Non-Arbitrage Models of Financial Markets

Gonchar Nicholas

doi:10.34257/GJSFRAVOL21IS4PG67

I. INTRODUCTION

In this paper, models of non-arbitrage markets are constructed on the basis of the invariance of a set of spot measures with respect to a certain class of evolution of risky assets. In the first part of the paper, models of complete non arbitrage markets are built on the basis of an analysis of conditions under which there is only one martingale measure. In the second part of the work, models of incomplete non-arbitrage realistic market models are built based on the same principles as in the first part of the work. For the introduced parametric models of the markets, estimates of parameters were obtained based on the observed real values of the evolution of risky assets. This opens up wide opportunities for hedging risks.

Historically the first model evolution of risky assets was suggested in Bachelier's work [4]. Then, in the famous works of Black F. and Scholes M. [5] and Merton R. S. [6] 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 [7], [8] for the finite probability space and in the general case in the paper [9]. In the continuous time evolution of risky asset, the proof of absence of arbitrage possibility see in [11]. 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 [12], [13], [14].

This work is a continuation of the works $[1]$ , $[19]$ , $[20]$ , $[21]$ . In paper $[1]$ , a new method for constructing and describing a family of martingale measures was proposed. This made it possible to build models of non-arbitrage markets. The construction of a realistic model of non-arbitrage markets has been an urgent problem since the moment when the concept of the absence of arbitrage appeared in the scientific literature as the most equitable model of the functioning of financial markets. What could be more attractive than a realistic model that can be built on the basis of observations of the evolution of the financial market. The main obstacle to this was the limited possibilities of constructing a risk-neutral martingale measure for a given evolution of risky assets in the case of a complete market and their complete description in the case of incomplete markets. In the case of discrete evolution of risky assets, the theoretical possibility of the existence of such non-arbitrage markets was established in $[7]$ , $[8]$ , $[9]$ , $[10]$ , $[11]$ , $[12]$ , $[13]$ , $[14]$ . But, there were no practically regular methods for constructing such non-arbitrage market models, although such attempts were made for some kind of models of the evolution of risky assets $[13]$ , $[14]$ . With the appearance of the work $[1]$ , which proposes a regular method for describing all martingale measures for a wide class of evolutions of risky assets $[22]$ , $[23]$ , $[24]$ that capture the phenomenon of price memory and clustering, it became possible to construct realistic models of non-arbitrage markets. Note that such efforts have been made in this direction, and more about this can be found in the monograph $[13]$ , $[14]$ . 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 $[15]$ , $[16]$ , $[17]$ , $[18]$ .

This work is the first step in constructing parametric models of non-arbitrage markets whose parameters can be estimated based on empirical data. In this paper, models of the evolution of risk assets on a discrete probabilistic space are considered. Such models can be used to approximate realistic models of the evolution of risky assets. The value of this model is that in this case the structure of the set of martingale measures is relatively simple.

In the case of incomplete non-arbitrage markets, the set of equivalent martingale measures has the cardinality of the continuum, but since they are a linear convex combination of a set of spot measures whose number is finite, this allows calculating the required characteristics using a finite number of operations. This allows a computer to be used to simulate non-arbitrage markets.

In the third section of the work, the necessary and sufficient conditions for the uniqueness of a martingale measure are established in terms of the law of evolution of risky assets, and the only martingale measure is found. Using the results of Section 3 in Section 4, a multi-parameter model of the complete financial market is built and parameter estimates are obtained through empirical data of the financial market. This will allow the model to be adapted to realistic financial markets to estimate the fair price of European-type derivatives with different payment functions.

Section 5 establishes the general structure of the family of equivalent martingale measures for a wide class of risky asset evolutions. The structure of spot measures is completely described, the formulas for the fair price of the super hedge and the range of non-arbitrage prices are established. Based on the results of Section 5, Section 6 builds a multi-parameter model of the incomplete non-arbitrage market. The estimates of the parameters of the model are obtained through empirical observations of the financial market. This will allow the computer to be used to model the financial market.

II. EVOLUTIONS OF RISKY ASSETS

In this section, a class of evolutions of risky assets is described which is used in this paper. This class is fairly wide and includes well known in the literature evolutions of risky assets. Let $\{\Omega_{N},\mathcal{F}_{N},P_{N}\}$ be a direct product of the probability spaces $\{\Omega_{i}^{0},\mathcal{F}_{i}^{0},P_{i}^{0}\}$ , $i=\overline{1,N}$ , $\Omega_{N}=\prod_{i=1}^{N}\Omega_{i}^{0}$ , $P_{N}=\prod_{i=1}^{N}P_{i}^{0}$ , $F_{N}=\prod_{i=1}^{N}F_{i}^{0}$ , 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 $\{\Omega_N, \mathcal{F}_N\}$ , 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 $G_i = \Omega_i^0$ for $i > n$ . We also introduce the probability spaces $\{\Omega_n, \mathcal{F}_n, P_n\}, n = \overline{1, N}$ , where $\Omega_n = \prod_{i=1}^{n} \Omega_i^0$ , $\mathcal{F}_n = \prod_{i=1}^{n} \mathcal{F}_i^0$ , $P_n = \prod_{i=1}^{n} P_i^0$ . 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 $\{\Omega_n, \mathcal{F}_n\}, n = \overline{1, N}$ . Therefore, we don't introduce new denotation for the $\sigma$ -algebra $\mathcal{F}_n$ of the measurable space $\{\Omega_n, \mathcal{F}_n\}$ , since it always will be clear the difference between the above introduced $\sigma$ -algebra $\mathcal{F}_n$ of filtration on the measurable space $\{\Omega_N, \mathcal{F}_N\}$ and the $\sigma$ -algebra $\mathcal{F}_n$ of the measurable space $\{\Omega_n, \mathcal{F}_n\}, n = \overline{1, N}$ .

We assume that the evolution of risky asset $\{S_n\}_{n=0}^N$ , given on the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , 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 $\{\Omega_n, \mathcal{F}_n\}, n = \overline{1, N}$ , we give the evolution of risky assets in the form $\{S_n(\omega_1, \ldots, \omega_n)\}_{n=0}^N$ , where $S_n(\omega_1, \ldots, \omega_n)$ is an $\mathcal{F}_n$ -measurable random variable, given on the measurable space $\{\Omega_n, \mathcal{F}_n\}$ . It is evident that such evolution is consistent with the filtration $\mathcal{F}_n$ on the measurable space $\{\Omega_N, \mathcal{F}_N, P_N\}$ .

Further, we assume that

where $\Delta S_{n} = S_{n}(\omega_{1},\ldots,\omega_{n}) - S_{n - 1}(\omega_{1},\ldots,\omega_{n - 1}), n = \overline{1,N}$ .

Let us introduce the denotations

In this paper we assume that

Further, in this paper, we assume that $P_{n}^{0}(\Omega_{n}^{0-}) > 0$ , $P_{n}^{0}(\Omega_{n}^{0+}) > 0$ , $n = \overline{1, N}$ . We also assume some technical suppositions: there exist subsets $B_{n,i}^{0-} \in F_{n}^{0}$ , $i = \overline{1, I_{n}}$ , $I_{n} > 1$ , and $B_{n,s}^{0+} \in F_{n}^{0}$ , $s = \overline{1, S_{n}}$ , $S_{n} > 1$ , satisfying the conditions

Below, we give the examples of evolutions $\{S_n(\omega_1,\ldots,\omega_n)\}_{n = 1}^N$ for which the representations (5) are true.

Suppose that the random values $a_{i}(\omega_{1},\ldots,\omega_{i})$ , $\eta_{i}(\omega_{i})$ satisfy the inequalities

If $S_{n}(\omega_{1},\ldots,\omega_{n})$ is given by the formula

then

Let us note that not only the evolutions given by the formula (8) provide the representation (5). In this work, we use the evolutions of the kind (8). Below we give examples of the evolution of risky assets that have the form (8). For example, if

where the random values $\sigma_{i}(\omega_{1},\ldots,\omega_{i-1})\geq\sigma_{i}^{0}>0,\ i=\overline{1,N}$ , and $P_{i}^{0}(\varepsilon_{i}(\omega_{i})<0)>0,\ P_{i}^{0}(\varepsilon_{i}(\omega_{i})>0)>0$ , then such an evolution has the form (8) with

satisfying needed conditions.

III. UNIQUENESS OF THE MARTINGALE MEASURE

In this section, the necessary and sufficient conditions in terms of the evolution of risky assets are obtained relative to the uniqueness of martingale measure. Under the fairly wide assumptions about the evolution of risky assets, an expression for a single martingale measure is found. Based on the explicit construction of the martingale measure and its invariance with respect to a certain type of evolutions, it is possible to construct the models of non arbitrage markets, both complete and incomplete.

In this and section 4, we put that $\Omega_i^0 = \{\omega_i^1,\omega_i^2\}$ . Denote by $\mathcal{F}_i^0$ the $\sigma$ -algebra of all subsets of the set $\Omega_i^0$ . Let $P_{i}^{0}$ be a probability measure on $\mathcal{F}_i^0$ . We assume that $P_{i}^{0}(\omega_{i}^{s}) > 0$ , $i = \overline{1,N}$ , $s = \overline{1,2}$ . As before, we put that the probability space $\{\Omega_N,\mathcal{F}_N,P_N\}$ is a direct product of the probability spaces $\{\Omega_i^0,\mathcal{F}_i^0,P_i^0\}$ , $i = \overline{1,N}$ , and we put $N < \infty$ . We also consider the probability spaces $\{\Omega_n,\mathcal{F}_n,P_n\}, n = \overline{1,N}$ , being the direct product of the probability spaces $\{\Omega_i^0,\mathcal{F}_i^0,P_i^0\}$ , $i = \overline{1,n}$ . We assume that the evolution of a risky asset is given by the formula

where the random values $a_{n}(\omega_{1},\ldots,\omega_{n - 1},\omega_{n})$ , $\eta_n(\omega_n)$ , $n = \overline{1,N}$ , given on the probability space $\{\Omega_n,\mathcal{F}_n,P_n\}$ , satisfy the conditions

So, for $\Delta S_{n}(\omega_{1},\ldots,\omega_{n - 1},\omega_{n})$ , $n = \overline{1,N}$ , the representation

is true, From these conditions, we obtain $\Omega_n^-$ = $\Omega_{n-1} \times \Omega_n^{0-}$ , $\Omega_n^+ = \Omega_{n-1} \times \Omega_n^{0+}$ , where $\Omega_n^{0-} = \{\omega_n \in \Omega_n^0, \eta_n(\omega_n) \leq 0\}$ , $\Omega_n^{0+} = \{\omega_n \in \Omega_n^0, \eta_n(\omega_n) > 0\}$ .

From the suppositions above, it follows that $P_{n}^{0}(\Omega_{n}^{0-}) > 0$ , $P_{n}^{0}(\Omega_{n}^{0+}) > 0$ . The measure $P_{n}^{0-}$ is a contraction of the measure $P_{n}^{0}$ on the $\sigma$ -algebra $\mathcal{F}_{n}^{0-} = \Omega_{n}^{0-} \cap \mathcal{F}_{n}^{0}$ , $P_{n}^{0+}$ is a contraction of the measure $P_{n}^{0}$ on the $\sigma$ -algebra $\mathcal{F}_{n}^{0+} = \Omega_{n}^{0+} \cap \mathcal{F}_{n}^{0}$ .

Let us introduce the following denotation. For every point $\{\omega_1, \ldots, \omega_{n-1}, \omega_n\} \in \Omega_n$ , we introduce the set $A(\omega_1, \ldots, \omega_{n-1}, \omega_n) \in \Omega_N$ , where

For fixed indexes $i_{1},\ldots,i_{n}$ we also use the denotation

It is evident that every set $A^{i_{1},\ldots,i_{n}}$ has the form

where indexes $i_s$ takes only one value from the set $\{1,2\}$ . Then, $A^{i_1,\dots,i_{n-1}} = A^{i_1,\dots,i_{n-1},1} \cup A^{i_1,\dots,i_{n-1},2} \in \mathcal{F}_{n-1}$ , where

If $P_{N}$ is a measure on $\mathcal{F}_N$ , then

We give an evident construction of martingale measure for risky assets evolution, given by the formula (11). Below, we assume that measures $P_{n}^{0}$ is concentrated at points $\omega_{n}^{1},\omega_{n}^{2}\in\Omega_{n}^{0}$ , where $\omega_{n}^{1}\in\Omega_{n}^{0-},\omega_{n}^{2}\in\Omega_{n}^{0+}$ and we have the representation $\Omega_{n}^{-}=\Omega_{n-1}\times\Omega^{0-}$ and $\Omega_{n}^{+}=\Omega_{n-1}\times\Omega^{0+}$ . So, we have $\eta_{n}(\omega_{n}^{1})<0,\eta_{n}(\omega_{n}^{2})>0$ .

Let us put $P_{n}^{0}(\omega_{n}^{1}) = p_{n}$ , $P_{n}^{0}(\omega_{n}^{2}) = 1 - p_{n}$ , where $0 < p_{n} < 1$ . Then, to satisfy the conditions (14 - 16), (see [1]) we need to put

and to require that

The next Lemma 1 is a consequence of results in [1].

Lemma 1. On the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , being the direct product of the probability spaces $\{\Omega_i^0, \mathcal{F}_i^0, P_i^0\}$ , for the evolution of risky asset given by the formula (11) only one spot measure $\mu_{\{\omega_1^1, \omega_1^2\}, \ldots, \{\omega_N^1, \omega_N^2\}}(A)$ exists, where $\{\omega_i^1, \omega_i^2\} \in \Omega_i^0$ , $i = \overline{1, N}$ . For it the representation

is true. This measure is martingale measure for the considered evolution of risky asset, where

Next Theorem 1 appeared first in [2](Theorem 1.4.1), where it was proved under the less general conditions.

Theorem 1. On the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , being the direct product of the probability spaces $\{\Omega_i^0, \mathcal{F}_i^0, P_i^0\}$ , suppose that the evolution of risky asset $\{S_n(\omega_1, \ldots, \omega_n)\}_{n=1}^N$ is given by the formula (11). The necessary and sufficient conditions of the uniqueness of martingale measure $\mu_0(A)$ , $A \in \mathcal{F}_N$ , are the inequalities

for every set of indexes $i_1, \ldots, i_{n-1}$ . For any martingale $\{m_n(\omega_1, \ldots, \omega_{n-1}, \omega_n)\}_{n=0}^N$ relative to the unique measure $\mu_0(A)$ the representation

is true, where

Proof. The necessity. Suppose that the evolution $\{S_{n}(\omega_{1},\ldots,\omega_{n})\}_{n=1}^{N}$ of the risky asset on the probability space $\{\Omega_{N},F_{N},P_{N}\}$ is such that the martingale measure $\mu_{0}(A)$ , $A\in F_{N}$ , being equivalent to the measure $P_{N}$ , is unique. Then, for every attainable contingent liability $m_{N}(\omega_{1},\ldots,\omega_{N})$ the representation (21) is true [11] for some $F_{k-1}$ -measurable finite valued random value $C_{k}(\omega_{1},\ldots,\omega_{k-1})$ , $k=\overline{1,N}$ , where $m_{n}(\omega_{1},\ldots,\omega_{n-1},\omega_{n})=E^{\mu_{0}}\{m_{N}(\omega_{1},\ldots,\omega_{N})|\mathcal{F}_{n}\}$ . For $m_{n}(\omega_{1},\ldots,\omega_{n-1},\omega_{n})$ and $S_{n}(\omega_{1},\ldots,\omega_{n-1},\omega_{n})$ the representations

are true. From the representation (21) and the equality (22) for $\{\omega_1, \ldots, \omega_{n-1}\} \in A^{i_1, \ldots, i_{n-1}}$ we obtain the equality

where $d_{i_1,\dots,i_{n - 1}}$ is finite. Since

we have

Further,

If to put

Then, the equality (26) is transformed into the equality

Due to that $S_{n}(\omega_{1},\ldots,\omega_{n})$ and $m_{n}(\omega_{1},\dots,\omega_{n})$ are martingales relative to the measure $\mu_0$ and $A^{i_1,\dots,i_{n - 1},1}$ , $A^{i_1,\dots,i_{n - 1},2}\in \mathcal{F}_n$ we have

Since $d_{i_{1},\ldots,i_{n-1}}$ is finite, then $R_{1}^{S_{N}}(\omega_{1}^{i_{1}},\ldots,\omega_{n-1}^{i_{n-1}})\neq0$ . The last means that inequality (20) takes place. This proves the equality

which means that (23) is true, where we introduced the denotation

This proves the necessity.

Proof of the sufficiency. Suppose that the inequalities (20) are true. Let us prove that the martingale measure $\mu_{0}$ is unique. For this purpose, we prove that for every martingale the representation (21) is true with validity of equalities (22), (23).

Let us note that the equality (26) is true if for $d_{i_1,\dots,i_{n-1}}$ to choose (37) since the equalities

are valid.

Taking into account the equality (26) and the equalities

Summing over all indexes $i_1, \ldots, i_{n-1}$ left and right hand sides of the equality (42) we obtain the equality

We proved that for every martingale the representation (21) is true, due to the conditions (20). Let us prove that the martingale measure is unique. Suppose that there are at most two martingale measures $\mu_{0}^{1}$ and $\mu_{0}^{2}$ . If to put $m(\omega_{1},\ldots,\omega_{N})=\chi_{A}(\omega_{1},\ldots,\omega_{N})$ , then

From this representation, we obtain the equalities $\mu_0^1 (A) = \mu_0^2 (A) = c_0$ , $A\in \mathcal{F}_N$ . Contradiction. The last proves Theorem 1.

Next Theorem is concerned the case as the set of martingale measures consists of one measure.

Theorem 2. On the probability space $\{\Omega_{N},\mathcal{F}_{N},P_{N}\}$ , being the direct product of the probability spaces $\{\Omega_{i}^{0},\mathcal{F}_{i}^{0},P_{i}^{0}\}$ , suppose that the evolution of risky asset is given by the formula (11), then the set of martingale measures, being equivalent to the measure $P_{N}$ , consists of one point

The fair price of contract with option $\varphi_{0}$ of European type with the payoff function $\varphi(\omega_{1},\ldots,\omega_{N})$ is given by the formula

Proof. Since

we have

From this, it follows that $\mu_{0}(A) > 0$ for every $A \in F_{N}$ . It means that $\mu_{0}(A)$ is equivalent to $P_{N}$ . The inequality

is true, since

due to the suppositions relative to the evolutions of risky asset, given by the formula (11). Thanks to Theorem 1, the martingale measure $\mu_0$ is unique.

To prove the rest statement of Theorem 2, we need to construct the self-financing strategy $\pi$ such that the capital corresponding this strategy on $(B, S)$ market satisfies the condition

Let us consider the martingale

Due to Theorem 1, for the finite martingale $\{m_n(\omega_1,\dots,\omega_{n - 1},\omega_n)\}_{n = 0}^N$ relative to the measure $\mu_0(A)$ the representation

is true, where $C_i(\omega_1, \ldots, \omega_{i-1})$ is $\mathcal{F}_{i-1}$ measurable random value, and $m_0 = E^{\mu_0}\varphi(\omega_1, \ldots, \omega_{n-1}, \omega_N)$ .

If to put $\pi = \{\beta_n,\gamma_n\}_{n = 0}^N$ , where

then it easy to see that $\pi$ is self-financed strategy. Really,

$\mathcal{F}_{n - 1}$ -measurability of $(\beta_{n},\gamma_{n})$ is evident.

It is easy to show that

IV. COMPLETE MARKET HEDGING

In this section, the securities market is constructed, the evolution of which occurs in accordance with Formula (11). Possible for this was the observation that with respect to a certain class of evolutions of risky assets, the family of martingale measures is invariant. This fact turned out to be crucial for the construction of models of non-arbitrage markets. In papers [10], [11], such a possibility of the existence of non-arbitrage markets is established on the basis of the Hahn-Banach Theorem. This beautiful result has the disadvantage that it does not provide an algorithm for constructing models of non-arbitrage markets. How to build them having the evolution of risky assets is practically a difficult problem.

In Proposition 1, we establish the form of measurable transformations relative to which the only measure is invariant. Using that, a model of the securities market is built, which is complete. This result is constructive in contrast to the existence theorem from [10], [11]. Our denotations in this section are the same as in the previous section. We consider the evolution of risky assets given by the formula (11) on the same probability space.

Proposition 1. On the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , being the direct product of the probability spaces $\{\Omega_i^0, \mathcal{F}_i^0, P_i^0\}$ , let the evolution of risky asset be given by the formula (11), with $a_i(\omega_1, \ldots, \omega_i) = b_i(\omega_1, \ldots, \omega_{i-1})f_i(\omega_1, \ldots, \omega_i)$ , where the random variables $f_i(\omega_1, \ldots, \omega_i)$ , $b_i(\omega_1, \ldots, \omega_{i-1})$ , satisfy the inequalities

For such an evolution, the unique martingale measure $\mu_{0}$ does not depend on the random variables $b_{i}(\omega_{1},\ldots,\omega_{i-1})$ , $i=\overline{1,N}$ , and it is given by the formula

where

Proof. Due to the representation (46) for the measure $\mu_{0}$ , to prove Proposition 1 it needs to prove that all $\psi_{n}(\omega_{1},\ldots,\omega_{n})$ , $n=\overline{1,N}$ , do not depend on the random variables $b_{i}(\omega_{1},\ldots,\omega_{i-1})$ , $i=\overline{1,N}$ , where

But,

Therefore,

The equalities (67), (68) prove Proposition 1.

Suppose that the market consists of d assets the evolutions of which are given by the law

where

and the random values $\eta_{i}(\omega_{i})$ , $f_{i}(\omega_{1},\ldots,\omega_{i})$ , $i=\overline{1,N}$ , do not depend on k, and satisfy inequalities

Proposition 2. On the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , being the direct product of the probability spaces $\{\Omega_i^0, \mathcal{F}_i^0, P_i^0\}$ , if the evolution of $d$ risky assets is given by the formula (69), (70), then such a market is complete non arbitrage one. The unique martingale measure does not depend on the random variables $b_i^k(\omega_1, \ldots, \omega_{i-1}), k = \overline{1,d}$ , $i = \overline{1,N}$ , and it is determined by the formula (58). For the contingent claims $\varphi_i(\omega_1, \ldots, \omega_N), i = \overline{1,d}$ , the fair prices $\varphi_0^i$ are given by the formulas

Corollary 1. (Cox, Ross, Rubinstein, see [3]) On the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , being the direct product of the probability spaces $\{\Omega_i^0, \mathcal{F}_i^0, P_i^0\}$ , let the evolution of risky asset is given by the formula

where the random values $\rho_i(\omega_i)$ , $i = \overline{1,N}$ , are such that $\rho_i(\omega_i^1) = a$ , $\rho_i(\omega_i^2) = b$ , and let the bank account evolution be given by the formula

Then, for the discount evolution of risky asset

the martingale measure $\mu_0$ is unique if $a < r < b$ . It is a direct product of measures $\mu_0^i (A)$ , $A \in \mathcal{F}_i^0$ , $i = \overline{1,N}$ , given on the measurable space $\{\Omega_i^0, \mathcal{F}_i^0\}$ , where $\mu_0^i (\omega_i^1) =$ $\frac{b - r}{b - a}$ , $\mu_0^i (\omega_i^2) = \frac{r - a}{b - a}$ . The fair price $\varphi_0$ of the contingent liability $\varphi_N(\omega_1,\dots,\omega_N)$ is given by the formula

Proof. For the discount evolution (75), the representation

is true, where $\eta_i(\omega_i) = \frac{\rho_i(\omega_i)) - r}{(1 + r)}$ . Due to Theorems 1, 2, since $\eta_i(\omega_i^1) = \frac{a - r}{1 + r} < 0$ , $\eta_i(\omega_i^2) = \frac{b - r}{1 + r} > 0$ , then the measure $\mu_0$ is unique.

Theorem 3. On the probability space $\{\Omega_{N},\mathcal{F}_{N},P_{N}\}$ , being the direct product of the probability spaces $\{\Omega_{i}^{0},\mathcal{F}_{i}^{0},P_{i}^{0}\}$ , let the evolution of risky asset be given by the formula

where the random values $\rho_{i}(\omega_{i})$ , $i = \overline{1,N}$ , are such that $\rho_{i}(\omega_{i}^{1}) = b_{i}^{1}$ , $\rho_{i}(\omega_{i}^{2}) = b_{i}^{2}$ , $i = \overline{1,N}$ , and let the bank account evolution be given by the formula

where the random values $r_i(\omega_i)$ , $i = \overline{1, N - 1}$ , are such that $r_i(\omega_i^1) = r_i^1$ , $r_i(\omega_i^2) = r_i^2$ , $i = \overline{1, N - 1}$ , $r_0 > 0$ . Then, for the discount evolution of risky asset

the martingale measure $\mu_0$ is unique, if $b_1^1 < r_0 < b_1^2$ , $b_i^1 < r_{i-1}^1 < r_{i-1}^2 < b_i^2$ , $i = \overline{2,N}$ . It is determined by the formula (58) with

The fair price $\varphi_0$ of the contingent liability $\varphi_N(\omega_1,\ldots,\omega_N)$ is given by the formula

Proof. To prove Theorem 3 it is necessary to prove the existence of unique spot measure. The discount evolution (80) can be represented in the form

where

It is evident that $\eta_{i}(\omega_{i}^{1})<0,\;\eta_{i}(\omega_{i}^{2})>0,\;f_{i}(\omega_{1},\ldots,\omega_{i})>0$ . Therefore, from the representation (83), (84) it follows that we can construct only one spot measure, which is martingale measure being equivalent to the initial measure $P_{N}$ . In accordance with Theorem 1, since $S_{n}(\omega_{1},\ldots,\omega_{n}^{1})\neq S_{n}(\omega_{1},\ldots,\omega_{n}^{2}),\;\{\omega_{1},\ldots,\omega_{n-1}\}\in\Omega_{n-1}$ such a measure is unique. Theorem 3 is proved.

Theorem 4. On the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , being the direct product of the probability spaces $\{\Omega_i^0, \mathcal{F}_i^0, P_i^0\}$ , let the evolution of risky asset be given by the formula

where the random values $\varepsilon_{i}(\omega_{i})$ , $i = \overline{1,N}$ , are such that $\varepsilon_{i}(\omega_{i}^{1}) < 0$ , $\varepsilon_{i}(\omega_{i}^{2}) > 0$ , $\sigma_{i}(\omega_{1},\ldots,\omega_{i - 1})\geq \sigma_{i}^{0} > 0$ , $i = \overline{1,N}$ , and let the bank account evolution be given by the formula

where the random values $r_i(\omega_i)$ , $i = \overline{1, N - 1}$ , are such that $r_i(\omega_i^1) = r_i^1$ , $r_i(\omega_i^2) = r_i^2$ , $i = \overline{1, N - 1}$ , $r_0 > 0$ . Then, for the discount evolution of risky asset

the martingale measure $\mu_0$ is unique, if

It is determined by the formula (58) with

The fair price $\varphi_0$ of the contingent liability $\varphi_N(\omega_1,\ldots,\omega_N)$ is given by the formula

Proof. For the discount evolution (87), the following representation

It is evident that $\eta_{i}(\omega_{i}^{1}) < 0$ , $\eta_{i}(\omega_{i}^{2}) > 0$ , $f_{i}(\omega_{1}, \ldots, \omega_{i}) > 0$ . From this, we obtain that the spot measure exists and it is unique. Theorem 4 is proved.

On the probability space $\{\Omega_{N},\mathcal{F}_{N},P_{N}\}$ , being the direct product of probability spaces $\{\Omega_{i}^{0},\mathcal{F}_{i}^{0},P_{i}^{0}\}$ , suppose that the market consists of d assets the evolution of which is given by the law

where

and the random values $\eta_{i}(\omega_{i})$ , $f_{i}(\omega_{1},\ldots,\omega_{i})$ , $i=\overline{1,N}$ , and constants $a_{i}^{k}$ satisfy the inequalities

Proposition 3. On the probability space $\{\Omega_N, \mathcal{F}_N, P_N\}$ , being the direct product of the probability spaces $\{\Omega_i^0, \mathcal{F}_i^0, P_i^0\}$ , let the evolution of risky assets be given by the formulas (93), (94), where constants $a_i^k$ $i = \overline{1,N}$ , $k = \overline{1,d}$ , satisfy the inequalities (95). For such an evolution of risky asset the martingale measure $\mu_0$ does not depend on $a_i^k$ and is unique. It is determined by the formula (58). For the contingent claims $\varphi_N^i(\omega_1, \ldots, \omega_N)$ , $i = \overline{1,d}$ , the fair prices $\varphi_0^i$ are given by the formulas

If $f_{i}(\omega_{1},\ldots,\omega_{i}) = 1$ , $i = \overline{1,N}$ , the unique martingale measure is a direct product of measures $\mu_0^i (A)$ , $A\in \mathcal{F}_i^0$ , given on the measurable space $\{\Omega_i^0,\mathcal{F}_i^0\}$ , $i = \overline{1,N}$ , where

The fair prices $\varphi_0^i$ , $i = \overline{1,N}$ , of the contingent liabilities $\varphi_N^i (\omega_1,\dots,\omega_N)$ , $i = \overline{1,N}$ , are given by the formula

Suppose that $\{g_k^i (X_N)\}_{k = 1}^N$ , $i = \overline{1,d}$ , are the mappings from the set $[0,1]^N$ into itself, where $X_{N} = \{x_{1},\ldots,x_{N}\}$ , $0\leq x_{k}\leq 1$ , $k = \overline{1,N}$ . If $S_0^i,S_1^i,\dots,S_N^i$ , $i = \overline{1,d}$ , are the samples of the processes (93), (94) let us denote the order statistics $S_{(0)}^{i},S_{(1)}^{i},\ldots,S_{(N)}^{i}$ , $i = \overline{1,d}$ , of this samples. Introduce also the denotation

Proposition 4. Suppose that $S_0^i, S_1^i, \ldots, S_N^i$ is a sample of the random processes (93), (94). Then, for the parameters $a_1^i, \ldots, a_N^i$ the estimation

is valid, if for $g_{N}^{i}([S^{i}]_{N}) > 0$ , $[S^{i}]_{N} \in [0,1]^{N}$ , the inequalities $g_{1}^{i}([S^{i}]_{N}) \geq g_{2}^{i}([S^{i}]_{N}) \geq \ldots \geq g_{N}^{i}([S^{i}]_{N})$ are true. If $\tau_{0}^{i}=0$ , then $a_{k}^{i}=1$ , $k=\overline{1,N}$ , $i=\overline{1,d}$ .

In the formulas (99) we put that $f_{k} = \max_{\{\omega_{1},\dots,\omega_{k - 1}\} \in \Omega_{k - 1}}f_{k}(\omega_{1},\dots,\omega_{k - 1},\omega_{k}^{1}),\quad k = \overline{1,N}$ .

V. MARTINGALE MEASURES ON DISCRETE PROBABILITY SPACE

This section presents all the necessary results for constructing a non-arbitrage incomplete market on a discrete probability space. The conditions under which the entire family of martingale measures is described for the considered class of evolution of risky assets are minimal. In particular, conditions are presented under which the family of martingale measures considered is equivalent to the original measure. They are minimal. The entire set of equivalent martingale measures is a convex combination of a finite number of spot martingale measures. On this basis, new formulas were found for the fair price of the super hedge.

In this section, we put that $\Omega_i^0 = \{\omega_i^1,\dots,\omega_i^M\}$ , $i = \overline{1,N}$ , and we assume that $2 < M < \infty$ , the $\sigma$ -algebra $\mathcal{F}_i^0$ consists from all subsets of $\Omega_i^0$ . We suppose that $P_{i}^{0}(\omega_{i}^{k}) > 0,\omega_{i}^{k}\in \Omega_{i}^{0},k = \overline{1,M}$ . As before, the probability space $\{\Omega_N,\mathcal{F}_N,P_N\}$ is a direct product of probability spaces $\{\Omega_i^0,\mathcal{F}_i^0,P_i^0\},i = \overline{1,N}$ . Sometimes, any elementary event $\omega_{i}^{k}\in \Omega_{i}^{0}$ it is convenient to denote by $\omega_{i}$ not indicating the index $k$ . Further, we use the both denotations. As in section 2, we introduce filtration $\mathcal{F}_n$ on the probability space $\{\Omega_N,\mathcal{F}_N,P_N\}$ . As before, it is convenient to introduce the family of probability spaces $\{\Omega_n,\mathcal{F}_n,P_n\},n = \overline{1,N}$ , being a direct product of probability spaces $\{\Omega_i^0,\mathcal{F}_i^0,P_i^0\},i = \overline{1,n}$ .

The evolution of risky assets is given by the formula (8) with the assumptions given in the section 2. In this case

where $\Omega_n^{0-} = \{\omega_n \in \Omega_n^0, \eta_n(\omega_n) \leq 0\}, \Omega_n^{0+} = \{\omega_n \in \Omega_n^0, \eta_n(\omega_n) > 0\}, P_n^0(\{\omega_n, \eta_n(\omega_n) > 0\}) > 0, P_n^0(\{\omega_n, \eta_n(\omega_n) < 0\}) > 0$ . Further, we also use the measurable space with measure

The measure $P_{n}^{0-}$ is a contraction of the measure $P_{n}^{0}$ on the $\sigma$ -algebra $\mathcal{F}_{n}^{0-} = \Omega_{n}^{0-} \cap \mathcal{F}_{n}^{0}$ , $P_{n}^{0+}$ is a contraction of the measure $P_{n}^{0}$ on the $\sigma$ -algebra $\mathcal{F}_{n}^{0+} = \Omega_{n}^{0+} \cap \mathcal{F}_{n}^{0}$ . Additionally, we assume

In this case, Lemma 1 (see [1]) is formulated as follows

Lemma 2. Suppose that for $\Omega_n^a$ , $a = -, +$ , $n = \overline{1, N}$ , the representations (100) are true. If the conditions

are true, then the set of bounded strictly positive random values $\alpha_{n}(\{\omega\}_{n}^{1};\{\omega\}_{n}^{2})$ , satisfying the conditions (14) - (16), (see [1]) is a nonempty set.

Lemma 3. Suppose that the conditions of Lemma 2 are true. For the measure $\mu_0(A)$ , $A \in \mathcal{F}_N$ , constructed by the recurrent relations (23) - (25), (see [1]) the representation

is true and $\mu_{0}(\Omega_{N})=1$ , that is, the measure $\mu_{0}(A)$ is a probability measure being equivalent to the measure $P_{N}$ , where we put

Proof. We only need to prove that $\psi_n(\omega_1,\ldots,\omega_n) > 0$ , $n = \overline{1,N}$ . Suppose that

where

Since

where

Therefore,

Analogously,