Mathematical model of a stock market

The aim of this paper is to propose a wide class of random processes to describe the evolution of a risk active price and to construct a mathematical theory of option pricing. For this purpose, a general mathematical model of evolution of a risk active price is proposed on a probability space constructed. On the probability space, an evolution of a risk active price is described by a random process with jumps that can have both finite and infinite number of jumps. We introduce a new notion of non-singular martingale and prove an integral representation for a wide class of local martingale by a path integral. This theorem is the basic result of the paper that permits us to introduce the important notion of an effective stock market. For an effective stock market the mathematical theory of European type options is constructed. As a result, the new formulas for option pricing, the capital investor and self-financing strategy corresponding to the minimal hedge are obtained.


Introduction
The aim of this paper is to propose a wide class of random processes to describe the evolution of a risk active price and to construct a mathematical theory of option pricing.For this purpose, a general mathematical model of evolution of a risk active price is proposed on a probability space constructed.On the probability space, an evolution of a risk active price is described by a random process with jumps that can have both finite and infinite number of jumps.We introduce a new notion of non-singular martingale and prove an integral representation for a wide class of local martingale by a path integral.This theorem is the basic result of the paper that permits us to introduce the important notion of an effective stock market.For an effective stock market the mathematical theory of European type options is constructed.As a result, the new formulas for option pricing, the capital investor and self-financing strategy corresponding to the minimal hedge are obtained.

Some auxiliary results
Hereafter we will use two elementary lemmas the proof of which is omitted.
Lemma 1.For any on the right continuous functions ϕ(x) and ψ(x), that have the By dϕ(y) and dψ(y) we denoted the charges, generated by functions ϕ(y) and ψ(y) correspondingly, ϕ − (x) = lim y↑x ϕ(y).φ(y)dα(y) + 1 (3) such that φ(a) = 0, φ(x) < ∞, x ∈ [a, b).The function F (x) is given by the formula Proof.The necessity.By definition we put F − (y) = lim x↑y F (x).If the representation (2) holds, then the following equality − 1 is valid.Therefore, the function is a positive, on the right continuous and monotonously non-decreasing solution of equation (3).The sufficiency.If there exists a solution to (3), satisfying conditions of lemma 3, then the function (4) satisfies equation . .
The latter means that dα(y) .
Lemma 3 is proved.
Let us give the necessary and sufficient conditions for the existence of a solution to equation (3) Lemma 4. Nonnegative solution to the equation (3) exists if and only if the series converges for all x ∈ [a, b).
Proof.The necessity.If there exists a non-negative solution to (3), then this solution is the solution to the equation dα(t n ) φ(x).
Arbitrariness of k, positiveness of every term of the series means the convergence of (5).The proof of sufficiency follows from the fact that if the series (5) converges then this series is evidently a solution to the equation (3).The lemma 4 is proved. .
Proof.First of all the product us verify that φ(x) is a solution to (3) in the case when all jump points of α(x) are isolated points.It is sufficient to prove that if φ(x) is the solution to (3) on a certain interval [a, x 0 ] and we prove that φ(x) is the solution to (3) on the interval (x 0 , x], x > x 0 then it will mean that φ(x) is the solution to (3) on the interval [a, x].We assume that the points x i , i = 1, 2, . . .are the jump points of the function α(x).To verify that φ(x) is the solution to the equation (3) let us assume that we have already proved that on the interval [a, x i ), where x i is the jump point of the function α(x), φ(x) is the solution to equation (3), that is, Let x be any point that satisfies the condition To complete the proof of the lemma it is necessary to note that on the interval [a, x 1 ) the solution to (3) is the function e α(x) .Let us prove lemma 5 in a general case.If α(x) satisfies the conditions to lemma 5, then , In the latter sum the summation comes over all jumps of α(x), where the jumps of α(x) are greater than m −1 .It is evident that on any interval [a, x] the set of such points is finite.Therefore φ m (x) satisfies the equation where var x∈[a,d] g(x) means a full variation of the function g(x).From these inequalities we have From the equality φ(y)dα(y) + 1 and from the preceding inequalities there follows the proof of the lemma 5.
is monotonously non-decreasing and on the right continuous function on [a, b) and such that then for the function ψ(x) the following representation is valid for a certain monotonously non-decreasing and on the right continuous function Proof.Let F (x) be the function, which is constructed in the lemma 3. Let us consider the product [1 − F (x)]ψ(x).Then for x < d < b From the lemma 3 . Therefore, f (y)dF (y) then, taking the limit in the equality (8), we obtain The theorem is proved.
Theorem 2. Let g(u) be a measurable function with respect to B([a, b)) and such that then the following formula is valid.
Proof.If we choose and use lemmas 1 and 2 we obtain the proof of the theorem 2.

Probability space
Hereafter we construct a probability space, in which the securities market evolution will be considered.Let Therefore, the set of intervals The number k(α) may be both finite and infinite.Further on, we consider the family of probability spaces Ω i = [a, b), i = 1, k(α).On every probability space Ω i a σ-algebra of events F 0 i is given.By definition the σ-algebra where we denoted by B([a α i , t]) the σ-algebra of subsets of [a, b) generated by the subsets of (c, d) ⊂ [a α i , t] and F 0,t i be the flow of the σ-algebras on the measurable space {Ω α , F 0 α }, that is the direct product of the σ-algebras F 0,t i , where Ωα be the direct sum of the probability spaces Ωα = {α, Ω α }.Elements of Ωα are the pairs {α, ω α }, where ω α ∈ Ω α Let us denote by F 0 α the σ-algebra of events of the kind Āα = {α, A α }, where is the flow of the σ-algebras from Ωα of the sets of the kind {α, A α }, where Let Σ be the σ-algebra of all subsets of X. Introduce a σ-algebra F 0 and the flow of the σ-algebras F 0 t in Ω.We assume that the σ-algebra F 0 in Ω is the set of the subsets of the kind This follows from the following inclusions By analogy with the construction of the σ-algebra F 0 , the flow of the σ-algebra F 0 t ⊆ F 0 is the set of the subsets of the type Further on we deal with the measurable space {Ω, F 0 } and the flow of the σalgebras F 0 t ⊆ F 0 on it.Hereafter we construct the probability space {Ω, F 0 , P }.Define a probability measure P α on the measurable space {Ω α , F 0 α }.For this purpose on every measurable space {Ω i , F 0 i } we determine the family of distribution functions Ω s is on the right continuous and non-decreasing function of the variable where < 1, it is on the right continuous and non-decreasing function of the variable ω α i on [a α i , a α i+1 ) at every fixed {ω α } i−1 ∈ Ω i−1 , moreover, it is a measurable function from the measurable space ) the measure constructed by the distribution function Let us determine a measure on the probability space {Ω α , F 0 α }, having determined it on the set of the type The function of the sets so defined can be extended to a certain measure P α on F 0 α due to Ionescu and Tulcha theorem [1].We put by definition that on the σ-algebra F 0 α the probability measure Pα is given by the formula Pα ( Āα ) = P α (A α ).Further on we consider both the probability spaces {Ω α , F 0 α , P α } and the probability spaces { Ωα , F 0 α , Pα }, that are isomorphic, and the flows of the σ-algebras F 0,α t ⊆ F 0 α and F 0,α t ⊆ F 0 α on the spaces Ω α and Ωα correspondingly.If µ(Y ) is a probability measure on Σ, we put that on the σ-algebra F 0 the probability measure P is given by the formula The latter integral exists, because Pα (B α ) is a measurable mapping from the measurable space {X, Σ} to the measurable space {R 1 , B(R 1 )}, where B(R 1 ) is the Borel σ-algebra on R 1 .
Further on we consider the probability space {Ω, F 0 , P } and the flow of the σ-algebras F 0 t ⊆ F 0 on it, the probability space {Ω, F , P } and the flow of the σalgebras F t ⊆ F , where F and F t are the completion of F 0 and F 0 t correspondingly with respect to the measure P. Then we use the same notation P for the extension of a measure P from the σ-algebra F 0 onto the σ-algebra F , where the σ-algebra F is the completion of the σ-algebra F 0 by the sets of zero measure with respect to the measure P given on the σ-algebra F 0 .

Random processes on the probability space
Definition 1.A consistent with the flow of the σ-algebras F 0 t measurable mapping ζ t ({α, ω α }) from the measurable space {Ω, F 0 } to the measurable space {R 1 , B(R 1 )} belongs to a certain class K if for ζ t ({α, ω α }) the representation Further we deal with the space X 0 that consists of sequences α = {a α i } k(α)+1 i=1 not having limiting points on the interval [a, x], ∀x < b, Σ is the σ-algebra of all subsets of X 0 .Hereinafter χ D (t) denotes the indicator function of the set D from [a, b).Definition 2. By K 0 we denote the subclass of the class K of measurable mappings, satisfying conditions: 1)

t) is an on the right continuous function of bounded variation of the variable t on any interval
) is a measurable and bounded mapping from the measurable space where , is monotonously non-decreasing and on the right continuous function of the variable t on the interval [a α i , a α i+1 ) at every fixed We denoted by γ i,α ({ω α } i−1 , dt) the measure on B([a α i , a α i+1 )), generated by the monotonously non-decreasing and on the right continuous function )) is the Borel σ-algebra on the interval [a α i , a α i+1 ).Lemma 6.Any on the right continuous and uniformly integrable martingale on the probability space {Ω, F , P } with respect to the flow F t is given by the formula where ) is a measurable and integrable function on the probability space {Ω, F , P }, that is, Proof.Further on, for the mapping g α ({ω Taking into account the σ-algebra Σ from X 0 consists of all subsets of X 0 to prove the lemma 6, it is sufficient to calculate the conditional expectation where g({α, ω α }) is a measurable and integrable function on the probability space {Ω, F , P }, M α {g({α, ω α })| F 0,α t } is the conditional expectation with respect to the flow of the σ-algebras F 0,α ).From this it follows that Due to the structure of the σ-algebra F 0,α t it follows that ϕ t i ({ω α }) depends only on variables {ω α } i and Granting this notation we have Really, Because of the fact that we prove the representation (14).Taking into account the definition of the conditional expectation we have Let us introduce the measurable mapping From this and (15) it follows that It is evident that between f α i ({ω α } i ) there exists the following relations The proof of the lemma 6 is completed.
Lemma 7. Let a measurable mapping ζ t ({α, ω α }) on the measurable space {Ω, F 0 } belong to the subclass K 0 , for every fixed α ∈ X 0 , i = 1, k(α), and there exists a constant A < ∞ such that for a certain probability measure µ on Σ.If f α i ({ω α } i ) 0, then on the measurable space {Ω, F 0 } there exist a measure P on the σ-algebra F 0 and a modification ζt ({α, ω α }) of the measurable mapping ζ t ({α, ω α }), such that ζt ({α, ω α }) is a local martingale on the probability space {Ω, F , P } with respect to the flow of the σ-algebra F t , where the σ-algebras F and F t are the completion of the σ-algebras F 0 and F 0 t correspondingly with respect to the measure P.
Proof.The proof of the lemma 7 follows from the theorem 1. Really, all conditions of the theorem 1 are valid.Therefore, there exists a family of distribution functions , with the properties, described at the introduction of the probability space {Ω, F , P }, that for ψ α i ({ω α } i−1 , t) the following representation on the probability space {Ω, F , P } consistent with the flow of the σ-algebras F t , where the σ-algebras F and F t are the completion of the σ-algebras F 0 and F 0 t correspondingly with respect to the measure P, generated by the family of distribution functions , α ∈ X 0 and the measure dµ(α).The measurable mapping ζ t ({α, ω α }) differ from the measurable mapping ζt ({α, ω α }) on the set Ω\Ω 0 , P (Ω\Ω 0 ) = 0. Let us construct the set Ω 0 .Consider the set Really, since the sequence of the sets Ω n α , the continuity of the probability measure P α , we obtain P α (Ω 0 α ) = 1.As far as there are no more than a countable set of α for which µ(α) > 0, then there exists a countable subset X 0 1 ⊆ X 0 such that the direct sum of the sets Ω 0 α , α ∈ X 0 1 forms the set Ω 0 ⊂ Ω, P (Ω 0 ) = 1.
From the condition of the lemma 7 we have the recurrent relations As far as then we have For every t 0 ∈ [a, b) let us introduce the measurable mapping from the measurable space {Ω, F } to the measurable space {R 1 , B(R 1 )}.
where i(t 0 , α) = max{i, a α i t 0 }.From the condition of lemma 7 Moreover, it is not difficult to see that The latter equality means that ζ t ({α, ω α }) is a local martingale since this equality is valid for any t 0 ∈ [a, b).Therefore, we can choose the sequence of stop moments t n 0 → b with probability 1 such that ζ t∧t n 0 ({α, ω α }) → ζ t ({α, ω α }) with probability 1.The lemma 7 is proved.
In a more general case, there holds Lemma 8. Let a measurable mapping ζ t ({α, ω α }) on the measurable space {Ω, F 0 } belong to the subclass K 0 .Suppose that for any t 0 ∈ [a, b), for a certain probability measure dµ(α) on Σ, where d α i = sup then on the measurable space {Ω, F 0 } there exist a measure P on the σ-algebra F 0 and a modification ζt ({α, ω α }) of the measurable mapping ζ t ({α, ω α }), such that ζt ({α, ω α }) is a local martingale on the probability space {Ω, F , P } with respect to the flow of the σ-algebra F t , where the σ-algebras F and F t are the completion of the σ-algebras F 0 and F 0 t correspondingly with respect to the measure P.
The proof of the lemma 8 is the same as the proof of the lemma 7.
As before, let {Ω, F , P } be the probability space with the flow of the σ-algebras F t ⊆ F on it.Suppose that ζ t ({α, ω α }) is a random process consistent with the flow of σ-algebras F t , where satisfying the conditions: ) is a local martingale.This assertion can be proved the same way as lemma 7 was proved.
Further on we connect with the local martingale ζ t ({α, ω α }) on {Ω, F , P } a stochastic process which is consistent with the flow of the σ-algebras F t , where We shall call the process ζ a t ({α, ω α }) as the process associated with the ζ t ({α, ω α }) process.
Definition 3. Realization of the assotiated process is no more than the countable set.Lemma 9. On the probability space {Ω, F , P } there always exists a non-singular local martingale.
Proof.To prove the lemma 9 we construct an example of a martingale on {Ω, F , P } that is non-singular.Let f α s (ω α s ) 0, s = 1, k(α), α ∈ X 0 be the measurable mapping with respect to the σ-algebra F 0 s , satisfying conditions: Then the local martingale is not singular, where .
The charge dξ 2 t ({α, ω α }) generated by realizations of the process Let us calculate Using (21) and the theorem 2 we have Therefore, Taking the limit t → a α i+1 we obtain , where Granting this and the definition of ψ k(α) (s|ω α ) we obtain ).The theorem 3 is proved.
Theorem 4. Let ξ 0 t ({α, ω α }) be a local martingale on {Ω, F , P }, satisfying conditions: The random process belongs to the subclass K 0 if the family of functions f α (x) > 0, x ∈ R 1 , α ∈ X 0 , are strictly monotonous, sup Proof.To prove the theorem 4 it is sufficient to verify the fulfillment of the conditions of definition 2. The condition 1 is valid, because is continuous on the right, where . Thus, is non-negative and monotonously non-decreasing on [a α i , a α i+1 ), where Further, ).The theorem 4 is proved.

Options and their pricing
We assume that {Ω, F , P } is a full probability space, generated by the family of distribution functions F α i (ω α i |{ω α } i−1 ), i = 1, k(α), α ∈ X 0 and a measure dµ(α) on the σ-algebra Σ.Further on we assume that X 0 is a space of possible hypothesis each of which may occur with probability µ(α), that is, an evolution of stock price can come by one of the possible scenario.This scenario is determined by sequence α and a probability space {Ω α , F 0 α , P α }.
) be a random value on the probability space {Ω, F , P }, satisfying conditions: 2) there exists satisfying conditions If the random process has got the form where a family of functions then there exists a measure P 1 on {Ω, F 0 }, generated by a certain family of distribution functions F α,1 i (ω α i |{ω α } i−1 ), i = 1, k(α), α ∈ X 0 , a probability measure dµ 1 (α) on the σ-algebra Σ and a modification ξt ({α, ω α }) of the the random process ξ t ({α, ω α }) such that ξt ({α, ω α }) is a local non-singular martingale on the probability space {Ω, F 1 , P 1 } with respect to the flow of the σ-algebras F 1 t , where the σ-algebras F 1 and F 1 t are the completion of the σ-algebras F 0 and F 0 t correspondingly with respect to the measure P 1 .Moreover, for the regular martingale M 1 {φ({α, ω α })|F 1  t } on the probability space {Ω, F 1 , P 1 } the representation is valid, where Proof.The conditions of the theorem 5 guarantee the monotonous of the conditions of the theorem 4. Therefore the random process ξ t ({α, ω α }) belongs to the subclass K 0 .Moreover, with probability 1 on the probability space {Ω, F , P }, since with probability 1.Further, where dµ Based on the lemma 7 there exists a measure P 1 on the σ-algebra F 0 , generated by a certain family of distribution functions F α,1 i (ω α i |{ω α } i−1 ), i = 1, k(α), α ∈ X 0 and the measure dµ 1 (α) on σ-algebra Σ such that the random process ξt ({α, is a modification of the random process ξ t ({α, ω α }) on the probability space {Ω, F 1 , P 1 }.Since ξ 0 t ({α, ω α }) is also a local non-singular martingale, then ξt ({α, ω α }) is also a local non-singular martingale because due to the strict monotony of f α (x), where ξi,α t,a ({ω ).To finish the proof of the theorem 5 it is sufficient to verify the monotonous of the conditions of the theorem 3. Really, because the first integral is finite and the second integral is finite since The theorem 5 is proved.Then we assume that interval [a, b) coincides with the interval [0, T ), that is a = 0, b = T.The time T is the terminal time of monotonous of the option.
Definition 5. A stock market is effective on the time interval [0, T ), if there is a certain probability space {Ω, F , P }, constructed above, a random process ξ 0 t ({α, ω α }) on it, describing the evolution of the average price of stocks such that ξ 0 t ({α, ω α })e −rt is a non-negative uniformly integrable and non-singular martingale on {Ω, F , P } with respect to the flow of the σ-algebras F t , where the σ-algebras F and F t are the completion of the σ-algebras F 0 and F 0 t with respect to the measure P on F 0 , generated by the family of distribution functions F α i (ω α i |{ω α } i−1 ).The random process ξ 0 t ({α, ω α }) has the form where r is an interest rate, the evolution of price of a stock being described by a certain random process for a certain family of functions f α (x) > 0, x ∈ R 1 , α ∈ X 0 , which are strictly fulfillment, sup The limit Let us consider an economic agent on the stock market, who acts an as investor, that is, he or she wants to multiply his or her capital using the possibilities of the stock market.We assume that the stock market is effective and the evolution of a stock price occurs according to the formula (25).We assume that the evolution of non-risky active price occurs according to the law where r is an interest rate, B 0 is an initial capital of the investor on a deposit.Definition 6.A stochastic process δ t ({α, ω α }) belongs to the class A 0 , if Let the capital of an investor X t ({α, ω α }) at time t equal where the stochastic processes β t ({α, ω α }) and γ t ({α, ω α }) belong to the class A 0 .
Definition 7. A financial strategy π t = {β t ({α, ω α }), γ t ({α, ω α })} of an investor is called self-financing if the random processes β t ({α, ω α }) and γ t ({α, ω α }) belong to the class A 0 , for the investor capital X π t ({α, ω α }) the representation is valid, the discounted capital This proves the lemma 10 in one direction.Applying the same argument in the inverse direction we obtain the proof of the lemma 10.Denote by SF R a set of self-financing strategies satisfying the conditions where R is a non-negative random value on {Ω, F 1 , P 1 }.
Lemma 12. Any strategy π t ∈ SF R , where R is non-negative and integrable random value on probability space, is not arbitrage strategy.
The proof of the lemma is analogous to the proof of the similar lemma in [2].Let ) be F 0 measurable random value on the probability space {Ω, F 0 , P }.Definition 9. A self-financing strategy π t ∈ SF R is (x α , φ T )-hedge for the European type option if the capital X π t ({α, ω α }), corresponding to this strategy is such that X π 0 (α) = x α and with probability 1 with respect to the measure P 1 Then we consider self-financing strategies, belonging to SF 0 , that is, in this case X π t ({α, ω α }) 0. Definition 10.Let H T (x α , φ T ) be the set of (x α , φ T )-hedges from SF 0 .Investment value is called the value where ∅ is the empty set.

Theorem 1 .
Let ψ(y) be an on the right continuous function of bounded variation on any interval [a, x], x ∈ [a, b), f (y) be a measurable mapping with respect to the Borel σ-algebra on [a, b) and bounded function on [a, x], x ∈ [a, b).If, moreover,