Description of Incomplete Financial Markets for Time Evolution of Risk Assets

Abstract

In the paper, the martingales and super-martingales relative to a regular set of measures are systematically studied. The notion of local regular super-martingale relative to a set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are founded. The regular set of measures play fundamental role for the description of incomplete markets. In the partial case, the description of the regular set of measures is presented. The notion of completeness of the regular set of measures have the important significance for the simplification of the proof of the optional decomposition for super-martingales. Using this notion, the important inequalities for some random values are obtained. These inequalities give the simple proof of the optional decomposition of the majorized super-martingales. The description of all local regular super-martingales relative to the regular set of measures is presented. It is proved that every majorized super-martingale relative to the complete set of measures is a local regular one. In the case, as evolution of a risk asset is given by the discrete geometric Brownian motion, the financial market is incomplete and a new formula for the fair price of super-hedge is founded.P ∈M E P |f m | < ∞, m = 1, ∞, and there exists an adapted nonnegative increasing random process {g m , F m } ∞ m=0 , g 0 = 0, sup P ∈M E P |g m | < ∞, m = 1, ∞, such that {f m + g m , F m } ∞ m=0 is a martingale relative to every measure from M. The next elementary Theorem 1 will be very useful later.

Assessment of Contingent Liabilities for Risk Assets Evolutions Built on Brownian Motion

“…In the next Lemma we give the sufficient condition of the existence of exhaustive decomposition. This Lemma is very important for the proof of the next Theorems [1].…”

“…The main aim is to describe the set of martingale measures for the evolution of risk asset given by the formula (2). This problem we solved in Theorem 8 [1] in the case as the random values ( ) ( ) 2) the ( ) 1 n + -th decomposition is a sub-decomposition of the n-th one, that is, for every j, 3) the minimal σ-algebra containing all , , , = Ω × Ω = ∞  ;…”

“…In this work, which is a continuation of the paper [1], we construct the random processes based on the discrete geometric Brownian motion which can describe the evolution of risky assets. A new method of the description of martingale measures for the introduced class of evolutions of risk assets is developed.…”

“…It is proved that every martingale measure can be represented as an integral on some measure on the set of extreme points of the set of martingale measures. This crucial fact is a base for the estimation of contingent liabilities in the incomplete financial markets with the evolution of risk assets introduced in [1].…”

“…This fact is unacceptable from the economic point of view. In the proposed paper, we generalize the class of evolutions of risk assets proposed in [1] and which contains a class of evolutions built on the discrete geometric Brownian motion. For this class of evolutions of risk assets the set of martingale measures is described and the representation for every martingale measure as integral over the set of extreme points is obtained.…”