Conditional Davis Pricing

Abstract

We study the set of marginal utility-based prices of a financial derivative in the case where the investor has a non-replicable random endowment. We provide an example showing that even in the simplest of settings -such as Samuelson's geometric Brownian motion model -the interval of marginal utility-based prices can be a non-trivial strict subinterval of the set of all no-arbitrage prices. This is in stark contrast to the case with a replicable endowment where non-uniqueness is exceptional. We provide formulas for the end points for these prices and illustrate the theory with several examples.

Erratum to: Utility maximization in incomplete markets with random endowment

“…We briefly recall the notation of that example which is the basis of the counterexample in [4]. The stock price process S = (S 0 , S 1 ) is defined by S 0 = 1 and by letting S 1 assume the value x 0 = 2 with probability p 0 = 1 − α and, for n ≥ 1, the value x n = 1 n with probability p n = α2 −n , for 0 < α < 1 sufficiently small.…”

“…Now comes the beautiful idea from [4]. Define the processS = (S 0 ,S 1 ) by setting S 0 = S 0 = 1 andS…”

“…Summing up, following [4], we have constructed an example where the value functionũ(·) fails to be differentiable.…”

Optimal investment with random endowments and transaction costs: duality theory and shadow prices

“…Remark 4.4 It should be possible to extend the framework of [23] for unbounded random endowments and then establish the duality theorem using the pair of bounded finitely additive measures which admits the Yosida-Hewitt decomposition Q i = Q i,r + Q i,s , i = 0, 1, and Q i,r is a countably additive measure. It then might be easier to check Assumption 4.1 in this extended framework.…”

“…When the market is incomplete, the problem becomes more delicate. In particular, to build the convex duality theorem treating unhedgeable random endowments demands new techniques, especially if endowments are unbounded, see for example [4], [12], [19] and [23]. The references [18], [30], [24] and [28] also study this problem when the intermediate consumption is considered.…”