Order the support
as K1<⋯<KN and put
si=Ki−Ki−1g(Ki)−g(Ki−1).
On the finite support,
g(ST)=g(K1)+∑i=2Nsi{(ST−Ki−1)+−(ST−Ki)+}.
This follows by telescoping: at
ST=Kj, only terms through
j survive and reconstruct successive increments of
g. The
static replication on a finite terminal support therefore has no-
arbitrage price
πt=g(K1)PtT+∑i=2NKi−Ki−1g(Ki)−g(Ki−1)(CtT,Ki−1−CtT,Ki).
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