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

This paper is a generalization of the results of the previous papers. Using these results a class of evolutions of risk assets based on the geometric Brownian motion is constructed. Among these evolutions of risk assets, the important class of the random processes is the random processes with parameters built on the basis of the discrete geometric Brownian motion. For this class of random processes the interval of non-arbitrage prices are found for the wide class of contingent liabilities. In particular, for the payoff functions of standard options call and put of the European type the fair prices of super-hedge are obtained. Analogous results are obtained for the put and call of arithmetical options of Asian type. For the parameters entering in the definition of random process the description of all statistical estimates is presented. Statistical estimate for which the fair price of super-hedge for the payoff functions of standard call and put options of European type is minimal is indicated. From the formulas found it follows that the fair price of super-hedge can be less than the price of the underlying asset. In terms of estimates the simple formula for the fair price of super-hedge is found. Every estimates can be realized in the reality. This depends on the distribution function of the observed dates in the financial market.


Introduction
In reality, all financial markets are incomplete and the evolution of risky assets is discrete. The question arises, what random process describes the evolution of risky assets in the financial markets? This problem is important both from the perspective of the risk asset price behavior and from the risk hedging behavior of How to cite this paper: Gonchar, N.S. (2020) Assessment of Contingent Liabilities for Risk Assets Evolutions Built on Brow-the risk asset.
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]. The problem of estimation of the range of non arbitrage prices was began in the papers [2], [3] for the Levy exponential processes and the diffusion processes with jumps describing evolution of risk assets. The upper estimate for the standard call option payoff function in this paper coincides with the price of underlying asset. 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. Having this representation the formulas for the lower and upper bounds of non arbitrage prices are found. It is showed that the upper bound for the payoff functions of standard call option of European type is less than the price of underlying asset. The statistical estimates of parameters entering entering in the introduced evolutions of risk assets are obtained. The statistic for which the fair price of super-hedge is minimal is indicated.
In terms of statistical estimates the simple formulas for the fair price of super-hedge are obtained. Every estimate can be realized in the reality. This depends on distribution function of the observed dates in the financial market.

A Wide Class of Non-Arbitrage Evolutions of Risky Assets
In this section, we generalize the results of the paper [1]. On the probability space where the random values ( ) 1 1 , , , satisfy the conditions ( ) . 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 , , , 1, n k A n k = ∞ , coincides with 1  .  3) the minimal σ-algebra containing all , n ks B , , , 1, n k s = ∞ , coincides with 1 2 ×   .
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]. Lemma 1. Let { } 1 1 , Ω  be a measurable space with a complete separable metric space 1 Ω and Borel σ-algebra 1  on it. Then { } 1 1 , Ω  has an exhaustive decomposition. , , , , , , The point 0 ω belongs to this ball and for every is true. Therefore , and the random values , are integrable ones relative to the measure 0 1 P . For this purpose, we introduce the denotations:  is a direct product of the σ- At last, let In the next Theorem 1, we assume that the random values  ) ( ) Due to Lemma 4, [1], this proves that the set M is a regular set of measure.  Let us introduce the denotations (see also [1] Note that the σ-algebra n  is generated by sets of the kind From this, for the measure Q the representation Since for the set 1 A the representation is true, where we introduced the denotations n P − and n P + which are the contractions of the measure n P onto the σ-algebras By the definition we put that for 1 n = the transformation 1 T is identical one.
Introduce the denotations  . ; It is evident that the expression (31) equals zero for every is a probability measure on the σ-algebra n  . Taking into account the denotation (26) and the formula (35), we obtain that the measure Since 0 1 Ω is a separable metric space, then it has an exhaustive decomposition. This is true for n Ω which is also separable metric space for every On the probability space { } , , n n n n n n P P ; n n f ω ω with probability one, as , m → ∞ since it is a regular martingale. It is evident that for those Then the contraction of the sequence of martingale measures m Q ε generated by sequence (44) on the σ-algebra n  is given by the formula From the equalities (45), (46) it follows that Let us denote From the inequalities (50), we obtain the inequalities Two cases are possible: a) for all 1 First, let us consider the case a).
Since the inequalities (51) are valid for every value ( ) ( ) 1 1 From the definition of 1 α , we obtain the inequalities ( ) ( ) ( ) Consider the case b). From the inequality (51), we obtain the inequalities The inequalities (57) give the inequalities . Then, from (57) we obtain the in- From the definition of 1 α , we have the inequalities The inequalities (60), (61) give the inequalities ( ) ( ) n n n n n n n n n n n n n n n n n n n n n n n n f f Taking into account the first part of the proof of Theorem 3 from the inequality where the constant 1 α is the same as in the first part of the proof of Theorem 3.
The inequalities (66) and Theorems 3, 4 [1], [17] prove Theorem 4. Ω be a complete separable metric space. Suppose that the evolution of the risk asset is defined by the formula (2) and the non risk asset evolve by the law 1 If the nonnegative payoff function N f is N  measurable integrable random value relative to every martingale measure and satisfying the conditions Theorem 16 from [17], then the fair price of super-hedge is given by the formula

Interval of Non-Arbitrage Prices for a Wide Class of Evolutions of Risky Assets
In the papers [2], [3] the range of non arbitrage prices are established. In the paper [2], for the Levy exponential model, the price of super-hedge for call option coincides with the price of the underlying asset under the assumption that the Levy process has unlimited variation, does not contain a Brownian component, with negative jumps of arbitrary magnitude. The same result is true obtained in the paper [3] if the process describing the evolution of the underlying asset is a diffusion process with the jumps described by Poisson jump process. In these papers the evolution is described by continuous processes. Below we consider the discrete evolution of risky assets that is more realistic from the practical point of view. 2) for every 1 t < < ∞ , Suppose that the evolution of risk asset is given by the formula (2) Proof. Due to Theorem 1 and 6 we have   To prove the inverse inequality we use the inequality Therefore, putting in the inequality (73) ( ) Let us prove the equality (69). Using Jensen inequality [18] we obtain Let us prove the inverse inequality Putting in this inequality ( ) 2) for every 1 t < < ∞ , Suppose that the evolution of risk asset is given by the formula (2) If, in addition, the nonnegative payoff function ( ) Therefore, putting in the inequality (81) ( ) Let us prove the equality (78). Due to convexity of payoff function ( ) f x , using Jensen inequality we obtain Let us prove the inverse inequality Putting in this inequality ( )   , let us consider the random values ( ) The random values ( ) , are independent between themselves.
The random values

Models of Evolution of Risky Assets Based on the Discrete Geometric Brownian Motion
Suppose that the set 0 For this purpose we could do it using the method, presented in section 2. But for further applications, it is more convenient to construct the simple probability space { } , ,P Ω  and the random process on it, which is equivalent in the wide sense to the process (86). Under the σ-algebra  on Ω , we understand the minimal σ-algebra generated by sets On the probability space { } , ,P Ω  , the random process given by the formula (88) is equivalent in the wide sense to the process (86), constructed above. Proof. The proof of Theorem 9 follows directly from Theorem 6.
If, in addition, the nonnegative payoff function ( ) Proof. As before, If, in addition, the nonnegative payoff function ( ) which follows from the identity: ( ) ( ) we obtain the inequality ( ) This proves Theorem 12. □ Theorem 13. On the measurable space { } , Ω  with the filtration n  , 0, n N = , on it, let the discount risk asset evolution is given by the formula (88), with 0 1 i a ≤ ≤ , For the payoff function ( ) ( ) The set of non arbitrage prices coincides with the set ( ) ( ) is true. Taking into account the inequality (96) of Theorem 11 we prove Theorem 13.
The set of non arbitrage prices coincides with the set Let us prove the inverse inequality. We have ( ) In the formula (121) we introduced the denotation ( ) The proof of Theorem 15 follows from the equality (121). □

Characteristic of the Random Processes Built on the Discrete Geometric Brownian Motion
On the probability space { } , ,P Ω  , with every sequence real numbers Theorem 16. The random process (123) is a non homogeneous Markov process with the transition probability function for the n-th step where the measure ( ) is a Lebesgue measure for those i for which

Estimation of the Parameters of the Considered Random Process
Suppose that is a sample of the process , , Theorem 17. Suppose that 0 1 , , , N S S S  is a sample of the random process (123). Then for the parameters 1 , , N a a  the estimation Proof. The estimation of the parameters 1 , , N a a  we do using the representation of random process It is evident that 0, 2, is also estimation of the parameters 1 is an estimation for the parameters 1 , , N a a  .
In the next Theorems we put 0 1 τ = . This corresponds to the fact that fair price of super-hedge is minimal for the considered statistic.
The fair price of super-hedge for the statistic (143), (144) is given by the formula The fair price of super-hedge is minimal one for the statistic (137) with − , and is given by the formula The set of non arbitrage prices coincides with the closed interval The fair price of super-hedge is minimal one for the statistic (137) with The set of non arbitrage prices coincides with the closed interval The fair price of super-hedge is minimal one for the statistic (137) with − , and is given by the formula The set of non arbitrage prices coincides with the closed interval ( ) The fair price of super-hedge is minimal one for the statistic (137) with   . This follows from the structure of random process (123), the transition probability function (126), the structure of the joint probability function (134) after Theorem 16.

Conclusions
In the paper, we generalize the results of the paper [1] [17]. In Section 2, we generalize an evolution of risk asset with memory proposed in the paper [1]. In Theorem 1, we describe completely the set of martingale measures for the considered evolution and prove that every martingale measure of this family is an integral over some measure on the set of extreme points of the set of martingale measures. In Theorem 3 the bound for every nonnegative n  measurable random value the mathematical expectation for which relative to every martingale measure is bounded by 1 is found. In Theorem 4, it is proved that every nonnegative super-martingale relative to the regular set of measures is a local regular one. The same statement, as in Theorem 4, it is proved in Theorem 5 in the case, as a super-martingale is bounded from below. Section 3 contains the application of the results obtained above to calculation of the interval of non-arbitrage prices for the wide class of evolutions of risky assets and payoff functions.
In Theorem 7, with the general assumptions about payoff functions and the evolution of risky assets, we found the non-arbitrage price interval and set the price of super-hedge. This set of payoff functions contains a payoff function of a standard European-type call option. Theorem 8 contains sufficient conditions regarding the evolution of risky assets (2) for which an interval of non-arbitrage prices has been found for a wide class of payoff functions. This class contains the payoff function for standard option put of European type. Section 4 contains the results about the interval of non-arbitrage prices for the class of evolutions of risky assets described by the random process with parameters built on the geometric Brownian motion and payoff functions for call and put options of standard type.
In Theorem 9, a formula for the fair price of super-hedge is found for the evolution of risky assets given by the formula (89). Theorem 10 contains the estimates for the value of a super-hedge for a particular class of payoff functions including a payoff function for a standard call option.
In Theorem 11, the estimates for the value of a super-hedge for a particular class of payoff functions are found. This class of payoff functions includes a standard put option payoff function.
Theorem 12 gives the interval for non-arbitrage prices and the price of super-hedge in the case of a standard call options. The peculiarity of this formula is that the price of a super-hedge is proportional to the price of the underlying asset with a ratio less than one.
In Theorem 13, the formula for the fair price of super-hedge is found for put option of standard type.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.