Mathematical model of a stock market

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1 Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp Mathematical model of a stock market N.S.Gonchar Bogolubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, 4 b Metrolohichna Str., Kyiv, Ukraine Received May 30, 2000 In this paper we construct a mathematical model of securities market. The results obtained are a good basis for an analysis of any stock market. Key words: random process, effective stock market, option pricing PACS: s, j Dedicated to prominent scientist Igor Yukhnovsky, initiator of my perspective research on economy.. 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. 2. Some auxiliary results Hereafter we will use two elementary lemmas the proof of which is omitted. Lemma. For any on the right continuous functions ϕ(x) and ψ(x), that have the c N.S.Gonchar 46

2 N.S.Gonchar bounded variation on [a, b), the following formula ϕ(d)ψ(d) ϕ(c)ψ(c) = ϕ (y)dψ(y)+ ψ(y)dϕ(y), (c,d] [a,b) () (c,d] (c,d] is valid, F (u) = lim v u F(v). 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). Lemma 2. The Radon-Nicodym derivative of the measure dg(y), generated by the function g(y) = ( F(y)), with respect to the measure df(y), where F(y) is on the right continuous and monotonouosly non-decreasing on [a, b) function and such that F(a) = 0, F(x) <,x [a,b),lim x b F(x) = is given by the formula dg(y) df(y) = ( F(y))( F (y)). Lemma 3. For on the right continuous and monotoneously non-decreasing function α(x) such that α(x) <, x [a,b), α(a) = 0, lim x b α(x) =, the representation α(x) = df(y) F (y) is valid for a certain F(x), that is on the right continuous and monotonously nondecreasingfunction, satisfying conditions:f(x) <, x [a,b),limf(x) =,F(a) = x b 0, if and only if there exists a positive, on the right continuous and monotoneously non-decreasing solution of equation φ(x) = φ(y)dα(y) + (3) such that φ(a) = 0, φ(x) <,x [a,b). The function F(x) is given by the formula (2) F(x) = φ(x). (4) φ(x) Proof.The necessity.bydefinitionweputf (y) = lim x y F(x).Iftherepresentation (2) holds, then the following equality 462 dα(y) F(y) = df(y) ( F(y))( F (y)) = F(x)

3 Mathematical model of a stock market is valid. Therefore, the function φ(x) = F(x) 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 dα(y) F(y) + = F(x). But The latter means that or df(y) ( F(y))( F (y)) + = F(x). dα(y) F(y) = dα(y) = From the latter equality it follows that α(x) = df(y) ( F(y))( F (y)), df(y) ( F (y)). df(y) ( F (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 φ(x) = + converges for all x [a,b). n= dα(t ) [a,t ] dα(t 2 )... [a,t n ] dα(t n ) (5) Proof. The necessity. If there exists a non-negative solution to (3), then this solution is the solution to the equation φ(x) = + k n= dα(t ) [a,t ] dα(t 2 )... [a,t n ] dα(t n ) 463

4 N.S.Gonchar + dα(t ) dα(t 2 )... dα(t k ) φ(t k+ )dα(t k+ ). [a,t ] [a,t k ] From the latter equality there follows the inequality + k n= dα(t ) [a,t ] dα(t 2 )... [a,t k ] [a,t n ] 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. Corollary. If α(x) is a continuous and monotonously non-decreasing function, α(x) <, x [a,b), limα(x) =, α(a) = 0, then the equation (3) has the x b solution φ(x) = e α(x). Corollary 2. If γ(x) is some measurable function on [a, b), which satisfies the inequality γ(y)dα(y)+ γ(x), x [a,b), then there exists a solution to equation (3). Lemma 5. The solution to the equation (3) exists if jumps of monotonously nondecreasing and on the right continuous function α(x) is such that α(s). It has the following form φ(x) = e α(x) e α(s) ( α(s)). {s x} If 0 α(s) <, s [a,b), then this solution is non-negative, on the right continuous and monotonously non-decreasing function, α(s) = α(s) α (s). Proof. First of all the product converges, because the estimate {s x} {s x} e α(s) ( α(s)) α(s) α(x) <, x < b is valid. Let 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. We assume that the points x i,i =,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 464

5 Mathematical model of a stock market 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, φ(y)dα(y) = φ (x i ) = e α e (x) α(s) ( α(s)). [a,x i ) {s<x i } Let x be any point that satisfies the condition x i < x < x i+. Since + φ(y)dα(y) = + φ(y)dα(y) + φ(y)dα(y) + φ(y)dα(y) +[e α(x) e α(x i) ] [a,x i ) [x i ] = φ (x i ) +e α(x i) α(x i ) {s x i } {s x i } e α(s) ( α(s)) + = eα(x) (x i,x] e α(s) ( α(s)) {s x} e α(s) ( α(s)) = φ(x). To completethe proof of thelemmaitisnecessary to note that onthe interval[a,x ) 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 α(x) = α c (x)+ α(s), {s x} where α c (x) is a continuous function on [a,b). Let us introduce the notation where φ m (x) = e αm(x) α m (x) = α c (x)+ {s x} e αm(s) ( α m (s)), {s x, α(s) m } α(s). In the latter sum the summation comes over all jumps of α(x), where the jumps of α(x) are greater than m. It is evident that on any interval the set of such points is finite. Therefore φ m (x) satisfies the equation φ m (x) = φ m (y)dα m (y)+. (6) Let d < b, then {s d, α(s) m } sup φ(x) φ m (x) sup x [a,d] ( α(s)) e αc(x) x [a,d] {s d, α(s) m } {s d} ( α(s)) ( α(s)) 465

6 Moreover, sup x [a,d] var [α(x) α m(x)] x [a,d] e αc(x) {s d, α(s)<m } {s d} α(s) ( α(s)) {s d, α(s)<m } 0, m. α(s) 0, m, N.S.Gonchar where var x [a,d] g(x) means a full variation of the function g(x). From these inequalities we have [φ m (y) φ(y)]dα m (y) sup φ m (x) φ(x) var α(x) 0, m, x [a,d] x [a,d] φ(y)d[α m (y) α(y)] sup φ(x) var [α m (y) α(y)] 0,m. x [a,d] x [a,d] From the equality φ m (x) = [φ m (y) φ(y)]dα m (y)+ φ(y)d[α m (y) α(y)]+ φ(y)dα(y) + and from the preceding inequalities there follows the proof of the lemma 5. Theorem. Let ψ(y) be an on the right continuous function of bounded variation on any interval, x [a,b), f(y) be a measurable mapping with respect to the Borel σ-algebra on [a,b) and bounded function on, x [a,b). If, moreover, α(x) = dψ(y) <, x [a,b) (7) ψ(y) f(y) is monotonously non-decreasing and on the right continuous function on [a, b) and such that ) 0 α(x) <, α(x) = α(x) α (x), x [a,b), 2) limα(x) =, α(a) = 0, x b 3) limψ(x)e α(x) = 0, x b 4) b a f(x) e α (x) dα(x) <, then for the function ψ(x) the following representation ψ(x) = f(x)df(x) ( F(x)) is valid for a certain monotonously non-decreasing and on the right continuous function F(x), such that F(a) = 0, F(x) <, x [a,b), lim x b F(x) =. 466 (x,b)

7 Mathematical model of a stock market Proof. Let F(x) be the function, which is constructed in the lemma 3. Let us consider the product [ F(x)]ψ(x). Then for x < d < b [ F(x)]ψ(x)+[ F(d)]ψ(d) = [ F (y)]dψ(y) ψ(y)df(y). (x,d] (x,d] From the lemma 3 df(y) dψ(y) = [ψ(y) f(y)] ( F (y)). Therefore, [ F(x)]ψ(x)+[ F(d)]ψ(d) = f(y)df(y). (8) (x,d] Since = [ F(d)]ψ(d) e α(d) ψ(d) 0, d b, f(y)df(y) f(y) df(y)= (x,d] (x,d] f(y) [ F (y)]dα(y) f(y) e α (y) dα(y) <, (x,d] (x,b) then, taking the limit in the equality (8), we obtain [ F(x)]ψ(x) = f(y)df(y). The theorem is proved. Theorem 2. Let g(u) be a measurable function with respect to B([a,b)) and such that g(y) df(y)<, (d,b) [a,b) (x,b) then the following formula g(y)df(y) ( F(d)) ( F(c)) is valid. = (c,d] ( F(u)) (u,b) (c,b) g(y)df(y) = g(y)df(y) g(u) df(u), (c,d] [a,b) (9) F (u) 467

8 N.S.Gonchar Proof. If we choose ϕ(x) = ( F(x)), ψ(x) = (x,b) g(u)df(u) and use lemmas and 2 we obtain the proof of the theorem Probability space Hereafter we construct a probability space, in which the securities market evolution will be considered. Let α = {a α i } + be a sequence from [a,b) R+, a < b, that satisfies conditions: a α i < a α i+, i =,, [a α i,a α i+) = [a,b), a α = a, a α + = b. Therefore, the set of intervals {[a α i,aα i+ ), i =,} forms a decomposition of interval [a,b) R+. The number may be both finite and infinite. Further on, we consider the family of probability spaces Ω i = [a,b), i =,. On every probabilityspace Ω i a σ-algebraofevents Fi 0 isgiven. Bydefinitiontheσ-algebraFi 0 is the set of subsets of Ω i = [a,b), that is generated by intervals (c,d) [a α i,a α i+). Let us determine the flow of the σ-algebras F 0,t i, t [a,b), F 0,t i Fi 0, by the formula F 0,t i = {,[a,b)}, a t a α i, B([a α i,t]), aα i < t < a α i+, t [a α i,aα i+ ) B([a α i,t]) = F 0 i, a α i+ t b, 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 B([a α i,t]) denotes the σ-algebra, that is the t [a α i,aα i+ ) union of the σ-algebras B([a α i,t]). Let {Ω α,fα 0 } be the direct product of measurable spaces {Ω i,fi 0 },i =,, and F0,α t = be the flow of the σ-algebras on the measurable space {Ω α,fα 0 }, that is the direct product of the σ-algebras F0,t i, where Ω α = Ω i,fα 0 =. Let us determine a certain measurable space Fi 0 {Ω,F 0 }. Denote by X a set of sequences α = {a α i } + from [a, b) that generate decomposition of [a,b). Let Ω = Ω α be the direct sum of the probability spaces α X Ω α = {α,ω α }. Elements of Ω α are the pairs {α,ω α }, where ω α Ω α Let us denote by F α 0 the σ-algebra of events of the kind Āα = {α,a α }, where A α Fα 0, {α,a α} = = {{α,ω α }, ω α A α }. Analogously, F0,α t is the flow of the σ-algebras from Ω α of the sets of the kind {α,a α }, where A α F 0,α t. It is evident that Ω α Ω β =,α β. 468 F 0,t i

9 Mathematical model of a stock market 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 C Y = α Y B α, Y Σ, B α F 0 α. This follows from the following inclusions C Yi = B α F 0, C Yi = B α F 0, C Y \C Y2 = B α F 0. α Y \Y 2 α Y i α Y i By analogy with the construction of the σ-algebra F 0, the flow of the σ-algebra F 0 t F0 is the set of the subsets of the type C Y = α Y B α, Y Σ, B α F 0,α t. Further on we deal with the measurable space {Ω,F 0 } and the flow of the σ- algebras Ft 0 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,fα 0 }. For this purpose on every measurable space {Ω i,fi 0 } we determine the familyof distribution functions Fi α(ωα i {ω, that at every fixed {ω α } i Ω i = i Ω s is on the right continuous and non-decreasing function of the variable ωi α [a,b), F α i (ωα i {ω = s= 0, a ω α i a α i, {ω α} i Ω i, φ α i (ω α i {ω α } i ), a α i < ω α i < a α i+, {ω α } i Ω i,, a α i+ ωα i < b, {ω α } i Ω i, where {ω α } i = {ω α,...,ωα i }, ω α = {ω α,...,ωα }. The function φ α i (ωα i {ω satisfies the conditions: 0 φ α i (ωα i {ω <, it is on the right continuous and non-decreasing function of the variable ωi α on [a α i,aα i+ ) at every fixed {ω α} i Ω i, moreover, it is a measurable function from the measurable space {Ω i, F i 0 } to the measurable space {[0,],B([0,])} at every fixed ωi α, where B([0,]) is the Borel σ-algebra on [0,], F0 i = i Fs. 0 Denote by Fi α(dωα i {ω the measure constructed by the distribution function Fi α (ωi α {ω α } i ) onthe σ-algebrafi 0 at every fixed {ω α } i Ω i. It isevident that Fi α(dωα i {ω is concentrated on the subset [a α i,aα i+ ) Ω i. Let us determine a measure on the probability space {Ω α,fα}, 0 having determined it on the set of the type A... A, A i Fi 0 by the formula =... A P α (A... A ) = A F α (dωα )Fα 2 (dωα 2 {ω α} )... F α (dωα {ω α} ). s= 469

10 N.S.Gonchar The function of the sets so defined can be extended to a certain measure P α on Fα 0 due to Ionescu and Tulcha theorem []. We put by definition that on the σ-algebra F α 0 the probability measure P α is given by the formula P α (Āα) = P α (A α ). Further on we consider both the probability spaces {Ω α,fα 0,P α} and the probability spaces { Ω α, F α, 0 P α }, that are isomorphic, and the flows of the σ-algebras F 0,α t Fα 0 and F 0,α t F α 0 onthespacesω α and Ω α correspondingly.ifµ(y)isaprobabilitymeasure on Σ, we put that on the σ-algebra F 0 the probability measure P is given by the formula P(C Y ) = P α (B α )dµ(α), C Y = B α, Y Σ, B α F α. 0 α Y Y The latter integral exists, because P α (B α ) is a measurable mapping from the measurablespace {X,Σ} to the measurablespace {R,B(R )}, where B(R ) is the Borel σ-algebra on R. 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 σ- algebras 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 Random processes on the probability space Definition. 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,B(R )} belongs to a certain class K if for ζ t ({α,ω α }) the representation ζ t ({α,ω α }) = χ [a α i, a α i+ ) (t)ζ i,α t ({ω α } i ), ζ i,α t ({ω α } i ) = f α i ({ω α} i )χ [a α i, t](ω α i )+ψα i ({ω α} i,t)χ (t, a α i+ )(ω α i ), t [a α i,a α i+) (0) is valid, where f α i ({ω α} i ) is a measurable mapping from the measurable space {Ω i, F 0 i } to the measurable space {R,B(R )} at every fixed α X 0, i =,, ψ α i ({ω α} i,t) is a measurable mapping from the measurable space {Ω i, F 0 i } to the measurable space {R,B(R )} at every fixed t [a α i,aα i+ ), α X0, i = 2,. Further we deal with the space X 0 that consists of sequences α = {a α i }+ not having limiting points on the interval, 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). 470

11 Mathematical model of a stock market Definition 2. By K 0 we denote the subclass of the class K of measurable mappings, satisfying conditions: ) ψi α({ω α} i,t) is an on the right continuous function of bounded variation of the variable t on any interval [a α i,τ],τ [aα i,aα i+ ) at every fixed {ω α} i Ω i, i = =,. 2) fi α({ω α} i ) is a measurable and bounded mapping from the measurable space {Ω i, F i 0 } to the measurable space {R,B(R )}. 3) The function γ i,α ({ω α } i,t) = [a α i, t] ψi α ({ω α } i,dτ) [ψi α({ω α} i,τ) fi α({ω, i =,, () α} i,τ)] where fi α ({ω α } i,τ) = fi α ({ω α } i ) ωi α = τ, is monotonously non-decreasing and on the right continuous function of the variable t on the interval [a α i,aα i+ ) at every fixed {ω α } i Ω i, α X 0, satisfying conditions: a) γ i,α ({ω α } i,t) <, {ω α } i Ω i, t [a α i,a α i+), γ i,α ({ω α } i,t) = γ i,α ({ω α } i,t) γ i,α ({ω α } i,t ), γ i,α ({ω α } i,t ) = limγ i,α ({ω α } i,s). s t b) lim γ i,α ({ω α } i,t) =, γ i,α ({ω α } i,a α t a α i ) = 0, {ω α } i Ω i,α X 0. i+ c) lim ψ t a α i α ({ω α } i,t)exp{ γ i,α ({ω α } i,t)} = 0, i+ d) fi α({ω α} i,t) exp{ γ i,α ({ω α } i,t )}γ i,α ({ω α } i,dt) <, [a α i, aα i+ ) {ω α } i Ω i,α X 0. We denoted by γ i,α ({ω α } i,dt) the measure on B([a α i,aα i+ )), generated by the monotonously non-decreasing and on the right continuous function γ i,α ({ω α } i,t) of the variable t at every fixed {ω α } i Ω i, B([a α i,a α i+)) is the Borel σ-algebra on the interval [a α i,aα i+ ). 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 m i,α M t ({α,ω α }) = χ [a α i, a α i+ ) (t)m i,α t ({ω α } i ), (2) t ({ω α } i ) = fi α ({ω α} i )χ [a α i, t](ωi α )+ψα i ({ω α} i,t)χ (t, a α i+ )(ωi α ), fi α ({ω α} i ) =... g α ({ω α } i,{ω α } [i+,] ) Ω i+ Ω ψ α i ({ω α } i,t) = Fi+(dω α i+ {ω α α } i )... F(dω α {ω α α } ), Fi α(t {ω fi α ({ω α } i )Fi α (dωi α {ω α } i ), (t, a α i+ ) i =,, α X 0, (3) 47

12 N.S.Gonchar g α ({ω α } i,{ω α } [i+,] ) is a measurable and integrable function on the probability space {Ω,F,P}, that is, X 0 M α g({α,ω α }) dµ(α) <. Proof. Further on, for the mapping g α ({ω α } i,{ω α } [i+,] ) we use one more notation g({α,ω α })=g α ({ω α } i,{ω α } [i+,] ), where {ω α } i ={ω α,...,ω α i }, {ω α } [i,] = {ω α i+,...,ωα }, ω α = {ω α,...,ωα } = {{ω α} i,{ω α } [i,] }. 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 M{g({β,ω β }) F t } β=α = M α {g({α,ω α }) F 0,α t }, where g({α,ω α }) is a measurable and integrable function on the probability space 0,α {Ω,F,P}, M α {g({α,ω α }) F t } is the conditional expectation with respect to the 0,α flow of the σ-algebras F t F α 0 on the probability space { Ω α, F α, 0 P α }. Suppose that t [a α i,aα i+ ). From this it follows that ϕ t i ({ω α}) = M α {g({α,ω α }) F 0,α t } 0,α is the measurable mapping from { Ω α, F t, P α } to {R,B(R 0,α )}, where F t = i Fs 0 F0,t i O s, O s = {,[a,b)}, s = i+,. Due to the structure s= s=i+ 0,α F t of the σ-algebra it follows that ϕ t i ({ω α}) depends only on variables {ω α } i and ϕ t i({ω α }) is a measurablemappingfrom{ω i, i Fs F 0 0,t i } to {R,B(R )}. Granting this notation we have s= ϕ t i ({ω α}) = Q t i ({ω α} i ) = ϕ α i ({ω α } i,t)χ [a α i, t](ω α i )+ψ α i ({ω α } i,t)χ (t, a α i+ )(ω α i ). (4) Really, Q t i({ω α } i ) = Q t i({ω α } i )χ [a α i, t](ω α i )+Q t i({ω α } i )χ (t, a α i+ )(ω α i ). Because of the fact that Q t i({ω α } i )χ [a α i,t](ωi α ) is a measurable mapping from {Ω i, i Fs 0 F0,t i } to {R,B(R )} it follows that Q t i ({ω α} i )χ (t,a α i+ )(ωi α ) is also the s= measurable mapping. But this is possible, when Q t i ({ω α} i ) does not depend on the variable ωi α (t,b), because the only measurable sets i B s A i belong to the σ-algebra i 472 Putting s= F 0 s F0,t i, when ω α i (t,b), where B s F 0 s, A i = [a,a α i ) (t,b). s= Q t i ({ω α} i ) = ϕ α i ({ω α} i,t), {ω α } i Ω i [a α i,t], Q t i({ω α } i ) = ψ α i ({ω α } i,t), {ω α } i Ω i {[a,a α i ) (t,b)},

13 Mathematical model of a stock market we prove the representation (4). Taking into account the definition of the conditional expectation we have... Q t i ({ω α} i )F α (dωα )... Fα i (dωα i {ω = B B i A = g({α,ω α })F α (dωα )... Fα (dωα {ω α} ), B A Ω i+ Ω B i B s Ω s, s =,i, A F 0,t i. (5) Let us introduce the measurable mapping fi α ({ω α} i ) =... Ω i+ Ω g α ({ω α } i,{ω α } [i+,] ) F α i+ (dωα i+ {ω α} i )... F α (dωα {ω α} ) from the measurable space {Ω i, F i 0} to the measurable space {R,B(R )}. It is evident that... fi α ({ω α} i ) F α (dωα )... Fα i (dωα i {ω M α g({α,ω α }) <. Ω Ω i From this and (5) it follows that ψ α i ({ω α} i,t) = ϕ α i ({ω α} i,t) = fi α ({ω α} i ), Fi α(t {ω fi α ({ω α} i )Fi α (dωα i {ω, (t, a α i+ ) i =,, α X 0. It is evident that between fi α({ω α} i ) there exists the following relations fi α ({ω α} i ) = fi+ α ({ω α} i+)fi+ α (dωα i+ {ω α} i ), i =,, α X 0. Ω i+ 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 =,, ψ α i ({ω α } i,a α i ) = f α i ({ω α } i ), {ω α } i Ω i and there exists a constant A < such that (a)dµ(α) A X 0 ψ α 473

14 N.S.Gonchar 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 ζ t ({α,ω α }) of the measurable mapping ζ t ({α,ω α }), such that ζ t ({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. Really, all conditions of the theorem are valid. Therefore, there exists a family of distribution functions Fi α(ωα i {ω, i =,, α X 0,{ω α } i Ω i,withtheproperties,described at the introduction of the probability space {Ω,F,P}, that for ψi α ({ω α } i,t) the following representation ψi α ({ω α } i,t) = Fi α(t {ω fi α ({ω α } i )Fi α (dωi α {ω α } i ), (t, a α i+ ) i =,, α X 0, is valid. Let us consider the measurable mapping ζ t ({α,ω α }) = χ [a α i, a α i+ ) (t) ζ i,α t ({ω α } i ), ζ i,α t ({ω α } i }) = fi α ({ω α} i )χ [a α i, t](ωi α ) + Fi α(t {ω fi α ({ω α} i )Fi α (dωα i {ω χ (t, a α i+ )(ωi α ), (t, a α i+ ) 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 Ft 0 correspondingly with respect to the measure P, generated by the family of distribution functions Fi α (ωi α {ω α } i ), i =,, α X 0 and the measure dµ(α). The measurable mapping ζ t ({α,ω α }) differ from the measurable mapping ζ t ({α,ω α }) on the set Ω\Ω 0, P(Ω\Ω 0 ) = 0. Let us construct the set Ω 0. Consider the set Ω 0 α = {ω α Ω α, a α i < ωi α < a α i+, i =,}, where ω α = {ω α,...,ωα } and show that P α (Ω 0 α ) =. Really, since the sequence of the sets Ω n α = {ω α Ω α, a α i < ω α i < a α i+, i =,n, a ωα i < b,i = n+,} has the probability, that is, P α (Ω n α) =, n =,2,..., and taking into account that Ω n α Ωn+ α, Ω 0 α = Ω n α, the continuity of the probability measure P α, we n= obtain P α (Ω 0 α ) =. 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 X0 such that the direct sum of the sets Ω 0 α,α X 0 forms the set Ω 0 Ω, P(Ω 0 ) =. 474

15 Mathematical model of a stock market From the condition of the lemma 7 we have the recurrent relations fi α ({ω = fi α ({ω α} i )Fi α (dωα i {ω [a α i, aα i+ ) As far as = fi α ({ω α} i )Fi α (dωα i {ω, i =,, α X 0. Ω i ψ(a) α = Ω f α ({ω α } )F α (dω α ) <, then we have ψ(a) α =... fi α ({ω α } i )F α (dω)... F α i α (dωi α {ω α } i ). Ω Ω i For every t 0 [a,b) let us introduce the measurable mapping from the measurable space {Ω,F} to the measurable space {R,B(R )}. g t0 ({α,ω α }) = f α i(t 0,α) ({ω α} i(t0,α)), where i(t 0,α) = max{i,a α i t 0 }. From the condition of lemma 7 Mg t0 ({α,ω α }) = ψ(a)dµ(α) α <. X 0 Moreover, it is not difficult to see that ζ t t0 ({α,ω α }) = M{g t0 ({α,ω α }) F t }. 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 such that ζ t t n 0 ({α,ω α }) ζ t ({α,ω α }) with probability. 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), X 0 d α i(t 0,α) dµ(α) < for a certain probability measure dµ(α) on Σ, where d α i = sup {{ω α} i Ω i } fi α({ω α} i ), i(t 0,α) = max{i,a α i t 0 }. If for every fixed i =,, α X 0, ψ α i ({ω α } i,a α i ) = f α i ({ω α } i ), {ω α } i Ω i, 475

16 N.S.Gonchar then on the measurable space {Ω,F 0 } there exist a measure P on the σ-algebra F 0 and a modification ζ t ({α,ω α }) of the measurable mapping ζ t ({α,ω α }), such that ζ t ({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 ζ t ({α,ω α }) = χ [a α i, a α i+ ) (t)ζ i,α t ({ω α } i ), ζ i,α t ({ω α } i }) = fi α ({ω α} i )χ [a α i, t](ωi α ) + Fi α(t {ω fi α ({ω α } i )Fi α (dωi α {ω α } i )χ (t, a α i+ )(ωi α ), (6) (t, a α i+ ) satisfying the conditions: X 0 f α i ({ω α} i ) = Ω i+ f α i+ ({ω α} i+)f α i+ (dωα i+ {ω α} i ),... fi α α ({α,ω α } iα ) F α (dωα )...Fα i α (dωi α α {ω α } iα )dµ(α) < Ω Ω iα for every t 0 [a,b), i α = i(t 0,α) = max{i,a α i t 0 }, then ζ t ({α,ω α }) 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 ζt({α,ω a α }) = χ [a α i, a α i+ ) (t)ζ i,α t,a({ω α } i ), which is consistent with the flow of the σ-algebras F t, where ζ i,α t,a({ω α } i ) = = fi α ({ω α } i,t) Fi α(t {ω fi α ({ω α } i )Fi α (dωi α {ω α } i ). (t, a α i+ ) We shall call the process ζ a t ({α,ω α}) as the process associated with the ζ t ({α,ω α }) process. 476

17 Mathematical model of a stock market Definition 3. Realization of the assotiated process ζ a t ({α,ω α}) is regular if the set is no more than the countable set. {t [a,b), ζ i,α t,a({ω α } i ) = 0, i =,} Definition 4. A local martingal ζ t ({α,ω α }) is non-singular on {Ω,F,P} if the set of regular realizations of the associated random process ζ a t ({α,ω α}) has got the probability. Lemma 9. On the probability space {Ω,F,P} there always exists a non-singular local martingale. Proof. To prove the lemma9 we construct an example of a martingaleon {Ω,F,P} that is non-singular. Let f α s (ωα s ) 0, s =,, α X0 be the measurable mapping with respect to the σ-algebra F 0 s, satisfying conditions: 0 < Ω s f α s (ω α s)f α s (dω α s {ω α } s ) <, s =,, α X 0, {ω α } s Ω s, (7) fs α (t) f Fs α s α (t {ω α } s ) (ωα s )Fα s (dωα s {ω α} s ) 0, (t, a α i+ ) Then the local martingale is not singular, where t [a α s,aα s+ ), s =,, α X0, {ω α } s Ω s. (8) ξt α ({α,ω α}) = χ [a α i, a α i+ ) (t)ξ i,α t ({ω α } i ) ξ i,α t ({ω α } i }) = g α i ({ω α } i )χ [a α i, t](ω α i ) + Fi α(t {ω gi α ({ω α} i )Fi α (dωα i {ω χ (t,a α i+ )(ωi α ), (t, a α i+ ) g α i ({ω α} i ) = i s= g 0,α s ({ω α } s ), g 0,α s ({ω α } s ) = fs α (ω s) α fs α(ωα s )Fα s (dωα s {ω α} s ). Ω s If, for example, f α s (ωα s ) > 0, s =,, α X0, are strictly monotonous on [a α s,a α s+), then the conditions (7), (8) are satisfied. The lemma 9 is proved. 477

18 N.S.Gonchar Theorem 3. For any local martingale ζ t ({α,ω α }) given by the formula (6) and satisfying conditions sup {ω α} i Ω i f α i ({ω α } i ) = β α i <, i =,, α X 0, ϕ α Fi α i (s {ω α } (ds {ω i ) F α [a α i, aα i+ ) i (s {ω α } i ) <, {ω α} i Ω i, α X 0, the following representation ζ t ({α,ω α }) = f α (ωα )Fα (dωα )+ Ω [a,t] ψ (s ω α )dξ s ({α,ω α }), t [a,b) is valid if the local martingale ξ t ({α,ω α }) is non-singular, sup {ω α} i Ω i g α i ({ω α} i ) = = δ α i <, i =,, α X 0, and [a α i, aα i+ ) ϕ 0,α Fi α (ds {ω α } i ) i (s {ω α } i ) Fi α(s {ω α } i ) <, {ω α} i Ω i, α X 0, where ξ t ({α,ω α }) = χ [a α i, a α i+ ) (t)ξ i,α t ({ω α } i ), ξ i,α t ({ω α } i }) = gi α ({ω α } i )χ [a α i, t](ωi α ) + Fi α(t {ω gi α ({ω α } i )Fi α (dωi α {ω α } i )χ (t, a α i+ )(ωi α ), (t, a α i+ ) ψ (s ω α ) = ϕ 0,α χ [a α i, a α i+ ) (s) ϕα i (s {ω α } i ) ϕ 0,α i (s {ω α } i ), i (s {ω α } i ) = = gi α ({ω α} i,s) Fi α(s {ω gi α ({ω α} i )Fi α (dωα i {ω, (s, a α i+ ) ϕ α i (s {ω α } i ) = = fi α ({ω α } i,s) Fi α(s {ω fi α ({ω α } i )Fi α (dωi α {ω α } i ). (s, a α i+ ) 478

19 Mathematical model of a stock market Proof. Let us consider on the right continuous version of the random processes ξt ({α,ω α}) = χ [a α i, a α i+ ) (t)ξ i,α, t ({ω α } i ), ξ i,α, t ({ω α } i ) = = Fi α(ωα i {ω gi α ({ω α} i )Fi α (dωα i {ω χ [a α i, t)(ωi α ) (ωi α, aα i+ ) + Fi α(t {ω gi α ({ω α} i )Fi α (dωα i {ω χ (t, a α i+ )(ωi α ), (t, a α i+ ) ξt 2 ({α,ω α}) = χ [a α i, a α i+ ) (t)ξ i,α,2 t ({ω α } i ), ξ i,α,2 t ({ω α } i ) = χ [a α i, t](ωi α ) gi α ({ω α} i ) Fi α(ωα i {ω gi α ({ω α} i )Fi α (dωα i {ω. (ωi α, aα i+ ) It is obvious that ξ t ({α,ω α }) = ξ t({α,ω α })+ξ 2 t({α,ω α }). All realizations of the random processes ξt i({α,ω α}), i =,2, have got a bounded variation on any interval [a,t], t < b. Denote by dξt({α,ω i α }), i =,2 and dξ t ({α,ω α }) the charges generated by these realizations on the σ-algebra B([a,b)). To prove the theorem 3, consider those realizations that are continuous at the points {a α i }+,α X 0. The set of realizations satisfying this condition have got the probability. The left and right limits at every point a α i, i =,, α X0 equal limξ t ({α,ω α }) = ξ i,α t a α a ({ω α} α i i ) i gi α({ω α} i )Fi α(dωα i {ω, a α i < ωi α < a α i+, {ω α} i Ω i, = [a α i, aα i+ ) gi α({ω α} i,a α i ), ωα i = a α i, {ω α} i Ω i, (9) limξ t ({α,ω α }) = g t a α i α ({ω, {ω α } i Ω i. (20) i Consider the set Ω 0 α = {ω α Ω α, a α i < ωi α < a α i+, i =,}, where ω α = {ω α,...,ωα } and show that P α(ω 0 α ) =. Really, since the sequence of the sets Ω n α = {ω α Ω α, a α i < ω α i < a α i+, i =,n, a ω α i < b,i = n+,} 479

20 N.S.Gonchar has got the probability, that is, P α (Ω n α ) =, n =,2,..., and taking into account that Ω n α Ω n+ α, Ω 0 α = Ω n α, the continuity of the probability measure P α, we n= obtain P α (Ω 0 α) =. As far as it is no more than a countable set of α for which µ(α) > 0, then there exists a countable subset X 0 X0 such that the direct sum of the sets Ω 0 α,α X0 forms the set Ω 0 Ω, P(Ω 0 ) =. Since gi α ({ω = gi α ({ω α} i )Fi α (dωα i {ω = [a α i, aα i+ ) = gi α ({ω α } i )Fi α (dωi α {ω α } i ), i =,, α X 0, Ω i then for every {α,ω α } Ω 0 the realization of a random process ξ t ({α,ω α }) is continuousat the points {a α i } +. The charge generated by realizations of the random process ξt ({α,ω α}) on the interval [a α i,aα i+ ) is absolutely continuous with respect to the measure Fi α (dt {ω α } i ) and the Radon-Nicodym derivative equals where dξ i,α, F α t ({α,ω α }) i (dt {ω = χ [a α i,ωα i ] (t) Fi α(t (2) {ω α } i ) Fi α(t {ω gi α ({ω α} i )Fi α (dωα i {ω gi α ({ω α} i,t), (t, a α i+ ) Fi α(t {ω α } i ) = lim τ t Fi α(τ {ω. The charge dξt({α,ω 2 α }) generated by realizations of the process ξt({α,ω 2 α }) on the interval [a α i,aα i+ ) is concentrated at the point t = ωα i dξt 2 ({α,ω α}) = dξ i,α,2 t ({ω α } i ) = δ(t ωi α )ϕ0,α i (ωi α {ω. (22) Let us calculate ϕ α i (s {ω ϕ 0,α i (s {ω α } i ) dξi,α s ({ω α} i ) = [a α i, t] + [a α i, t] [a α i, t] ϕ α i (s {ω ϕ 0,α i (s {ω α } i ) dξi,α, s ({ω α } i ) ϕ α i (s {ω ϕ 0,α i (s {ω α } i ) dξi,α,2 s ({ω α } i ) = K i,α, t ({ω α } i )+K i,α,2 t ({ω α } i ). Using (2) and the theorem 2 we have K i,α, t ({ω α } i ) = i,ωα i ] (s)ϕ α i (s {ω Fi α (ds {ω Fi α(s {ω α } i ) 480 [a α i, t] χ [aα

21 Mathematical model of a stock market = Fi α(ωα i {ω fi α ({ω α} i )Fi α (dωα i {ω χ [a α i, t)(ωi α ) (ωi α, aα i+ ) + Fi α(t {ω fi α ({ω α} i )Fi α (dωα i {ω χ (t, a α i+ )(ωi α ) (t, a α i+ ) i (dωi α {ω α } i ). (a α i, aα i+ ) f α i ({ω α } i )F α Further, K i,α,2 t ({ω α } i ) = χ [a α i, t](ωi α ) fi α ({ω α } i ) Fi α(ωα i {ω fi α ({ω α } i )Fi α (dω α i {ω α } i ). (ωi α, aα i+ ) Therefore, [a α i, t] ϕ α i (s {ω ϕ 0,α i (s {ω α } i ) dξi,α s ({ω α } i ) = = fi α ({ω α } i )χ [a α i, t](ωi α ) +χ (t, a α i+ )(ωi α ) Fi α(t {ω fi α ({ω α } i )Fi α (dωi α {ω α } i ) (t, a α i+ ) i (dωα i {ω. [a α i, aα i+ ) f α i ({ω α} i )F α Taking the limit t a α i+ we obtain ϕ α i (s {ω ϕ 0,α i (s {ω α } i ) dξi,α s ({ω α } i ) = where [a α i, aα i+ ) = f α i ({ω α } i ) f α i ({ω α } i ), i =,, f0({ω α α } 0 ) = f({ω α α } )F α (dω) α = (a α, aα 2 ) Ω f α ({ω α } )F α (dω α ). Granting this and the definition of ψ (s ω α ) we obtain f α (ωα )Fα (dωα )+ ψ (s ω α )dξ s ({α,ω α }) = ζ i,α t ({ω α } i ) Ω The theorem 3 is proved. [a,t] = ζ t ({α,ω α }), t [a α i,aα i+ ). 48

22 N.S.Gonchar Theorem 4. Let ξ 0 t ({α,ω α}) be a local martingale on {Ω,F,P}, ξt 0 ({α,ω α}) = ξ i,α χ [a α i, a α i+ ) (t)ξ i,α t ({ω α } i ), t ({ω α } i }) = gi α ({ω α} i )χ [a α i, t](ωi α ) + Fi α(t {ω gi α ({ω α} i )Fi α (dωα i {ω χ (t, a α i+ )(ωi α ), (t, a α i+ ) satisfying conditions: where sup gi α ({ω α} i ) = βi α <, i =,, α X 0, {ω α} i Ω i Fi α (ds {ω α } i ) i (s {ω α } i ) Fi α(s {ω α } i ) <, [a α i, aα i+ ) ϕ 0,α {ω α } i Ω i, i =,, α X 0, ϕ 0,α i (s {ω α } i ) = = gi α ({ω α} i,s) Fi α(s {ω gi α ({ω α} i )Fi α (dωα i {ω. (s, a α i+ ) The random process ξ t ({α,ω α }) = χ [a α i, a α i+ ) (t)f α (ξ i,α t ({ω α } i )) belongs to the subclass K 0 if the family of functions f α (x) > 0, x R, α X 0, are strictly monotonous, sup f α (x) = fα <, inf x R x R f α (x) = fα 2 > 0, moreover, sup i sup sup {ω α} i Ω i s [a α i,aα i+ ) Fi α (s {ω α } i ) Fi α(s {ω α } i ) < fα 2. f α Proof. To prove the theorem 4 it is sufficient to verify the fulfillment of the conditions of definition 2. The condition is valid, because ψ α i ({ω α} i,t) = f α (T α i (t {ω ) is continuous on the right, where Ti α (t {ω α } i ) = Fi α(t {ω gi α ({ω α } i )Fi α (dωi α {ω α } i ). (t, a α i+ ) 482

23 Mathematical model of a stock market Moreover, var t [a α i, τ]ψα i ({ω α } i,t) f α 0,α ϕ [a α i, τ] Fi α i (s {ω α } i ) (ds {ω Fi α(s {ω α } i ) <, τ [a α i,aα i+ ), {ω α} i Ω i, α X 0. The Radon-Nicodym derivative of the charge ψ α i ({ω α } i,dt), generated by ψ α i ({ω α} i,t), equals Thus, ψi α({ω α} i,dt) Fi α(dt {ω = f α (Tα i (t {ω )ϕ 0,α i (t {ω α } i ) Fi α(t. {ω α } i ) γ i,α ({ω α } i,t) = = [a α i, t] [a α i, t] ψ α i ({ω α} i,dτ) ψ α i ({ω α} i,τ) f α (g α i ({ω α} i,τ)) f α(t i α (τ {ω α } i ))Fi α (dτ {ω α } i ) U({ω α } i,τ)[ Fi α(τ {ω α } i )] is non-negative and monotonously non-decreasing on [a α i,a α i+), where U({ω α } i,τ) = Further, γ i,α ({ω α } i,t) fα f α 2 0 f α(g α i ({ω α } i,τ)+z[t α i (τ {ω α } i ) g α i ({ω α } i,τ)])dz. sup i lim γ i,α ({ω α } i,t) fα 2 t a α i+ f α lim t a α i sup sup {ω α} i Ω i lim t a α i+ [a α i, t] s [a α i,aα i+ ) Fi α(s {ω Fi α(s {ω α } i ) <. Fi α(dτ {ω Fi α(τ {ω α } i ) =, γ i,α ({ω α } i,t) = 0, {ω α } i Ω i, α X 0. (c) is evident from (b) and boundedness of ψ α i ({ω α} i ). At last [a α i,aα i+ ) f α (g α i ({ω α} i,t)) exp{ γ i,α ({ω α } i,t )}γ i,α ({ω α } i,dt) The theorem 4 is proved. e(β α i fα +f α(0)). 483

24 N.S.Gonchar 5. Options and their pricing We assume that {Ω,F,P} is a full probability space, generated by the family of distribution functions Fi α (ωi α {ω α } i ), i =,, 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,fα 0,P α}. Theorem 5. Let φ({α,ω α }) = φ α ({ω α } ) be a random value on the probability space {Ω,F,P}, satisfying conditions: ) φ α ({ω α } ) C α <, α X 0, C α dµ(α) < ; X 0 2) there exists t α i [a α i,aα i+ ) such that φ α ({ω α } i,s,{ω α } [i+,] ) φ α ({ω α } i,s 2,{ω α } [i+,] ) C α i Fα i (s {ω α } i ) F α i (s 2 {ω α } i ) εi α, ε i α > 0, s, s 2 [t α i,aα i+ ), C α i <, i =,, {ω α } i Ω i, {ω α } [i+,] Ω i, α X 0. Further, let ξ 0 t({α,ω α }) be a local non-singular martingale on {Ω,F,P}, ξt({α,ω 0 α }) = ξ i,α χ [a α i, a α i+ ) (t)ξ i,α t ({ω α } i ), t ({ω α } i }) = gi α ({ω α} i )χ [a α i, t](ωi α ) + Fi α(t {ω gi α ({ω α} i )Fi α (dωα i {ω χ (t, a α i+ )(ωi α ), (t, a α i+ ) satisfying conditions sup gi α ({ω α } i ) = βi α <, i =,, α X 0, {ω α} i Ω i 0,α Fi α ρi (s {ω α } i ) (ds {ω F α [a α i, aα i+ ) i (s {ω α } i ) <, {ω α} i Ω i, α X 0, ρ 0,α i (s {ω α } i ) = = gi α ({ω α} i,s) Fi α(s {ω gi α ({ω α} i )Fi α (dωα i {ω, (s, a α i+ ) If the random process has got the form 484 ξ t ({α,ω α }) = χ [a α i, a α i+ ) (t)f α (ξ i,α t ({ω α } i )),

25 Mathematical model of a stock market where a family of functions f α (x) > 0, x R, α X 0, is such that each of the functions f α (x) is strictly fulfillment, sup f α(x) = f α <, inf α(x) = f α x R x R f 2 > 0, and Fi α sup sup sup (s {ω i {ω α} i Ω i s [a α i,aα i+ ) Fi α(s {ω α } i ) < fα 2, f α then there exists a measure P on {Ω,F 0 }, generated by a certain family of distribution functions F α, i (ωi α {ω α } i ), i =,, α X 0, a probability measure dµ (α) on the σ-algebra Σ and a modification ξ t ({α,ω α }) of the the random process ξ t ({α,ω α }) such that ξ t ({α,ω α }) is a local non-singular martingale on the probability space {Ω,F,P } with respect to the flow of the σ-algebras Ft, where the σ-algebras F and Ft are the completion of the σ-algebras F 0 and Ft 0 correspondingly with respect to the measure P. Moreover, for the regular martingale M {φ({α,ω α }) Ft } on the probability space {Ω,F,P } the representation M {φ({α,ω α }) F t } = Mα φα ({ω α } )+ ψ (s ω α )d ξ s ({α,ω α }), t [a,b) is valid, where = f α (g α i ({ω α} i,s)) [a,t] ψ (s ω α ) = χ [a α i, a α i+ ) (s) ϕα i (s {ω α } i ) ϕ 0,α i (s {ω α } i ), F α, ϕ 0,α i (s {ω α } i ) = f α (gi α i (s {ω α } i ) ({ω α} i ))F α, i (dωi α {ω, (s, a α i+ ) ϕ α i (s {ω α } i ) = = φ α i ({ω α} i,s) F α, φ α i i (s {ω α } i ) ({ω α} i )F α, i (dωi α {ω, (s, a α i+ ) φ α i ({ω α } i ) =... φ α ({ω α } i,{ω α } [i+,] ) Ω i+ Ω F α, i+ (dωα i+ {ω α} i )... F α, (dωα {ω α} ). 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, ψi α ({ω α } i,a α i ) = f α (ξ i,α a ({ω α} α i i )) = f α i (dωi α {ω α } i ) = (a α i, aα i+ ) g α i ({ω α } i )F α 485

26 N.S.Gonchar = f α (g α i ({ω ) = f α i ({ω with probability on the probability space {Ω,F,P}, since gi ({ω α α } i ) = gi α ({ω α } i )Fi α (dωi α {ω α } i ) (a α i, aα i+ ) with probability. Further, X 0 f α (v(α))dµ (α) <, where dµ (α) = u(α)dµ(α), v(α) = (a α i, aα i+ ) g α (ω α )F α (dω α ), u(α) = χ A(α)+[f α (v(α))] χ X 0 \A(α), D D = {χ A (α)+[f α (v(α))] χ X0\A(α)}dµ(α), X 0 χ A (α) is the characteristic function of the set A = {α, f α (v(α)) }. Based on the lemma 7 there exists a measure P on the σ-algebra F 0, generated by a certain family of distribution functions F α, i (ωi α {ω, i =,, α X 0 and the measure dµ (α) on σ-algebra Σ such that the random process + F α, ξ t ({α,ω α }) = χ [a α i, a α i+ ) (t) ξ i,α t ({ω α } i ), ξ i,α t ({ω α } i ) = f α (gi α ({ω α } i ))χ [a α i, t](ωi α ) f α (gi α i (t {ω α } i ) ({ω α} i ))F α, i (dωi α {ω χ (t, a α i+ )(ωi α ), (t, a α i+ ) i =,, α X 0, is a modification of the random process ξ t ({α,ω α }) on the probability space {Ω,F,P }. Since ξ 0 t ({α,ω α}) is also a local non-singular martingale, then ξ t ({α,ω α }) is also a local non-singular martingale because 486 {t [a,b), ξi,α t,a({ω α } i ) = 0, i =,} {t [a,b), ξ i,α t,a({ω α } i ) = 0, i =,},

27 Mathematical model of a stock market due to the strict monotony of f α (x), where = f α (g α i ({ω α } i )) F α, ξ i,α t,a({ω α } i ) = f α (g α α, i ({ω α } i ))Fi (dωi α {ω α } i ) i (t {ω α } i ) (t, a α i+ ) = f α (gi α ({ω α} i )) f α (Ti α (t {ω ), Ti α (t {ω α } i ) = Fi α(t {ω gi α ({ω α } i )Fi α (dωi α {ω α } i ), (t, a α i+ ) ξ i,α t,a({ω α } i ) = g α i ({ω α } i ) T α i (t {ω α } i ). To finish the proof of the theorem 5 it is sufficient to verify the monotonous of the conditions of the theorem 3. Really, [fα ]2 f α 2 +C α i 0,α ϕi (s {ω α } i ) [a α i, aα i+ ) F α, i (ds {ω α } i ) F α, i (s {ω α } i ) 0,α Fi α ρi (s {ω α } i ) (ds {ω F α [a α i, aα i+ ) i (s {ω α } i ) <, {ω α } i Ω i, α X 0. [a α i, aα i+ ) ϕ α i (s {ω α} i ) 2C α [a α i, tα i ] ( Fi α (s {ω ) εi α [t α i, aα i+ ) F α, i (ds {ω α } i ) F α, i (s {ω α } i ) F α, i (ds {ω α } i ) F α, i (s {ω α } i ) F α, i (ds {ω α } i ) F α, i (s {ω α } i ) <, because the first integral is finite and the second integral is finite since γ i,α ψi α ({ω α } i,dτ) ({ω α } i,t) = ψi α({ω α} i,τ) f α (gi α({ω α} i,τ)) = [a α i, t] = [a α i, t] [a α i, t] F α, i (dτ {ω α } i ) F α, i (τ {ω α } i ) f α (Tα i (τ {ω )Fi α(dτ {ω U({ω α } i,τ)[ Fi α(τ {ω α } i )]. 487

28 N.S.Gonchar Therefore ( Fi α (s {ω ) εi α [t α i, aα i+ ) F α, i (ds {ω α } i ) F α, i (s {ω α } i ) fα. ε i αf2 α 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 marketis effective on the time interval [0,T), if there is a certain probability space {Ω,F,P}, constructed above, a random process ξt 0({α,ω α}) on it, describing the evolution of the average price of stocks such that ξt({α,ω 0 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 Ft 0 with respect to the measure P on F 0, generated by the family of distribution functions Fi α(ωα i {ω. The random process ξt({α,ω 0 α }) has the form ξt 0 ({α,ω α}) = B 0 e rt χ [a α i, a α i+ ) (t)ξ i,α t ({ω α } i ), (23) ξ i,α t ({ω α } i }) = gi α ({ω α } i )χ [a α i, t](ωi α ) + Fi α(t {ω gi α ({ω α} i )Fi α (dωα i {ω χ (t, a α i+ )(ωi α ), (24) (t, a α i+ ) where r is an interest rate, the evolution of price of a stock being described by a certain random process S t ({α,ω α }) = S 0e rt V α χ [a α i, a α i+ ) (t)f α (ξ i,α t ({ω α } i )), V α = f α (M α φ α ({ω α } )) (25) for a certain family of functions f α (x) > 0, x R, α X 0, which are strictly fulfillment, sup f α (x) = fα <, inf x R x R f α (x) = fα 2 > 0, moreover, sup i sup sup {ω α} i Ω i s [a α i,aα i+ ) Fi α(s {ω Fi α(s {ω α } i ) < fα 2. (26) f α The limit φ α ({ω α } )) = limξt 0 ({α,ω t T α})b0 e rt satisfies the conditions: ) φ α ({ω α } ) C α <, α X 0, C α dµ(α) < ; X 0 2) there exists t α i [a α i,aα i+ ) such that 488 φ α ({ω α } i,s,{ω α } [i+,] ) φ α ({ω α } i,s 2,{ω α } [i+,] )

29 Mathematical model of a stock market C α i Fα i (s {ω α } i ) F α i (s 2 {ω α } i ) εi α, ε i α > 0, s, s 2 [t α i,aα i+ ), C α i <, i =,, {ω α } i Ω i, {ω α } [i+,] ) Ω i, α X 0. 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 B(t) = B 0 e rt, (27) 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 b α, δ t ({α,ω α }) = χ [a α i, a α i+ ) (t)δ i,α t ({ω α } i ), δ i,α t ({ω α } i ) = b α, i ({ω α } i,t)χ [a α i, t](ω α i )+bα,2 i ({ω α } i,t)χ (t, a α i+ )(ω α i ), i ({ω α } i,t) is a measurable mapping from the measurable space {Ω i, F i} 0 to the measurable space {R,B(R )} at every fixed t from the interval [0,T), b α,2 i ({ω α } i,t) is a measurable mapping from the measurable space {Ω i, F i 0 } to the measurable space {R,B(R )} at every fixed t [0,T). Moreover, b α, i ({ω α } i,t) is a bounded measurable mapping from the measurable space {[0,T),B([0,T))} to the measurable space {R,B(R )} at every fixed {ω α } i Ω i, b α,2 i ({ω α } i,t) is a bounded measurable mapping from {[0,T),B([0,T))} to {R,B(R )} at every fixed {ω α } i Ω i. Let the capital of an investor X t ({α,ω α }) at time t equal X t ({α,ω α }) = B(t)β t ({α,ω α })+γ t ({α,ω α })S t ({α,ω α }), (28) where the stochastic processes β t ({α,ω α }) and γ t ({α,ω α }) belong to the class A 0. The pair π t = {β t ({α,ω α }),γ t ({α,ω α })} is called the financial strategy of the investor. The capital of the investor with the financial strategy π t will be denoted by X π t ({α,ω α }). 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 Xt π({α,ω α}) the representation Xt π ({α,ω α}) = X0 π (α)+ β τ ({α,ω α })db(τ)+ γ τ ({α,ω α })ds τ ({α,ω α }) (29) [0,t] [0,t] is valid, the discounted capital Y π t ({α,ω α}) = Xπ t ({α,ω α}) B(t) 489

30 N.S.Gonchar belongs to the class of local martingale on the probability space {Ω,F,P } with respect to the flow of the σ-algebras F t, M X π t ({α,ω α }) <, where F,P and F t are constructed in the theorem 5. A class of self-financing strategy is denoted by SF. Lemma 0. Let a financial strategy π t = {β t ({α,ω α }),γ t ({α,ω α })} be self-financing, then for the investor capital the representations Xt π ({α,ω α}) = X0 π (α)+ β τ ({α,ω α })db(τ)+ γ τ ({α,ω α })ds τ ({α,ω α }) (30) [0,t] X π t ({α,ω α}) = e rt X π 0 (α)+b 0e rt [0,t] γ τ ({α,ω α })ds 0 τ ({α,ω α}) (3) are equivalent, where [0,t] St 0 ({α,ω α}) = S 0 χ [a α B 0 V i, a α i+ ) (t)f α (ξ i,α t ({ω α } i )). (32) α Proof. Since Xt π({α,ω α}) is a process of a bounded variation on any interval [0,t], therefore from (3) and lemma Xt π ({α,ω α}) = X0 π (α)+ X0 α +B 0 γ τ ({α,ω α })dsτ 0 ({α,ω α}) de rt Since = X0(α)+ π [0,t] +B 0 [0,t] [0,t] [0,t] e rτ γ τ ({α,ω α })ds 0 τ ({α,ω α}) X τ ({α,ω α }) db(τ) B(τ) + [0,t] S t ({α,ω α }) = B(t)S 0 t ({α,ω α}), B(τ)γ τ ({α,ω α })ds 0 τ({α,ω α }). ds t ({α,ω α }) = S 0 t ({α,ω α})db(t)+b(t)ds 0 t ({α,ω α}), (33) therefore, taking into account (28) and (33), we obtain Xt π ({α,ω α}) = X0 π (α)+ db(τ) β τ ({α,ω α })db(τ)+ B(τ) γ τ({α,ω α })S τ ({α,ω α }) + [0,t] γ τ ({α,ω α })ds τ ({α,ω α }) [0,t] γ τ ({α,ω α })S 0 τ ({α,ω α})db(τ) [0,t] [0,t] 490

31 Mathematical model of a stock market = X π 0 (α)+ β τ ({α,ω α })db(τ)+ γ τ ({α,ω α })ds τ ({α,ω α }). [0,t] This proves the lemma 0 in one direction. Applying the same argument in the inverse direction we obtain the proof of the lemma 0. Denote by SF R a set of self-financing strategies satisfying the conditions [0,t] M {Y π t ({α,ω α}) F t } M {R F t }, M R <, where R is a non-negative random value on {Ω,F,P }. Lemma. Let π t = {β t ({α,ω α }),γ t ({α,ω α })} be a self-financing strategy, that is, π t SF R, then {Yt π,ft,t [0,T]} is a supermartingale and for any stop time τ and τ 2 such that P (τ τ 2 ) = the inequality is valid. M {Y π τ 2 ({α,ω α }) F τ } Y π τ ({α,ω α }) The proof is similar to the proof of the analogous lemma in [2]. Corollary 3. If π t SF R, then for any stop time τ 0, P (τ < ) = M Y π τ ({α,ω α}) Y π 0 (α) = Xπ 0(α) B 0. Definition 8. A self-financing strategy π t is an arbitrage strategy on [0,T], if from that X π 0(α) 0, X π T({α,ω α }) 0 it follows that X π T ({α,ω α}) > 0 with a positive probability. Lemma 2. 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 φ T = φ T ({α,ω α }) = φ α T ({ω α} )bef 0 measurablerandomvalueontheprobability space {Ω,F 0,P}. Definition 9. A self-financingstrategy 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 with respect to the measure P X π T({α,ω α }) φ T ({α,ω α }). (x α,φ T )-hedge πt SF R is called minimal if for any (x α,φ T )-hedge π t SF R the inequality XT({α,ω π α }) XT π ({α,ω α }) is valid. 49

32 N.S.Gonchar Then we consider self-financing strategies, belonging to SF 0, that is, in this case X π t ({α,ω α }) 0. Definition 0. 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. C α T(φ T ) = inf{x α 0, H T (x α,φ T ) }, α X 0, The main problem is to calculate C α T (φ T) and to find an expression for the portfolio of an investor π t at every moment of time t the initial capital of which is xα. Further on we assume that T <, then lim S t({α,ω α }) = S T ({α,ω α }) = S 0e rt f α (φ({α,ω α })). t T V α Theorem 6. Let a stock market be effective, the evolution of a risky active price comes according to the formula (25) and the evolution of non-risky active price occur by (27). If f(x) is a certain function such that f(x ) f(x 2 ) C x x 2 and the paying function at terminal time T is given by the formula f T ({α,ω α }) = f(s T ({α,ω α })), moreover, the conditions fc α α dµ(α) <, V α X 0 X 0 f α (0) V α dµ(α) <, are valid, then the minimal hedge π t exists, evolution of the capital investor X t ({α,ω α}), option price X 0 (α) and self-financial strategy {β t ({α,ω α}), γ t({α,ω α })} corresponding to the minimal hedge π t are given by the formulas X t({α,ω α }) = e r(t T) M {f(s T ({α,ω α })) F t}, (34) where X 0(α) = e rt M αf(s T ({α,ω α })), γ t({α,ω α }) = ψ (t {ω} α ), (35) βt({α,ω α }) = X t ({α,ω α}) γt ({α,ω α})s t ({α,ω α }), (36) B(t) F α, ψ (s ω α ) = χ [a α i, a α i+ ) (s) ϕα i (s {ω ϕ 0,α i (s {ω α } i ), ϕ 0,α i (s {ω α } i ) = = f α (gi α ({ω α} i,s)) f α (g α α, i ({ω α } i ))Fi (dωi α {ω α } i ), i (s {ω α } i ) (s, a α i+ ) 492