A Binomial Asset Pricing Model in a Categorical Setting
aa r X i v : . [ q -f i n . M F ] D ec A BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING
TAKANORI ADACHI, KATSUSHI NAKAJIMA AND YOSHIHIRO RYU
Abstract.
Adachi and Ryu introduced a category
Prob of probability spaces whose objectsare all probability spaces and whose arrows correspond to measurable functions satisfying anabsolutely continuous requirement in [Adachi and Ryu, 2019]. In this paper, we develop abinomial asset pricing model based on
Prob . We introduce generalized filtrations with whichwe can represent situations such as some agents forget information at some specific time. Weinvestigate the valuations of financial claims along this type of non-standard filtrations. Introduction
Adachi and Ryu introduced the category
Prob as an adequate candidate of the category ofprobability spaces with good arrows. They show the existence of the conditional expectationfunctor from
Prob to Set , which is a natural generalization of the classical notion of conditionalexpectation ([Adachi and Ryu, 2019]).,In this paper, we develop a binomial asset pricing model based on the category
Prob . Gener-alized filtrations defined in this setting change not only σ -algebras but also probability measuresand even underlying sets throughout time. We introduce a few types of generalized filtrations.Each of them represents a subjective filtration of an agent. In other words, each agent has notonly her subjective probability measure but also her own subjective filtration. For example,some filtration represents the situation in which she forgets the information generated at a spe-cific time. This paper investigate the valuations of financial claims along these non-standardfiltrations.First, in Section 2, we review the concept of categorical probability theory and introducegeneralized filtrations and adapted processes and martingales along them. In this setting, ourprobability spaces are changing as time goes on. For example, we may have a bigger underlyingset in future than that in past. This case allows us to have unknown future elementary events.Section 3 is the heart of this paper in which we develop a concrete binomial asset pricing modeland investigate a few generalized filtrations and possibility of valuations along them. We alsoprovide a complete form of a replication strategy making the valuation possible.2. Generalized Filtrations
In this section, we introduce some basic concepts of categorical probability theory which weremainly introduced in [Adachi and Ryu, 2019] as a preparation for Section 3.Let ¯ X = ( X, Σ X , P X ), ¯ Y = ( Y, Σ Y , P Y ) and ¯ Z = ( Z, Σ Z , P Z ) be probability spaces through-out this paper. Date : December 17, 2019.2010
Mathematics Subject Classification.
Primary 91B25, 16B50; secondary 60G20, 91Gxx .
Key words and phrases. binomial asset pricing model, categorical probability theory, generalized filtration .This work was supported by JSPS KAKENHI Grant Number 18K01551.
Definition 2.1. [Null-preserving functions [Adachi and Ryu, 2019] ] A measurable function f : ¯ Y → ¯ X is called null-preserving if f − ( A ) ∈ N Y for every A ∈ N X , where N X := P − X (0) ⊂ Σ X and N Y := P − Y (0) ⊂ Σ Y . Definition 2.2. [Category
Prob [Adachi and Ryu, 2019] ] A category
Prob is the categorywhose objects are all probability spaces and the set of arrows between them are defined by
Prob ( ¯ X, ¯ Y ) := { f − | f : ¯ Y → ¯ X is a null-preserving function. } , where f − is a symbol corresponding uniquely to a function f .We write Id X for an identity measurable function from ¯ X to ¯ X , while writing id X for anidentity function from X to X . Therefore, the identity arrow of a Prob -object ¯ X is Id − X . Definition 2.3. [Generalized Filtrations] Let T be a fixed small category which we sometimescall the time domain . A T -filtration is a functor F : T →
Prob . T F (cid:15) (cid:15) t i / / i ◦ i t i / / t i / / . . . Prob
F t F i / / F ( i ◦ i )= F i ◦ F i ; ; F t F i / / F t F i / / . . . Figure 2.1. T -filtrationWhen we say filtrations in the classical setting, we keep using a same underlying set Ωthroughout time. This situation can be represented by the following diagram. T t / / t / / t / / . . . F t Id − / / F t Id − / / F t Id − / / . . . Ω Ω Id Ω o o Ω Id Ω o o . . . Id Ω o o However, in our new setting, the filtration can change not only σ -fields but also probabilitymeasures and underlying sets as the following diagram shows. T F (cid:15) (cid:15) t / / t / / t / / . . . Prob ¯ X t f − / / ¯ X t f − / / ¯ X t f − / / . . .X t X t f o o X t f o o . . . f o o One of the implications of this generalization is that we can think possibly distorted filtrationsby using adequate null-preserving function f t .Actually, the biggest aim of this paper is to investigate this kind of non-standard filtrationsby using, as a first example, a simple binomial asset pricing model. BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING 3
Let
Meas be the category of measurable spaces that consists of all measurable spaces as itsobjects and all measurable functions between them as arrows.(2.1) U : Prob → Meas is a forgetful functor that is it maps a object ¯ X to ( X, Σ X ) by dropping its probability measure,and an arrow f : ¯ X → ¯ Y to f : ( X, Σ X ) → ( Y, Σ Y ).Later, we will investigate a modification of a given filtration F to another filtration G suchthat U ◦ F = U ◦ G , that is, the situation when they share their measurable space nature.Before going into our concrete example, we will define adapted processes and martingalesover this generalized filtrations.Let F be a fixed T -filtration throughout this section. Definition 2.4. [ F -Adapted Processes] An F -adapted process is a collection of naturaltransformations(2.2) τ := { τ s : T ( s, − ) ˙ → L ◦ F } s ∈ Obj ( T ) For a
Prob -arrow ϕ : ¯ X → ¯ Y , there exists a measurable function f : Y → X such that ϕ = f − by its definition. We write ϕ + for this f . That is, ( ϕ + ) − = ϕ .Now Let τ be an F -adapted process and i : s → t be a T -arrow. Then, we have the followingcommutative diagram. Id s ∈ ✤ / / ❴ (cid:15) (cid:15) τ s,s ( Id s ) ∈ ❴ (cid:15) (cid:15) s i (cid:15) (cid:15) T ( s, s ) τ s,s / / T ( s,i ) (cid:15) (cid:15) L ( F s ) L ( F i ) (cid:15) (cid:15) t T ( s, t ) τ s,t / / L ( F t ) i ∈ ✤ / / τ s,t ( i ) = L ( F i )( τ s,s ( Id s )) ∈ For s ∈ Obj ( T ) pick a random variable v s satisfying [ v s ] ∼ P F s = τ s,s ( Id s ). Then, we have(2.3) τ s,t ( i ) = [ v s ◦ ( F i ) + ] ∼ P F t . That is, τ s,t ( i ) is ( F i )-measurable.
Proposition 2.5.
Let AP ( F ) be the set of all F -adapted processes. Then, (2.4) AP ( F ) ∼ = Y t ∈ Obj ( T ) L ( F t ) . Proof.
By Yoneda Lemma, we have for t ∈ Obj ( T ),(2.5) y t : Nat( T ( t, − ) , L ◦ F ) ∼ = ( L ◦ F ) t. Then, Q t ∈ Obj ( T ) y t is an isomorphism denoting (2.4). (cid:3) For x ∈ AP ( F ), we sometimes write(2.6) x = { x t } t ∈ Obj ( T ) T. ADACHI, K. NAKAJIMA AND Y. RYU where(2.7) x t := x ( t ) ∈ L ( F t ) . Remark 2.6.
For an arrow i : s → t in T , in general, F s and
F t are different probabilityspaces. So we cannot (for example) add two random variables x s ∈ L ( F s ) and x t ∈ L ( F t )whose domains are ˜
F s and ˜
F t . s i / / tF s F i / / F t ˜ F s x s (cid:15) (cid:15) ˜ F t ( F i ) + o o x t (cid:15) (cid:15) x s ◦ ( F i ) + (cid:15) (cid:15) R R
In order to import x s into L ( F t ), we take x s ◦ ( F i ) + as its proxy. This fact allows us to treat L ( F t ) as a vector space containing all preceding random variables x s ∈ L ( F s ) with s ≤ t .Next, we go into the definition of martingales. In order to make it possible, we need a conceptof conditional expectations in the category Prob which was introduced in [Adachi and Ryu, 2019].
Theorem 2.7. [Conditional Expectation [Adachi and Ryu, 2019]]
Let f − : ¯ X → ¯ Y be a Prob -arrow. For all v ∈ L ( ¯ Y ) and A ∈ Σ X , there exists u ∈ L ( ¯ X ) satisfying the following equation. (2.8) Z A u d P X = Z f − ( A ) v d P Y . We call u a conditional expectation along f − and denote it by E f − ( v ) . Theorem 2.8. [Conditional Expectation Functor [Adachi and Ryu, 2019]]
There exists a func-tor E : Prob op → Set as following: X ¯ X f − (cid:15) (cid:15) ✤ E / / E ¯ X := L ( ¯ X ) ∋ [ E f − ( v )] ∼ P X Y f O O ¯ Y ✤ E / / E ¯ Y := E f − O O L ( ¯ Y ) ∋ [ v ] ∼ P Y . ❴ E f − O O We call E a conditional expectation functor . Definition 2.9. [ F -Martingales] Let F : T →
Prob be a functor. An F -martingale is an F -adapted process x ∈ AP ( F ) such that for every T -arrow i : s → t ,(2.9) ( E ◦ F ) i ( x ( t )) = x ( s ) .s i (cid:15) (cid:15) ✤ F / / F s
F i (cid:15) (cid:15) ✤ E / / E ( F s ) := L ( F s ) ∋ x s = [ E F i ( v )] ∼ P F s t ✤ F / / F t ✤ E / / E ( F t ) := E ( F i ) O O L ( F t ) ∋ x t = [ v ] ∼ P F t . ❴ E ( F i ) O O Figure 2.2. F -martingale BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING 5 A Binomial Asset Pricing Model
In this section, we introduce a binomial asset pricing model based on the category
Prob .3.1.
Filtration B . First, we define a general scheme of our model by introducing a filtration B . Definition 3.1. [Filtration B ] Let ω be the category whose objects are all integers startingwith 0 and for each pair of integers m and n with m ≤ n there is a unique arrow ∗ m,n : m → n .That is, ω is the category corresponding to the integer set N with the usual total order. Let p := { p i } i =1 , ,... be an infinite sequence of real numbers p i ∈ [0 , ω -filtration B := B p : ω → Prob in the following way.For an object n of ω , B n is a probability space ¯ B n := ( B n , Σ n , P n ) whose components aredefined as follows:(1) B n := { , } n , the set of all binary numbers of t digits,(2) Σ n := 2 B n ,(3) for a := d d . . . d n ∈ B n where d i ∈ { , } ( i = 1 , , . . . n ). P n : Σ n → [0 ,
1] is theprobability measure defined by(3.1) P n ( { a } ) := n Y i =1 p d i i (1 − p i ) − d i . For integers m and n with m < n , we define(3.2) B ( ∗ m,n ) := f − m,n := ( f m ◦ f m +1 ◦ · · · ◦ f n − ) − where f n := ( B ( ∗ n,n +1 )) + is a predefined null-preserving function from B n +1 to B n .The filtration B is called non-trivial if there exists i such that 0 < p i < B n is measurable since Σ n is a powerset of B n . ω B (cid:15) (cid:15) i / / i / / . . . i n − / / n i n / / n + 1 i n +1 / / . . . Prob ¯ B f − / / ¯ B f − / / . . . f − n − / / ¯ B n f − n / / ¯ B n +1 f − n +1 / / . . . As we introduced, the functor B is a generalized filtration, representing a filtration over theclassical binomial model, for example developed in [Shreve, 2005].The classical version requires the terminal time horizon T for determining the underlyingset Ω := { , } T while our version does not require it since the time variant probability spacescan evolve without any limit. That is, our version allows unknown future elementary events,which, we believe, shows a big philosophical difference from the Kolmogorov world.In order to see a variety of filtrations, we introduce two candidates of f n . Definition 3.2. [Candidates of f n ](1) f fulln B n +1 f fulln / / ∈ B n ∈ d . . . d n d n +1 ✤ f fulln / / d . . . d n T. ADACHI, K. NAKAJIMA AND Y. RYU (2) f dropn B n +1 f dropn / / ∈ B n ∈ d . . . d n − , d n d n +1 ✤ f dropn / / d . . . d n − f dropn can be interpreted to forget what happens at time n .Note that the function f fulln is always null-preserving while f dropn is null-preserving if and onlyif p n = 0. Example 3.3. [Filtrations] As we mentioned in Definition 3.1, all we need to determine thefiltration is to specify f n : B n +1 → B n . We have three examples of filtration B . For j =1 , , . . . , n ,(1) Classical filtration: f n := f fulln . (2) Drop- k : f n := ( f dropn if n = k,f fulln if n = k. (3) Elderly person: For fixed numbers k , k ∈ N , f n := ( f dropn if k ≤ n ≤ T − k f fulln if 0 ≤ n < k or T − k < n ≤ T. Proposition 3.4.
For a
Prob -arrow f − n : ¯ B n → ¯ B n +1 , v ∈ L ( ¯ B n +1 ) and a ∈ B n , (3.3) E f − n ( v )( a ) P n ( { a } ) = X b ∈ f − n ( a ) v ( b ) P n +1 ( { b } ) . Especially, with the classical filtration, we have(3.4) f − n ( a ) = ( f fulln ) − ( a ) = { a , a } . Hence E f − n ( v )( a ) = v ( a P n +1 ( { a } ) P n ( { a } ) + v ( a P n +1 ( { a } ) P n ( { a } )= v ( a − p n +1 ) + v ( a p n +1 . (3.5) Definition 3.5. [ B -Adapted Process ξ n ] For n = 1 , , . . . define a B -adapted process ξ n by B n ξ n / / ∈ R ∈ d d . . . d n ✤ ξ n / / d n − Proposition 3.6.
For a ∈ B n with P n ( a ) = 0 , E f − n ( ξ n +1 )( a ) = X e ∈ I n (1 ,a ) P n +1 ( e ) P n ( a ) − X e ∈ I n (0 ,a ) P n +1 ( e ) P n ( a )= f − n ( a )) p n +1 − I n (0 , a ) BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING 7 where I n ( j, a ) := { e ∈ f − n ( a ) | ( e ) n +1 = j } for j = 0 , , and A denotes the cardinality of the set A . Arbitrage Strategies.
Now we define two instruments tradable in our market.
Definition 3.7. [Stock and Bond Processes] Let µ, σ, r ∈ R be constants such that σ > µ > σ − r > − stock process S n : B n → R over B is defined by(3.6) S ( hi ) := s , S n +1 := ( S n ◦ f n )(1 + µ + σξ n +1 )where hi ∈ B is the empty sequence, and s > bond process b n : B n → R over B is defined by(3.7) b ( hi ) := 1 , b n +1 := ( b n ◦ f n )(1 + r ) . The condition µ > σ − B , S, b ) a market But, it does not mean that the market willnot contain other instruments.
Proposition 3.8.
For any a ∈ B n ,(1) E f − n ( S n +1 ) = S n (cid:0) (1 + µ ) E f − n (1 B n +1 ) + σE f − n ( ξ n +1 ) (cid:1) . (2) E f − n (1 B n +1 )( a ) = P n +1 ( f − n ( a )) P n ( a ) . (3) b n ( a ) = (1 + r ) n . Definition 3.9. [Strategies] A strategy is a sequence ( φ, ψ ) = { ( φ n , ψ n ) } n =1 , ,... , where(3.8) φ n : B n − → R and ψ n : B n − → R . Each element of the strategy ( φ n , ψ n ) is called a portfolio . The value V n of the portfolio attime n is determined by:(3.9) V n := ( S φ + b ψ if n = 0 S n ( φ n ◦ f n − ) + b n ( ψ n ◦ f n − ) if n = 1 , , . . . Definition 3.10. [Gain Processes] A gain process of the strategy ( φ, ψ ) is the process { G ( φ,ψ ) n } n =0 , , ,... defined by(3.10) G ( φ,ψ ) n := ( − ( S φ + b ψ ) if n = 0( S n ( φ n ◦ f n − ) + b n ( ψ n ◦ f n − )) − ( S n φ n +1 + b n ψ n +1 ) if n = 1 , , . . . Lemma 3.11.
Let n be an object of ω such that (3.11) S n φ n +1 + b n ψ n +1 = 0 . Then, we have (3.12) S n +1 ( φ n +1 ◦ f n ) + b n +1 ( ψ n +1 ◦ f n ) = ( µ + σξ n +1 − r )(( S n φ n +1 ) ◦ f n ) . Proof.
LHS =( S n ◦ f n )(1 + µ + σξ n +1 )( φ n +1 ◦ f n ) + ( b n ◦ f n )(1 + r )( ψ n +1 ◦ f n )=(1 + µ + σξ n +1 )(( S n φ n +1 ) ◦ f n ) + (1 + r )(( b n ψ n +1 ) ◦ f n )=(1 + µ + σξ n +1 )(( S n φ n +1 ) ◦ f n ) − (1 + r )(( S n φ n +1 ) ◦ f n ) = RHS. (cid:3)
Definition 3.12. [Arbitrage Strategies]
T. ADACHI, K. NAKAJIMA AND Y. RYU (1) A strategy ( φ, ψ ) is called a B - arbitrage strategy if P n (cid:0) G ( φ,ψ ) n ≥ (cid:1) = 1 for every n ,and P n (cid:0) G ( φ,ψ ) n > (cid:1) > n .(2) The market is called non-arbitrage or NA if it does not allow B -arbitrage strategies. Proposition 3.13.
If the market ( B , S, b ) with a non-trivial filtration B is non-arbitrage, then | µ − r | < σ. Proof.
Assuming that r ≤ µ − σ or r ≥ µ + σ , we will construct an arbitrage strategy ( φ, ψ )by using the following algorithm. for n = 0 , 1 , 2 , ...:o b s e r v e S ( n ) and b ( n )if r <= mu - sigma :phi ( n +1) > 0 In the above code, ‘*’ is the function composition operator.By Lemma 3.11, we have(3.13) G ( φ,ψ ) n = ( µ + σξ n − r )(( S n − φ n ) ◦ f n − ) . So we have G ( φ,ψ ) n ≥ r ≤ µ − σ or r ≥ µ + σ .By the way, since our filtration is non-trivial, there exists a number n such that 0 < p n < P n ( G ( φ,ψ ) n > > , which concludes that ( φ, ψ ) is an arbitrage strategy. (cid:3) Risk-Neutral Filtrations.
In this subsection, we assume that | µ − r | < σ .Let us consider about the discounted stock process(3.15) S ′ n := b − n S n . Definition 3.14. [Risk-neutral filtrations] An ω -filtration C is called a risk-neutral filtrationhaving the same shape of B if U ◦C = U ◦B and discounted stock process becomes a C -martingale,that is, for any arrow of the form ∗ n,n +1 : n → n + 1 in the category ω (3.16) ( E ◦ C ) ∗ n,n +1 ( S ′ n +1 ) = S ′ n . We want to find a risk-neutral filtration C . Here is the shape of the filtration whose detailwe will determine. Definition 3.15. [Filtration C ] Let n be an object of the category ω .(1) Q n : Σ n → [0 ,
1] is a probability measure of ( B n , Σ n ),(2) ¯ C n := ( B n , Σ n , Q n ),(3) g n := f n .We define an ω -filtration C by for n ∈ Obj ( ω ),(3.17) C ( n ) := ¯ C n , C ( ∗ n,n +1 ) := g − n . BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING 9 ω C (cid:15) (cid:15) i / / i / / . . . i n − / / n i n / / n + 1 i n +1 / / . . . Prob ¯ C g − / / ¯ C g − / / . . . g − n − / / ¯ C n g − n / / ¯ C n +1 g − n +1 / / . . . Figure 3.1.
Filtration C With these notations, (3.16) is reduced to the form(3.18) E g − n ( S ′ n +1 ) = S ′ n . Theorem 3.16.
A process S ′ n is a C -martingale , that is, for n ∈ N , E g − n ( S ′ n +1 ) = S ′ n if andonly if for all n ∈ N and a ∈ B n , (3.19) Q n ( { a } ) = c Q n +1 ( I n (1 , a )) + c Q n +1 ( I n (0 , a )) where for j = 0 , I n ( j, a ) := { e ∈ f − n ( a ) | ( e ) n +1 = j } and (3.21) c := 1 + µ + σ r , c := 1 + µ − σ r . Proof.
For a ∈ B n S ′ n ( a ) Q n ( { a } ) = E g − n ( S ′ n +1 )( a ) Q n ( { a } )= X e ∈ f − n ( a ) S ′ n +1 ( e ) Q n +1 ( { e } )= X e ∈ f − n ( a ) b − n +1 ( e )( S n ◦ f n )( e )(1 + µ + σξ n +1 ( e )) Q n +1 ( { e } )= X e ∈ f − n ( a ) (1 + r ) − ( n +1) S n ( a )(1 + µ + σξ n +1 ( e )) Q n +1 ( { e } )= S ′ n ( a ) X e ∈ f − n ( a ) µ + σξ n +1 ( e )1 + r Q n +1 ( { e } ) . if and only if Q n ( { a } ) = X e ∈ I n (1 ,a ) µ + σ r Q n +1 ( { e } ) + X e ∈ I n (0 ,a ) µ − σ r Q n +1 ( { e } )= c Q n +1 ( I n (1 , a )) + c Q n +1 ( I n (0 , a )) . (cid:3) In order to determine more detail of C , we need the following condition for Q n . Proposition 3.17.
The following conditions for Q n are equivalent.(1) for all n ∈ N , a ∈ B n , (3.22) Q n +1 ( { a , a } ) = Q n ( { a } ) (2) for all n ∈ N , f fulln is measure-preserving w.r.t. Q n , that is, (3.23) Q n = Q n +1 ◦ ( f fulln ) − . (3) there exists a sequence of functions { q k : B k → [0 , } k =1 , ,... such that for all n = 1 , , . . . and d j = 0 , , (3.24) Q n ( { d d . . . d n } ) = n Y k =1 q k ( d d . . . d k ) such that for every a ∈ B n − , q n ( a
0) + q n ( a
1) = 1 . In the following discussion, we assume the following assumption which is the condition (2)of Proposition 3.17.
Assumption 3.18.
For all n ∈ N , f fulln is measure-preserving w.r.t. Q n .By Assumption 3.18 and (3) of Proposition 3.17, we have Q n +1 ( { d d . . . d n d n +1 } ) = Q n ( { d d . . . d n } ) q n +1 ( d d . . . d n +1 ) . In the rest of this subsection, we will investigate the shape of Q n under the assumption that S ′ n is C -martingale.3.3.1. Classical Filtration.
First, we prepare a lemma for for the proof of the following propo-sitions.
Lemma 3.19. If c x + c (1 − x ) , then (3.25) x = 12 + r − µ σ and − x = 12 − r − µ σ . Proposition 3.20.
For a fixed n ∈ N , assume that f n = f fulln . Then for a ∈ B n with Q n ( { a } ) =0 , we have q n +1 ( a
1) = 12 + r − µ σ ,q n +1 ( a
0) = 12 − r − µ σ . Note that the resulting probability depends neither on a nor on n . Proof.
By observing the following diagram a a ✐✐✐✐✐✐✐✐✐✐✐ ❯❯❯❯❯❯❯❯❯❯❯ a B n ∈ B n +1 f fulln o o ∈ a ad n +1 ✤ f fulln o o we have ( f fulln ) − ( a ) = { a , a } I n (1 , a ) = { a } ,I n (0 , a ) = { a } By (3.19) Q n ( { a } ) = c Q n +1 ( I n (1 , a )) + c Q n +1 ( I n (0 , a ))= c Q n +1 ( { a } ) + c Q n +1 ( { a } ) BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING 11
Now since Q n +1 ( { ad n +1 } ) = Q n ( { a } ) q n +1 ( ad n +1 )and Q n ( { a } ) = 0, we have 1 = c q n +1 ( a
1) + c q n +1 ( a . Hence by Lemma 3.19, we have q n +1 ( a
1) = 12 + r − µ σ , q n +1 ( a
0) = 12 − r − µ σ . (cid:3) Corollary 3.21. If B is the classical filtration, then for any n ∈ N and a ∈ B n we have (3.26) Q n ( a ) = (cid:0)
12 + r − µ σ (cid:1) n (1 ,a ) (cid:0) − r − µ σ (cid:1) n (0 ,a ) where (3.27) n ( j, a ) := { k | ( a ) k = j } . Drop- k Filtration.
Proposition 3.22.
For a fixed n (= 1 , , . . . ) , assume that f n = f dropn . Then for a ∈ B n − with Q n − ( { a } ) = 0 , we have q n ( a
1) = 0 ,q n ( a
0) = 1 ,q n +1 ( a
01) = 12 + r − µ σ ,q n +1 ( a
00) = 12 − r − µ σ . Proof.
By observing the following diagram a a ❢❢❢❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳❳❳❳ a a ♠♠♠♠♠♠♠♠♠♠♠ ◗◗◗◗◗◗◗◗◗◗◗ a a ❢❢❢❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳❳❳❳ a B n − ∈ B nf fulln − o o ∈ B n +1 f dropn o o ∈ a a ✤ f fulln − o o ad n d n +1 ✤ f dropn o o we have ( f dropn ) − ( a
1) = ∅ I n (1 , a
1) = I n (0 , a
1) = ∅ ( f dropn ) − ( a
0) = { a , a , a , a } I n (1 , a
0) = { a , a } I n (0 , a
0) = { a , a } By (3.19) Q n ( { a } ) = c Q n +1 ( I n (1 , a c Q n +1 ( I n (0 , a . Now since Q n ( { ad n } ) = Q n − ( { a } ) q n ( ad n ) and Q n − ( { a } ) = 0, we have q n ( a
1) = 0 , q n ( a
0) = 1 − q n ( a
1) = 1 . Next, again by (3.19) Q n ( { a } ) = c Q n +1 ( I n (1 , a c Q n +1 ( I n (0 , a c (cid:0) Q n +1 ( { a } ) + Q n +1 ( { a } ) (cid:1) + c (cid:0) Q n +1 ( { a } ) + Q n +1 ( { a } ) (cid:1) By dividing both hands by Q n − ( { a } ) = 0, q n ( a
0) = c (cid:0) q n ( a q n +1 ( a
01) + q n ( a q n +1 ( a (cid:1) + c (cid:0) q n ( a q n +1 ( a
00) + q n ( a q n +1 ( a (cid:1) Then, since q n ( a
1) = 0 and q n ( a
0) = 1,1 = c q n +1 ( a
01) + c q n +1 ( a . Hence, by Lemma 3.19, we have q n +1 ( a
01) = 12 + r − µ σ , q n +1 ( a
00) = 12 − r − µ σ . (cid:3) We have to check that both f fulln and f dropn are null-preserving w.r.t. Q n . B ndrop n (cid:15) (cid:15) ∋ d . . . d n − d n ❴ (cid:15) (cid:15) B n +1 f fulln qqqqqqqqqq f dropn & & ▼▼▼▼▼▼▼▼▼▼ B n ∋ d . . . d n − Q n ( d . . . d n − = 0, then drop n is null-preserving, and so is f dropn since f fulln is measure-preserving. a a ❢❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳❳ a a ♠♠♠♠♠♠♠♠♠♠♠ ◗◗◗◗◗◗◗◗◗◗◗ a a ❢❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳❳ a B n − B nf fulln − o o B n +1 f dropn o o Figure 3.2. f dropn Remark 3.23.
We have the following remarks for Figure 3.2.(1) Since the agent evaluates stock and bond along the function f dropn , she can recogniseonly the nodes a a
01 and a
00 and can not recognise the nodes a a
11 and a
10. Weinterpret these nodes a a
11 and a
10 as invisible.
BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING 13 (2) The values q n +1 ( a ∈ [0 ,
1] can be arbitrarily selected, and q n +1 ( a
10) is computed by1 − q n +1 ( a Q n +1 is not determined uniquely, so isnot the risk-neutral filtration C .(3) The probability measure Q n is not equivalent to the original measure P n . Therefore, itis not an EMM. a a ❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳ a a ♠♠♠♠♠♠♠♠♠♠ ◗◗◗◗◗◗◗◗◗◗ a a ❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳ a a ③③③③③③③③③③③③③③③ ❉❉❉❉❉❉❉❉❉❉❉❉❉❉ a a ❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳ a a ♠♠♠♠♠♠♠♠♠♠ ◗◗◗◗◗◗◗◗◗◗ a a ❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳ a B k − B k − f fullk − o o B kf fullk − o o B k +1 f dropk o o Figure 3.3. drop- k filtration Remark 3.24.
Let C : ω → Prob be a risk-neutral filtration, and Y : B T → R be a payoff attime T . Then, for the agent who has a drop- k filtration as her subjective filtration, the priceof Y at time n with a unique arrow i n : n → T is given by Y n := E C i n ( b − T Y ) .n i n (cid:15) (cid:15) ✤ C / / ¯ C n C i n (cid:15) (cid:15) ✤ E / / E ( ¯ C n ) := L ( ¯ C n ) ∋ Y n = E C i n ( b − T Y ) T ✤ C / / ¯ C T ✤ E / / E ( ¯ C T ) := E ( C i n ) O O L ( ¯ C T ) ∋ b − T Y ❴ E ( C i n ) O O For a ∈ B n − , you can see in Figure 3.4 that at time n − Y n ( a
1) is discardedand use only the value of Y n ( a
0) for computing Y n − ( a ).3.3.3. Replication Strategies.
Let us investigate the situation where a given strategy ( φ, ψ )becomes a replication strategy of the payoff Y at time T . Definition 3.25. [Self-Financial Strategies] A self-financial strategy is a strategy ( φ, ψ )satisfying(3.28) S n φ n +1 + b n ψ n +1 = V n for every n = 1 , , . . . . Y n +1 ( a Y n ( a ❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳ Y n +1 ( a Y n − ( a ) := Y n ( a ❥❥❥❥❥❥❥❥❥❥❥❥❥ ❚❚❚❚❚❚❚❚❚❚❚❚❚ Y n +1 ( a Y n ( a ❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳ Y n +1 ( a B n − B nf fulln − o o B n +1 f dropn o o Figure 3.4.
Valuation along f dropn For a self-financial strategy ( φ n , ψ n ) n =1 , ,... , we have: V n +1 = S n +1 ( φ n +1 ◦ f n ) + b n +1 ( ψ n +1 ◦ f n )= ( S n ◦ f n )(1 + µ + σξ n +1 )( φ n +1 ◦ f n ) + b n +1 ( b − n ( V n − S n φ n +1 ) ◦ f n )= (1 + µ + σξ n +1 )(( S n φ n +1 ) ◦ f n ) + (1 + r )(( V n − S n φ n +1 ) ◦ f n )= ( µ − r + σξ n +1 )(( S n φ n +1 ) ◦ f n ) + (1 + r )( V n ◦ f n ) . Therefore, for a ∈ B n and d n +1 ∈ { , } ,(3.29) V n +1 ( ad n +1 ) = ( µ − r + σ (2 d n +1 − S n ( b n ) φ n +1 ( b n ) + (1 + r ) V n ( b n )where(3.30) b n := f n ( ad n +1 ) . Now let us assume that there exists a function g n : B n → B n such that f n = g n ◦ f fulln . B n B n +1 f n qqqqqqqqqq f fulln & & ▼▼▼▼▼▼▼▼▼▼ B ng n O O Then f n ( ad n +1 ) = g n ( a ) for every a ∈ B n and d n +1 ∈ { , } . So the equation (3.29) becomes(3.31) V n +1 ( ad n +1 ) = ( µ − r + σ (2 d n +1 − S n ( g n ( a )) φ n +1 ( g n ( a )) + (1 + r ) V n ( g n ( a )) . Hence, we have: φ n +1 ( g n ( a )) = V n +1 ( a − V n +1 ( a σS n ( g n ( a ))(3.32) V n ( g n ( a )) = ( σ − µ + r ) V n +1 ( a
1) + ( σ + µ − r ) V n +1 ( a σ (1 + r ) . (3.33)Therefore, we can determine the appropriate strategy ( φ n +1 , ψ n +1 ) on g n ( B n ) ⊂ B n by (3.32).We actually do not care the values of ( φ n +1 , ψ n +1 ) on B n \ g n ( B n ).For example, in the case of f n = f dropn , the function g n : B n → B n satisfies(3.34) g n ( d . . . d n − d n ) = d . . . d n − BINOMIAL ASSET PRICING MODEL IN A CATEGORICAL SETTING 15 for all d . . . d n − d n ∈ B n . Looking at Figure 3.4, values in the region B n \ g n ( B n ) are notnecessary for computing Y n − ( a ). Hence, determining the values of ( φ n +1 , ψ n +1 ) in g n ( B n ) isenough for making the practical valuation.4. Concluding Remarks
We formulated an infinitely growing sequence of binomial probability spaces in the category
Prob . We gave some concrete (possibly distorted) filtrations. We determined the shape ofthe risk-neutral filtrations to the above examples. We showed the valuations of claims given attime T through the distorted filtrations, and provided a replication strategy implementing thevaluation. References [Adachi and Ryu, 2019] Adachi, T. and Ryu, Y. (2019). A category of probability spaces.
J. Math. Sci. Univ.Tokyo , 26(2):201–221.[Capi´nski and Kopp, 2012] Capi´nski, M. and Kopp, E. (2012).
Discrete Models of Financial Markets . CambridgeUniversity Press.[MacLane, 1997] MacLane, S. (1997).
Categories for the Working Mathematician . Number 5 in Graduate Textsin Mathematics. Springer-Verlag, New York, 2nd edition.[Shreve, 2005] Shreve, S. E. (2005).
Stochastic Calculus for Finance I; The Binomial Asset Pricing Model .Springer-Verlag, New York.
Graduate School of Management, Tokyo Metropolitan University, 1-4-1 Marunouchi, Chiyoda-ku, Tokyo 100-0005, Japan
E-mail address : Takanori Adachi
College of International Management, Ritsumeikan Asia Pacific University, 1-1 Jumon-jibaru, Beppu, Oita, 874-8577 Japan
E-mail address : Katsushi Nakajima
Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu,Shiga, 525-8577 Japan
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