financing, the wealth process
with inital capital
and a predictable strategy
follows
belongs to a space of adimissible strategy given by
,
Let be a given claim.
Problem [SP]. Given a fixed constant , maximize success probability, i.e. find
Problem [SR] Given a fixed constant , maximize success ratio, i.e. find
Problem [CP] Given a constant , minimize the cost
to beat
with probability
, i.e. find
Problem [CR] Given a constant , minimize the cost
to beat
with ratio
, i.e. find
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