Abstract and Applied Analysis

Abstract and Applied Analysis / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5568636 | https://doi.org/10.1155/2021/5568636

Sergey N. Smirnov, Andrey Yu. Zanochkin, "Guaranteed Deterministic Approach to Superhedging: Case of Binary European Option", Abstract and Applied Analysis, vol. 2021, Article ID 5568636, 18 pages, 2021. https://doi.org/10.1155/2021/5568636

Guaranteed Deterministic Approach to Superhedging: Case of Binary European Option

Academic Editor: Lucas Jodar
Received16 Jan 2021
Accepted31 May 2021
Published18 Jun 2021

Abstract

For the superreplication problem with discrete time, a guaranteed deterministic formulation is considered: the problem is to guarantee coverage of the contingent liability on sold option under all admissible scenarios. These scenarios are defined by means of a priori defined compacts dependent on price prehistory: the price increments at each point in time must lie in the corresponding compacts. In a general case, we consider a market with trading constraints and assume the absence of transaction costs. The formulation of the problem is game theoretic and leads to the Bellman–Isaacs equations. This paper analyses the solution to these equations for a specific pricing problem, i.e., for a binary option of the European type, within a multiplicative market model, with no trading constraints. A number of solution properties and an algorithm for the numerical solution of the Bellman equations are derived. The interest in this problem, from a mathematical prospective, is related to the discontinuity of the option payoff function.

1. Introduction

1.1. Literature Review

One of the first publications to develop a guaranteed deterministic approach is an article by Kolokoltsov [1], published in 1998. To the best of our knowledge, this was the first work to explicitly articulate this approach to pricing and hedging contingent clams. Implicitly, however, some mathematical tools for a guaranteed deterministic approach were already present in 1994 in the first edition of the book by Dana and Jeanblanc-Picqué [2] (Sections 1.1.6 and 1.2.4). The result of the first part of [1] (the case of a single risky asset and a convex payout function on European option) follows from [2]. The guaranteed deterministic approach is closely related to a class of market models called interval models in [3], especially to the ideas and results of Kolokoltsov published in [3] (Chapters 11–14), including the independent discovery of the game-theoretic interpretation of risk-neutral probabilities under the assumption of no trading constraints; we find this interpretation to be quite important from an economic point of view. One can also consider guaranteed deterministic approach to be a Merton-type approach, which goes back to 1973; see [4] (no reference probability measure is used in this seminal work). Note that we share an idea, suggested in an unpublished work of Carassus and Vargiolu about 15 years ago and finally published in [5]: in order to get a meaningful theory, it is reasonable to assume the boundedness of price increments.

Formally, from the contemporary point of view (the guaranteed deterministic approach was developed by us in the late 90s (although at that period we were not aware of Kolokoltsov’s paper), but published (primarily in Russian) only in the last three years, together with some recent new results), the guaranteed deterministic approach to the superhedging problem can be classified as a specific pathwise (or pointwise) approach addressing uncertainty in market modelling by defining a set of deterministic market scenarios (described in detail in the next section), a result of an agent’s beliefs. Or it can be formally described in terms “quasisure” approach (we refer to [6, 7] for these two robust modelling approaches and for detailed review of large literature focusing on robust approach to mathematical finance), by the choice of a collection of probabilistic models (possible priors) for the market. In our case, all these probabilities initially (but can be enlarged to a family of probabilities which is a mixed extension of pure “market” strategies) are Dirac measures (but certainly not all of them). However, it is to stress that we adopt an alternative interpretation to the common robust approach to pricing of contingent liabilities. Our interpretation, as already mentioned above, is game theoretic: we deal with a deterministic dynamic two-player zero-sum game of “hedger” against “market.” A family of probabilities appears as a secondary notion, thanks to the introduction of mixed strategies of the “market.”

We deem to be related to our approach a formulation of the upper hedging price based on the game-theoretic probability, presented in [8].

1.2. Problem Statement

The present paper joins a series of publications (in particular, [9] describes the market model in detail and provides a literature review) [915] that develop a financial market model consistent with an uncertain deterministic price evolution with discrete time: asset prices evolve deterministically under uncertainty described using a priori information about possible price increments. Namely, they are assumed to lie in the given compacts that depend on the prehistory of the prices (such a model is an alternative to the traditional probabilistic market model (in our proposed deterministic approach, the reference probability measure is not initially set, as it is supposed in the probabilistic approach, see, e.g., [16])).

The proposed approach allows us to simplify the mathematical technique to a certain extent and make the formulation of statements more understandable for economists. The advantages of the approach include game-theoretic interpretation (in the absence of trading constraints, this interpretation provides an economically important explanation for the emergence of risk-neutral probabilities as one of the properties of the most unfavourable mixed market strategies).

The market model described above explores the problem of option pricing, by which we mean nondeliverable (for the risk management purposes, mainly nondeliverable contracts are used) over-the-counter contracts whose payoffs depend on the evolution of underlying asset prices up to the time of expiration. The writer of an option assumes a contingent liability that, unlike contingent liabilities on insurance policies, can be protected from market risk by hedging in markets (by means of transactions in underlying assets and risk-free assets). One of the most important ways to hedge the contingent liability of a sold option is through superreplication (this term originated because conditional liabilities cannot be replicated in incomplete markets (this is only possible in complete markets)) or in other words superhedging (we prefer to use the second of the two equivalent terms). The problem of option pricing in superhedging is to determine the minimum level of funds at the initial moment required by the seller (in other words, it is the premium charged to the buyer of the option if the seller uses superhedged pricing), which guarantee, if an appropriate hedging strategy is chosen, the coverage of the contingent liability under the option sold (remind that the corresponding payments under depend on the prehistory of prices). In general, we consider American-style options (American-style options) in which the seller’s counterparty (the option holder) can exercise the option (i.e., demand payment in accordance with the rules set out in that contract) at any time, up to the expiration of the option. Note that European- and Bermuda-type options can be seen as a case of American options, subject to certain regularity conditions, including “no arbitrage” condition, in a certain sense.

Let us now formalize the above construction for the superhedging problem. The main premise of the proposed approach is to specify “uncertain” price dynamics by assuming a priori information about price movements at time , namely, that the increments (the increments are taken “backward,” i.e., , where is the vector of discounted prices at time ; the -th component of this vector represents the unit price of the -th asset) of discounted prices (we assume that the risk-free asset has a constant price equal to 1) lie in a priori defined compacts (the dot denotes the variables describing the evolution of prices. More precisely, this is the prehistory for , while for the functions and introduced below, this is the history ) , where the point denotes the prehistory of prices up to and including time , . We denote by the infimum of the portfolio value at time , at a known prehistory that guarantees, given some choice of an acceptable hedging strategy, the coverage of the current and future liabilities arising with respect to possible payoffs on the American option.

The corresponding Bellman–Isaacs equations in discounted prices arise directly from an economic sense by choosing, at step , the “best” admissible hedging strategy (vector describes the size of positions taken in assets, i.e., the -th component of this vector represents the number of units of the -th asset being bought or sold) for the “worst-case” scenario of (discounted) prices increments for given functions , describing the potential option payoff. Thus, we obtain the following recurrence relations (the sign denotes the maximum, and is the scalar product of vector on vector ):

where describes the prehistory with respect to the present moment . The conditions for the validity of (1) are formulated in Theorem 3.1 of [17].

Multivalued mappings and , as well as functions , are assumed to be given for all , . Therefore, the functions are given by equation (1) for all . In equation (1), the functions , as well as the corresponding suprema and infima, take values in the extended set of real numbers , a two-point compactification (the neighbourhoods of points and are , and , , respectively) of .The derivation of equation (1) is easily obtained by a kind of “engineering reasoning.” In informal economic language, this can be explained as follows. Assuming for simplicity that suprema and infima in equation (1) are attained, let ; by the current (present) time , we know the (discounted) price history . The portfolio value when hedging the contingent liability of a sold American option should first be no less than the current liability, equal to the potential payout , to guarantee its coverage. Second, the portfolio value at the next moment (here, the strategy is formed at moment and can only depend on the prehistory of prices ) should provide a guaranteed coverage of the contingent claim under any scenario of price movements at step ; hence, it should be not less than . Thus, to cover future liabilities, the portfolio value when an admissible hedging strategy is used should be no less than under the worst-case scenario of price movements at step , i.e., for that maximizes the expression . The resulting value is minimized by choosing a strategy ) to evaluate the required reserves to cover future potential payoffs. It remains to put equal to the maximum amount of current liabilities and the amount of reserves for future potential payments.

We deem a trajectory on the time interval of asset prices to be possible if . Let us denote by the set of possible trajectories of asset prices on the time interval ; thus,

One of the conditions for the validity of (1) is the assumption of boundedness of payoff functions formulated in Theorem 3.1 from [9], due to which the functions are bounded from above. The assumption is as follows.

Throughout the following, we will assume that the assumptions listed in Theorem 3.1 of [9] as well as those listed in (2) of Remark 3.1 of [9] are met.

This paper considers the problem of superhedging pricing of a binary option (European type) for a multiplicative one-dimensional market model, under the assumption of no trading constraints. A number of solution (1) properties are obtained, in particular, continuity except a single point. In addition, an algorithm for obtaining a “semi-implicit” solution (1), represented in the form of a piecewise rational function, is proposed. The interest to this problem is caused by the fact that the payout function is discontinuous, and therefore, the results concerning the case of continuous payout functions given in [12, 13] are not applicable here.

2. Auxiliary Results

Throughout the discussion below, we refer only to discounted prices. The price of the risk-free asset (after discounting, see [9]) is identically equal to 1. According to the terminology proposed in [9], for risky assets, the price dynamics (trading constraints) belongs to the Markov type if (respectively ) depend only on the price value at the previous moment, i.e., (respectively ) can be represented in the following form: respectively, for .Let us formulate some simple but useful statements.

Proposition 1. If price dynamics and trade constraints are of the Markov type and the payoff functions depend only on the current price, i.e., for are represented in the form then the solutions of the Bellman–Isaacs equation (1) also depend only on the current price, i.e., for , they can be represented in the following form:

Proof. It follows directly from the form of the Bellman–Isaacs equation (1). ☐

Proposition 2. Let assumptions (2) and (4) be satisfied, trading constraints be absent (in this case, condition (5) is obviously fulfilled), i.e., , and the condition NDAO of no arbitrage opportunities be satisfied (in this case, the NDAO condition is equivalent to a geometric one: lies in the relative interior of convex hull of ; see [10]). Then, for European options, the solutions of the Bellman–Isaacs equation (1) are monotonically decreasing in time, i.e.,

Proof. When there are no trading constraints and the condition NDAO of no arbitrage opportunities is fulfilled, we can assume that this is a special case of American options with payout functions (in principle, a weaker condition NDSA of no guaranteed arbitrage is sufficient for this; see [10])

Using Proposition 1 and the theorem proved in [11], we obtain for that the representation (7) holds and the following equality is valid where is the set of probability measures on with a finite support (in fact, it is sufficient to consider the set of measures with the number of support points not exceeding ) satisfying the martingality condition (more precisely, the price increments form a martingale difference sequence): . In particular, , where is a probability measure centred at point 0, and thus,

Proposition 3. Let for the one-dimensional model (that is, for a model with one risky asset (and one riskless asset)) the assumptions of Proposition 1 be satisfied and the payoff functions be monotonically nondecreasing (respectively, monotonically nonincreasing). Then, the solutions of the Bellman–Isaacs equations are also monotonically nondecreasing (respectively, monotonically nonincreasing).

Proof. This follows directly from the form of the Bellman–Isaacs equation (1). ☐

Further, we consider a one-dimensional market model, where, in a multiplicative representation, the dynamics of the discounted price of a risky asset are described by the following relations (according to the terminology proposed in [9], in this case, the price dynamics refer to a multiplicative-independent type): where (here, the prices and multipliers are considered as “uncertain” values (a deterministic analogue of random variables)) the multiplier The trading constraints are absent and the condition NDAO of no arbitrage opportunities is fulfilled, which in our case is equivalent to the following inequalities:

A model of this kind was first proposed by Kolokoltsov [1]. (1)If the function satisfies the Lipschitz condition on some interval , then the function also satisfies the Lipschitz condition on the (narrower) interval , and on this interval, the Lipschitz constant for does not exceed the Lipschitz constant for on the interval (2)If there is an upper estimate of the Bellman function for , then for (3)If the payoff functions are upper semicontinuous, then the strict inequality for entails a strict inequality for (4)If , , and , then for the inequality holds, where

Proposition 4. Let the model of price dynamics be described by relations (12), (13), and (14); we fix . Then, the following statements hold for the European option.

Proof. (1)Let us use the multiplicative analogue of formula (10) for the European option: where is the set of probability measures on with a finite support (it is sufficient to consider the set of measures with the number of support points not exceeding ) satisfying the multiplicative martingality condition: . Denote the Lipschitz constant for on the interval by . Since for the inclusion holds, for any points and such that we have the following inequalities: (2)Given the inclusion for , for any we havewhence, according to (16), we obtain . (3)Under the assumptions made, because the supremum in (16) is attained (see [13]) for some measure , then(4)For , choose and ; we have then . Consider a measure concentrated at points (the probabilities of these points are uniquely determined from the normalization and martingality conditions; therefore, depends on , and ) and . Thanks to the choice of constants and in (15), the functions and coincide at the points of the support of measure , and we obtain the following equality:whence, using (16), we obtain the required inequality. ☐

3. Binary Option of European Type

3.1. General Case of the Support of Distribution of Uncertain Multiplier

Within the framework of the price dynamics model described by relations (12), (13), and (14), we are interested in the superhedging problem within the guaranteed deterministic approach for a European-type binary option. Without limiting the generality, we can assume that the strike price is equal to 1. Let us consider a binary call option (the case of a binary put option can be investigated using similar methods) whose payoff function at the expiration moment is equal to where is an indicator function of set . Note that Proposition 2 is applicable in the case of our model, and thus, the solutions of the Bellman–Isaacs equation (1) are monotonically nonincreasing over time. By virtue of the condition of the absence of NDAO arbitrage opportunities, as noted above, the European option superhedging problem is reduced to the American option superhedging problem, with the payoff functions described by (9), i.e., with zero payoff functions except for the expiration moment (21). Thus, Proposition 1 is applicable, and a representation of the form (7) holds for the solution of the corresponding Bellman–Isaacs equations. Hereinafter, we will consider our problem as a superhedging of an American option with zero payoff functions except for the expiration moment. Since the terminal payoff function is monotonically nondecreasing, Proposition 3 is applicable. Thus, the solutions to the corresponding Bellman–Isaacs equations are also monotonically nondecreasing, or equivalently, by notation (26), the functions are monotonically nondecreasing. Therefore, these functions can have discontinuities of the first kind (jumps) only. In addition, as the payoff function is upper semicontinuous and the multivalued mappings and are continuous, the solutions to the Bellman–Isaacs equations are also upper semicontinuous; see [12]. For monotonically nondecreasing functions, upper semicontinuity is equivalent to their right continuity. Since the solutions of the Bellman–Isaacs equations are upper semicontinuous, a game equilibrium takes place (at each time step); see [14]. In this case, according to the results of [14], for the saddle point, the most unfavourable mixed strategies are achieved in the class of distributions concentrated in no more than two points. To find the solution to the Bellman equations (after separating the pricing problem from the hedging problem), it is sufficient (see [15]) to construct at each step on the interval (upper semicontinuous) concave envelope of Bellman function and set .

3.2. Cox–Ross–Rubinstein Assumption about the Endpoints of the Uncertain Multiplier Support

The general case of parameters and is quite difficult to analyse owing to the chaotic behaviour (including the mutual position) of the products of the form , where and are nonnegative integers, unless and are rationally commensurable. We choose the simplest case of rational commensurability of and , proposed in the Cox–Ross–Rubinstein model [18], namely, we apply

In this case, the condition of no arbitrage opportunities (14) is automatically satisfied for . Note that assumption (22) simplifies significantly the analysis: if, at step , point , the price value at the previous time, lies in an interval of the form , then the endpoints of the interval of the possible values of the uncertain value given , i.e., points and , lie in the adjacent intervals and , respectively. We will say that the points form a skeleton at step . The most unfavourable mixed market strategies in step for a given price in the previous step may be nonunique. For example, if , any distribution with the support contained in and the barycentre would be such, and if , any distribution with the support contained in and the barycentre would be such. At points where there is a nonuniqueness of the most unfavourable mixed market strategy, we adopt a convention to choose a distribution with barycentre that has the minimum number of support points to fix the unique “optimal” mixed market strategy. There will never be more than two such points, and hence, given the martingality condition, the corresponding distribution is defined in the only way possible. Due to this convention, the conditional distribution of price given , concentrated in no more than two points, will be chosen as the most unfavourable mixed market strategy at step (when the maximum in (16) is attained). We call the support of the distribution a scenario. When the scenario is a one-point set, , where denotes the probability measure concentrated at a point . When the scenario is a set of two points, has the following form: where . Given a scenario, the probabilities and are uniquely defined from the normalization condition and price martingality condition, whence

For convenience, we shall use the following notations:

In particular, , where is given by (21). The recurrence relations for are

In what follows, using notation (26), we will investigate the properties of the solution of the European binary call option superhedging problem, with the payoff function at the expiration moment given by (21), for the market described using relations (12), (13), (14), and (22).

3.3. Solutions of the Bellman Equations for the First Two Steps

For , the function is identically equal to zero because the interval is contained in , where the function is zero. For , the function is identically equal to 1 because the (upper semicontinuous) concave envelope of the function on at is equal to 1.

Note that in the first step, for , the most unfavourable mixed market strategy can be a conditional distribution concentrated at two points and , with probabilities and , respectively. Formula (25) in this case takes the form and by (27), the values of function on the interval are given by the expression

Thus, in the interval , the scenario is realized, and function has a hyperbolic form which is strictly monotonically increasing and (strictly) convex. At point , the function has a single discontinuity (jump), is right-continuous, and

On the right endpoint of interval by (31), we have so that function is continuous at point 1.

Note that the line passing through the points in the plane of the hyperbola (31) corresponding to the arguments and 1, i.e., passing through the points with coordinates and , is defined by which has a root , i.e.,

In particular, for , we obtain that the line passing through the points of the hyperbola (31) corresponding to the arguments and 1 have the root . To complete the geometric image, we also note that the tangent at point to the restriction of the function to the interval , given by the function has a root .

The graph of the function for is shown in Figure 1.

It follows from (33) and (34) that for , the line segment defined by function (23), connecting points with coordinates and , is a (upper semicontinuous) concave envelope of function on the interval , and thus, for . At the right endpoint of the hyperbola (36), the equality holds.

Note that in the second step, for , the most unfavourable mixed market strategy (note that when the most unfavourable mixed market strategy is not unique: any distribution with barycentre concentrated at no more than three points: , , and 1, i.e., a distribution represented as a mixture , is “optimal”) can be represented as a conditional distribution of the form . Formula (25) in this case takes the form and by (27), the function on the interval , taking into account (31), is an affine function, namely,

Specifically, Given (37), the function is therefore continuous at point . The function is not only continuous at : it turns out that at this point there exists a derivative equal to so that the function is differentiable at . It is easily seen that for the function is identically equal to zero, and for , the function is identically equal to one. Because (39) implies , the function is continuous at point 1, and hence, the function is continuous at .

The graph of the function for is shown in Figure 2.

3.4. Solutions of the Bellman Equations: Recurrence Properties

We now fix .

Proposition 5. Outside the interval , the function takes the following values:

Proof. The relations in (41) are obtained through induction, given the property noted in the previous section, and the endpoints of the interval for lie in adjacent intervals, that is, and . For , this property is established as described in the previous section. Suppose (28) is valid for , let us show its validity for . The function is identically equal to zero for , as the interval is contained in , where the function is equal to zero. For , the function is identically equal to 1 because the (upper semicontinuous) concave envelope of the function on at is equal to 1. ☐

Proposition 6. The function has a discontinuity (jump) at point , in which is right continuous, and on the interval , the function satisfies the property of self-similarity (owing to the properties of function , on the interval , the function is strictly monotonically increasing and strictly convex):

Proof. When , (43) is an identity. Let us make the inductive assumption that (43) holds for and check that it holds for . Substituting in (29) and expression (31) for , we have for

From geometric similarity considerations, it is clear that for , the concave envelope of the function on the interval is the line segment connecting the points with coordinates и¸ given by where , and hence, for which follows from formula (43) for . Using Proposition 5, we have , and putting in (46), we get Thus, has a jump at point (where is right continuous).

Theorem 7. (1)For , the function is convex on each of the intervals (2)For , it is sufficient to consider only four scenarios, i.e., the variants of the point locations as and introduced in Section 3.2: (I)Scenarioand(II)Scenarioand(III)Scenarioand(IV)ScenarioandMoreover, the number of possible switching scenarios on the intervals does not exceed 2. (3)For , the function is piecewise rational on the interval or, more precisely, rational on at most adjacent intervals, which we shall call rationality intervals (in particular, for , the function is infinitely differentiable within intervals of rationality interior), with endpoints ; all points of type are endpoints of rationality intervals for the function . The partitioning into rationality intervals for the function is a refinement of the partitioning into rationality intervals for the function . For the given intervals of rationality of the rational functions represented in the form of an irreducible fraction of polynomials, the degree of polynomials does not exceed , and this degree on intervals and equals zero; if scenario I is realized, the degree equals 1, whereas is scenario IV is realized, the degree does not exceed (4)For , the derivative of the function is positive (at points that are endpoints of rationality intervals, a jump in the derivative of the function may occur, but not necessarily so, as seen in the example of the function ). In particular, the function is strictly monotone on the interval

Proof. For convenience, we write out for scenarios I, II, III, and IV the specific formulas given in the general case by (23), (25), and (27). Note that for those points for which one of the scenarios I, II, III, and IV holds, the points and belonging to the support of distribution given by (16), and hence, the probabilities and are independent of , and thus, for these scenarios the carrier points and probabilities will have omitted. ☐

For scenario I, when , , and , the probabilities and take the form of affine functions and the values of the function are expressed through the values of the function by the formula Thus, in the case of scenario I on the interval , the function is affine, and in the case of this scenario , for some , the function values match:

In addition, in the case of this scenario, for , for some , the following “matching” relations take place:

For scenario II, when , , and , the probabilities and take the form and the values of the function are expressed through the values of the function by the formula In this scenario for , for some , the “matching” relations take place: and in the case of this scenario for , for some , the “matching” relations take place:

For scenario III, when , а , and , the probabilities and are as follows: and the values of the function are expressed through the values of the function by the formula In this scenario, for , for some , the “matching” relations take place: and in this scenario, for , for some , the “matching” relations take place:

For scenario IV, when , , and , the probabilities and are as follows: and the values of the function are expressed through the values of the function by the formula and in this scenario, for , for some , the “matching” relations take place: and in this scenario, for , for some , the “matching” relations take place:

Let us show by induction that for the function satisfies the four properties from the formulation of the theorem. For , these properties are satisfied (note that for the function , scenario II takes place on the interval , whereas on the interval , scenario I takes place). Suppose that this property is satisfied for . Let us check its fulfilment for . For from the interval , when , this follows from formula (43). If and point lies in an interval of the form , as mentioned above, the endpoints of the interval , that is, points and , lie in adjacent intervals and , respectively. At the points , a positive jump is in principle possible (below, we prove that continuity takes place at these points), and continuity to the right takes place. In case when there is a jump at the point , the function preserves the convexity on the closed interval if it is convex on the interval . Owing to the convexity of the function on the interval , for , one may not consider any point of the open interval as a “candidate” to be a point of the support of the most unfavourable mixed market strategy; it is sufficient to consider only the extreme points и from the interval .Next, we fix the numbers and consider a distribution concentrated at points and with probabilities and , respectively, satisfying the condition ; subject to normalization, whence

Let us show that the integral considered as a function of , i.e., the function , is monotonically nondecreasing on , where and are given by (64) and are considered as functions of the variable . We shall need the following result from a mathematical analysis. If functions and are absolutely continuous on the interval and and are their derivatives (defined almost everywhere with respect to the Lebesgue measure), then functions and are summable (in this case, the product is absolutely continuous on the interval , which can be verified directly by definition, given the boundedness of functions and ) and for ; see Theorem 5 of Section 7 of Chapter IX in [19]. Let us add to this that the convex function is absolutely continuous and one can choose for its derivative an equivalent (it is a function coinciding with the original at almost all points (with respect to the Lebesgue measure)) monotonically nondecreasing at all points; see, e.g., [20], Theorem 24.2, as well as Corollary 24.2.1 and Theorem 24.1.

Using the equality we obtain that for

Owing to the convexity (by an inductive assumption) of the function , the expression in square brackets under the integral in (68) is nonnegative almost everywhere, and thus, we obtain that function as monotonically nondecreasing. Thus, as a “candidate” for the point of the support of distribution from the interval , we can consider only one point, . Similarly, consider a “candidate” point for the support of the distribution on the left side, i.e., on the interval . Let us now fix the numbers , and consider a distribution concentrated at points and , with probabilities and , respectively, satisfying the condition , subject to the normalization, whence

Let us show that the integral considered as a function of , i.e., the function , is monotonically nonincreasing on , where and are given by (69) and are considered as functions of the variable . Using equality it is easy to see that for

Thanks to the convexity of the function , the expression in square brackets under the integral in (