Appl Math Optim (2013) 68:145–155
Pricing of Claims in Discrete Time with Partial
Kristina Rognlien Dahl
Published online: 28 March 2013
© Springer Science+Business Media New York 2013
Abstract We consider the pricing problem of a seller with delayed price information.
By using Lagrange duality, a dual problem is derived, and it is proved that there is no
duality gap. This gives a characterization of the seller’s price of a contingent claim.
Finally, we analyze the dual problem, and compare the prices offered by two sellers
with delayed and full information respectively.
Keywords Mathematical ﬁnance · Lagrange duality · Delayed information · Pricing
We consider the pricing problem of a seller of a contingent claim B in a ﬁnancial
market with a ﬁnite scenario space Ω and a ﬁnite, discrete time setting. The seller
is assumed to have information modeled by a ﬁltration (G
which is generated by
a delayed price process, so the seller has delayed price information. This delay of
information is a realistic situation for many ﬁnancial market traders. Actually, traders
may pay to get updated prices.
The seller’s problem is to ﬁnd the smallest price of B, such that there is no risk
of her losing money. We solve this by deriving a dual problem via Lagrange duality,
and use the linear programming duality theorem to show that there is no duality gap.
A related approach is that of King , where the fundamental theorem of mathemati-
cal ﬁnance is proved using linear programming duality. Vanderbei and Pilar also
use linear programming to price American warrants.
A central theorem of this paper is Theorem 3.1, which describes the seller’s price
of the contingent claim. This generalizes a pricing result by Delbaen and Schacher-
mayer to a delayed information setting (see , Theorem 5.7). Contrary to what one
might guess, this characterization does not involve martingale measures. We can how-
ever get an idea of the seller’s price by comparing it to that of an unconstrained seller,
K. Rognlien Dahl (
Department of Mathematics, University of Oslo, Pb. 1053 Blindern, 0316 Oslo, Norway