Mathematical Model of Banking Operation

2021

GJSFR

Self Cite

In the first part of the paper, we construct the models of the complete non-arbitrage financial markets for a wide class of evolutions of risky assets.This construction is based on the observation that for a certain class of risky as set evolutions the martingale measure is invariant with respect to these evolutions. For such a financial market model the only martingale measure being equivalent to an initial measure is built. On such a financial market,formulas for the fair price of contingent liabilities are presented. A multi-parameter model of the financial market is proposed, the martingale measure of which does not depend on the parameters of the model of the evolution of risky assets and is the only one.

“…We remind that the evolution of risky asset is given by the formula (8). Therefore, in this case the condition (16) (see [1]) is formulated, as follows:…”

Section: Proof Let Us Prove Thatmentioning

confidence: 99%

“…Valuable is the fact that there is a wide range of models for the evolution of risky assets for which it is possible to build parametric models of non-arbitrage markets whose parameters can be estimated based on statistical data. Problems of risk estimates was considered in papers [15], [16], [17], [18].…”

mentioning

confidence: 99%

Non-Arbitrage Models of Financial Markets

2021

GJSFR

Self Cite

“…The optional decomposition for super-martingales plays the fundamental role for the risk assessment in incomplete markets [2], [3], [6], [8], [9], [10], [11]. Considered in the paper problem is a generalization of the corresponding one that appeared in mathematical finance about the optional decomposition for a super-martingale and which is related with the construction of the super-hedge strategy in incomplete financial markets.…”

Section: Introductionmentioning

confidence: 99%

Description of Incomplete Financial Markets for Time Evolution of Risk Assets

2019

APM

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.

“…The optional decomposition for super-martingales plays the fundamental role for the risk assessment in incomplete markets [1], [2], [5], [7], [8], [9], [10]. Considered in the paper problem is a generalization of the corresponding one that appeared in mathematical finance about the optional decomposition for a super-martingale and which is related with the construction of the superhedge strategy in incomplete financial markets.…”

Section: Introductionmentioning

confidence: 99%

Martingales and Super-Martingales Relative to a Convex Set of Equivalent Measures

2018

APM

In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex 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 description of all local regular super-martingales relative to a convex set of equivalent measures is presented. The notion of the complete set of equivalent measures is introduced. We prove that every bounded in some sense super-martingale relative to the complete set of equivalent measures is local regular. A new definition of the fair price of contingent claim in an incomplete market is given and the formula for the fair price of Standard Option of European type is found. The proved Theorems are the generalization of the famous Doob decomposition for super-martingale onto the case of super-martingales relative to a convex set of equivalent measures.