I. INTRODUCTION

The study of non-arbitrage markets was begun for the first time in Bachelier's work [1]. Then, in the famous works of Black F. and Scholes M. [2] and Merton R. S. [3] the formula was found for the fair price of the standard call option of European type. The absence of arbitrage in the financial market has a very transparent economic sense, since it can be considered reasonably arranged. The concept of non arbitrage in financial market is associated with the fact that one cannot earn money without risking, that is, to make money you need to invest in risky or risk-free assets. The exact mathematical substantiation of the concept of non arbitrage was first made in the papers [4], [5] for the finite probability space and in the general case in the paper [6]. In the continuous time evolution of risky asset, the proof of absent of arbitrage possibility see in [7]. The value of the established Theorems is that they make it possible to value assets. They got a special name "The First and The Second Fundamental Asset Pricing Theorems." Generalizations of these Theorems are contained in papers [8], [9], [10].

If the martingale measure is not the only one for a given evolution of a risky asset, then a rather difficult problem of describing all martingale measures arises in order to evaluate, for example, derivatives.

Assessment of risk in various systems was begun in papers [11], [12], [13], [14].

Statistical studies of the time series of the logarithm of the price ratio of risky assets contain heavy tails in distributions with strong elongation in the central region. The temporal behavior of these quantities exhibits the property of clustering and a strong dependence on the past. All this should be taken into account when building models for the evolution of risky assets.

In this paper, we generalize the results of the papers $[15]$, $[16]$, $[17]$ and construct the evolution of risky assets for which we completely describe the set of equivalent martingale measures.

The aim of this study is to describe the family of martingale measures for a wide class of risky asset evolutions. The paper proposes the general concept for constructing the family of martingale measures equivalent to a given measure for a wide class of evolutions of risky assets. In particular, it also contains the description

First, we show an integral inequality for a nonnegative random variable under the inequality for this nonnegative random variable with respect to the constructed family of measures. Further, using this integral inequality for the non-negative random variable, a pointwise system of inequalities is obtained for this non-negative random variable for a particular case. After that, the pointwise system of inequalities is obtained for the non-negative random variable in the general case. Then, using the resulting pointwise system of inequalities, an inequality is established for this non-negative random variable whose right-hand side is such that its conditional mathematical expectation is equal to one.

On the basis of the results of Section 4, in Section 5, we prove the optional decomposition for the non negative super-martingales. In Section 6, we introduce the spot measures by the recurrent relations and find the representation for them. Using these facts under certain conditions we prove integral representation for every martingale measure over the set of spot measures.

First, the optional decomposition for diffusion processes super-martingale was opened by El Karoui N. and Quenez M. C. [21]. After that, Kramkov D. O. and Follmer H. [22], [23] proved the optional decomposition for the nonnegative bounded super-martingales. Folmer H. and Kabanov Yu. M. [24], [25] proved analogous result for an arbitrary super-martingale. Recently, Bouchard B. and Nutz M. [26] considered a class of discrete models and proved the necessary and sufficient conditions for the validity of the optional decomposition.

Section 7 contains applications of the results obtained. A class of random processes is considered, which contains well-known processes of the type ARCH and GARCH ones. Two types of random processes are considered, those for which the price of an asset cannot go down to zero and those for which the price can go down to zero during the period under consideration. The first class of processes describes the evolution of well-managed assets. We will call these assets relatively stable. For the evolution of relatively stable assets in the period under consideration, the family of martingale measures is one and the same. The family of martingale measures for the evolution of risky assets whose price can drop to zero is contained in the family of martingale measures for the evolution of relatively stable assets. Each of the martingale measures for the considered class of evolutions is an integral over the set of spot martingale measures.

The interval of non-arbitrage prices is found for a wide class of payoff functions in the case when evolution describes relatively unstable assets. This range is quite wide for the payoff functions of standard put and call options. The fair price of the super hedge is in this case the starting price of the underlying asset. The estimates are found for the fair price of the super-hedge for the introduced class of evolutions with respect to stable assets. The formulas are found for the fair price of contracts with call and put options for the evolution of assets described by parametric processes.

The same formulas are found for Asian-type put and call options. A characteristic feature of these estimates is that for the evolution of relatively stable assets the fair price of the super hedge is less than the price of the underlying asset.

In Section 8, the estimates of the parameters of risky assets included in the evolution are obtained. The formulas are found for the fair price of contracts with call and put options for the obtained parameter estimates, and the interval of non-

Let $\{{\mathrm{\Omega }}_N,{\mathcal{F}}_N,P_N\}$ be a direct product of the probability spaces $\{{\mathrm{\Omega }}_i^0,{\mathcal{F}}_i^0,P_i^0\}$, $i=\overline{1,N}$, ${\mathrm{\Omega }}_N=\prod _{i=1}^N{\mathrm{\Omega }}_i^0$, $P_N=\prod _{i=1}^N{P}_i^0$, ${\mathcal{F}}_N=\prod _{i=1}^N{\mathcal{F}}_i^0$, where the $\sigma$ -algebra ${\mathcal{F}}_N$ is a minimal $\sigma$ -algebra, generated by the sets $\prod _{i=1}^N{G}_i$, $G_i\in {\mathcal{F}}_i^0$. On the measurable space $\{{\mathrm{\Omega }}_N,{\mathcal{F}}_N\}$, under the filtration ${\mathcal{F}}_n$, $n=\overline{1,N}$, we understand the minimal $\sigma$ -algebra generated by the sets $\prod _{i=1}^N{G}_i$, $G_i\in {\mathcal{F}}_i^0$, where $G_i={\mathrm{\Omega }}_i^0$ for . We also introduce the probability spaces $\{{\mathrm{\Omega }}_n,{\mathcal{F}}_n,P_n\}$, $n=\overline{1,N}$, where ${\mathrm{\Omega }}_n=\prod _{i=1}^n{\mathrm{\Omega }}_i^0$, ${\mathcal{F}}_n=\prod _{i=1}^n{\mathcal{F}}_i^0$, $P_n=\prod _{i=1}^n{P}_i^0$. There is a one-to-one correspondence between the sets of the $\sigma$ -algebra ${\mathcal{F}}_n$, belonging to the introduced filtration, and the sets of the $\sigma$ -algebra ${\mathcal{F}}_n=\prod _{i=1}^n{\mathcal{F}}_i^0$ of the measurable space $\{{\mathrm{\Omega }}_n,{\mathcal{F}}_n\}$, $n=\overline{1,N}$. Therefore, we don't introduce new denotation for the $\sigma$ -algebra ${\mathcal{F}}_n$ of the measurable space $\{{\mathrm{\Omega }}_n,{\mathcal{F}}_n\}$, since it always will be clear the difference between the above introduced $\sigma$ -algebra ${\mathcal{F}}_n$ of filtration on the measurable space $\{{\mathrm{\Omega }}_N,{\mathcal{F}}_N\}$ and the $\sigma$ -algebra ${\mathcal{F}}_n$ of the measurable space $\{{\mathrm{\Omega }}_n,{\mathcal{F}}_n\}$, $n=\overline{1,N}$.

We assume that the evolution of risky asset $\{S_n{\}}_{n=0}^N$, given on the probability space $\{{\mathrm{\Omega }}_N,{\mathcal{F}}_N,P_N\}$, is consistent with the filtration ${\mathcal{F}}_n$, that is, $S_n$ is a ${\mathcal{F}}_n$ -measurable. Due to the above one-to-one correspondence between the sets of the $\sigma$ -algebra ${\mathcal{F}}_n$, belonging to the introduced filtration, and the sets of the $\sigma$ -algebra ${\mathcal{F}}_n$ of the measurable space $\{{\mathrm{\Omega }}_n,{\mathcal{F}}_n\},n=\overline{1,N}$, we give the evolution of risky assets in the form $\{S_n({\omega }_1,\dots ,{\omega }_n){\}}_{n=0}^N$, where $S_n({\omega }_1,\dots ,{\omega }_n)$ is an ${\mathcal{F}}_n$ -measurable random variable, given on the measurable space $\{{\mathrm{\Omega }}_n,{\mathcal{F}}_n\}$. It is evident that such evolution is consistent with the filtration ${\mathcal{F}}_n$ on the measurable space $\{{\mathrm{\Omega }}_N,{\mathcal{F}}_N,P_N\}$.

Further, we assume that

where $\mathrm{\Delta }{S}_n=S_n({\omega }_1,\dots ,{\omega }_n)-S_{n-1}({\omega }_1,\dots ,{\omega }_{n-1}),n=\overline{1,N}$.

The number $N_n$ may be finite or infinite. Since ${\mathrm{\Omega }}_n^{-}\cup {\mathrm{\Omega }}_n^{+}={\mathrm{\Omega }}_n$, ${\mathrm{\Omega }}_n^{-}\cap {\mathrm{\Omega }}_n^{+}=\mathrm{\varnothing }$, and , , we have

Further, in this paper, we assume that , , $n=\overline{1,N}$, $k=\overline{1,N_n}$. We also assume some technical suppositions: there exist subsets $B_{n,i}^{0,k-}\in F_n^0$, $i=\overline{1,I_n}$, , and $B_{n,s}^{0,k+}\in F_n^0$, $s=\overline{1,S_n}$, , satisfying the conditions

Below, we give the examples of evolutions $\{S_n({\omega }_1,\dots ,{\omega }_n){\}}_{n=1}^N$ for which the representations (5) are true.

Suppose that the random values $a_i({\omega }_1,\dots ,{\omega }_i)$, ${\eta }_i({\omega }_i)$ satisfy the inequalities , $1+{\eta }_i({\omega }_i)\ge 0$, , , $i=\overline{1,N}$. If $S_n({\omega }_1,\dots ,{\omega }_n)$ is given by the formula