Generalized Filtrations and Its Application to Binomial Asset Pricing Models
aa r X i v : . [ q -f i n . M F ] N ov GENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIALASSET PRICING MODELS
TAKANORI ADACHI, KATSUSHI NAKAJIMA AND YOSHIHIRO RYU
Abstract.
We introduce generalized filtration with which we can represent situations suchas some agents forget information at some specific time. The filtration is defined as a functorto a category
Prob whose objects are all probability spaces and whose arrows correspond tomeasurable functions satisfying an absolutely continuous requirement [Adachi and Ryu, 2019].As an application of a generalized filtration, we develop a binomial asset pricing model, andinvestigate the valuations of financial claims along this type of non-standard filtrations. Introduction
It is well known that in stochastic process theory and theories developed on it such as sto-chastic differential equation theory and stochastic control theory, the concept of filtration thatexpresses increasing information along time is important. The idea that the world’s informationgrows over time seems to be quite natural, but in a sense it is a divine perspective of omniscienceand almighty, and it would be a little different if we say that the amount of information that anindividual has always increases with time. People forget and misunderstand. The transition ofsuch individuals’ information may therefore be reduced, and may be remembered as a differentform of experience than objective information. The purpose of the first half of this paper is topropose a kind of subjective filtration that expresses the transition of such information.In this way, we generalize the concept of filtration so that we can handle subjective situations,but the purpose of generalized filtration is not limited to that. For example, consider a situationin which Black Swan, who no one had imagined up to a certain point in time, was falling. Thefinancial crisis that hit the world in 2008 and the COVID-19 pandemic in 2020 are typicalexamples. When Black Swan suddenly appeared, which was not included among the possiblefuture world lines, we could not give a probability for that event and we were greatly upset.Of course, God could have a sufficiently large set of primitive events to take into account suchpossibilities, it would have been possible to give a probability to an event that ordinary peopledid not expect. But can such an idealized perspective really create a theory that averts therisk of Black Swan?The generalized filtration formulated in this paper allows even the underlying set of proba-bility space, which is the set of primitive events, to change over time. And it allows the suddenappearance of Black Swan to be incorporated into the theory in a natural way.In the second half of this paper, we consider two types of filtration on the binomial assetprice model as an application of generalized filtration. In particular, we show that there is arisk-neutral filtration associated with subjective filtration that a person who has lost memoryfor a certain period of time, and use it to price securities. This indicates that people with alack of memory can price securities.Finally, in summary, other applications of generalized filtration and future development di-rections are described.
Date : November 18, 2020.2010
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. Generalized Filtrations
In this section, we define generalized filtration by gradually extending the classical filtration.2.1.
Time Domains.
A filtration represents a set of information that increases with time.The set of times here is called a time domain and is represented by T . Typical T has thefollowing forms.(1) T := { , , , . . . , T } , (2) T := { , , , . . . } , (3) T := [0 , T ] , (4) T := [0 , + ∞ ) , where T is a time horizon. In general time domain may be a totally ordered set having theminimum element 0.2.2. Classical Filtrations.
Let ¯Ω := (Ω , F , P ) be a probability space. Let { t n } be an increas-ing sequence in a time domain T . Then, an increasing sequence of σ -fields F t ⊂ F t ⊂ · · · ⊂ F t n ⊂ F t n +1 ⊂ · · · with F t n ⊂ F is called a classical filtration . In other words, a filtration is a family ofset-inclusion relations like {F s ⊂ F t } s ≤ t . Now let ¯Ω t := (Ω , F t , P )be probability spaces whose σ -fields are changing per time t . Then, for s ≤ t in T , the conditionthat the function below defined as an identity function i s,t is measurable is equivalent to thecondition F s ⊂ F t ¯Ω s ∈ ¯Ω ti s,t o o ∈ ω ω ✤ o o . In other words, the filtration can be identified with a family of measurable functions { ¯Ω s ¯Ω t } s ≤ ti s,t o o . Therefore, in the following, instead of using the σ -field F t , filtration will be considered as afamily of measurable functions.2.3. Generalization of Filtrations.
As we see in the previous section, a filtration can beseen as a family of identity functions i s,t as measurable functions. Now what if we generalizethem to arbitrary measurable functions like the following? { ¯Ω s ¯Ω t } s ≤ tf s,t o o satisfying f t,t = Id ¯Ω t and f s,t ◦ f t,u = f s,u for any s ≤ t ≤ u in T , where Id ¯Ω t is an identity function on ¯Ω t . However, this definition istoo general for a random variable X : ¯Ω t → R to define its conditional expectation E f s,t ( X ) :¯Ω s → R satisfying Z A E f s,t ( X ) d P = Z f − s,t ( A ) Xd P ( ∀ A ∈ F s ) . ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 3
In order to make it possible, we need to add an extra condition to the measurable function f s,t called null-preserving , that is, for any A ∈ F s , P ( A ) = 0 implies P ( f − s,t ( A )) = 0[Adachi, 2014]. In fact, if f s,t is null-preserving, as we will see later, we can define a conditionalexpectation E f s,t ( X ) : ¯Ω s → R . Note that when the identity function is generalized to a null-preserving function, the corresponding sequence of the σ -fields is not necessarily monotonicallyincreasing.In order to give a further generalization, we consider that the probability space at each timefluctuates not only with the σ -fields but also with probability measures and underlying sets. Inother words, the probability space ¯Ω t at time t is redefined as follows.¯Ω t := (Ω t , F t , P t ) . Along with this, the definition of null-preserving functions is extended as follows.
Definition 2.1.
Let ¯Ω = (Ω , F , P ) and ¯Ω ′ = (Ω ′ , F ′ , P ′ ) be two probability spaces and f : ¯Ω → ¯Ω ′ be a measurable functions between them. Then f is called null-preserving if P ◦ f − ≪ P ′ (absolutely continuous) . Definition 2.2. A generalized filtration is a family of null-preserving functions { ¯Ω s ¯Ω t } s ≤ tf s,t o o satisfying f t,t = Id ¯Ω t and f s,t ◦ f t,u = f s,u for all triples s ≤ t ≤ u in T .Then, we obtain a following theorem. Theorem 2.3. ([Adachi and Ryu, 2019])
For any random variable X on ¯Ω t and any null-preserving function f : ¯Ω t → ¯Ω s , there exists a random variable Y on ¯Ω s such that for every A ∈ F s , (2.1) Z A Y d P s = Z f − ( A ) Xd P t . We write E f ( X ) for the random variable Y , and call it a conditional expectation of X along f .Proof. Define a measure X ∗ on (Ω t , F t ) as in the following diagram. D ✤ / / ∈ X ∗ ( D ) := ∈ R D X d P t F s f − / / P s : : F t X ∗ / / P t / / R Then, since X ∗ ≪ P t and f is null-preserving, we have X ∗ ◦ f − ≪ P t ◦ f − ≪ P s . Therefore, we get a following Radon-Nikodym derivative. Y := ∂ ( X ∗ ◦ f − ) /∂ P s . With this Y we obtain for every A ∈ F s , Z A Y d P s = Z A d ( X ∗ ◦ f − ) = ( X ∗ ◦ f − )( A ) = X ∗ ( f − ( A )) = Z f − ( A ) X d P t . T. ADACHI, K. NAKAJIMA AND Y. RYU T F / / Prob s Id s % % F ( s ) F ( Id s )= Id F ( s ) := ¯Ω s t ι Ns,t O O Id t $ $ F ( t ) f s,t =: F ( ι Ns,t ) O O F ( Id t )= Id F ( t ) u ι Nt,u O O Id u & & ι Ns,t ◦ ι Nt,u ] ] F ( u ) f t,u =: F ( ι Nt,u ) O O F ( Id u )= Id F ( u ) F ( ι Ns,t ◦ ι Nt,u )= F ( ι Ns,t ) ◦ F ( ι Nt,u )= f s,u \ \ := ¯Ω u Figure 2.1.
Filtration F : T →
Prob (cid:3)
Henceforth, generalized filtration will be referred to simply as filtration.2.4.
Filtration is a Functor.
In this subsection, we will try to redefine the filtration intro-duced in Section 2.3 using Category Theory [MacLane, 1997].
Definition 2.4. [Two Categories
Prob and T ](1) All probability spaces and all null-preserving functions between them form a category .This category is denoted by Prob .(2) A time domain T can be regarded as a category if we consider its elements as objects ,and if two objects s and t have one and only one arrow from t to s when there is arelation s ≤ t .Then, the filtration introduced in Section 2.3 can be regarded as a functor F : T →
Prob (Figure2.1). We sometimes call F a T -filtration in order for clarifying its time domain.Classical Filtration T t ≤ t ≤ t ≤ · · · F F t ⊂ F t ⊂ F t ⊂ · · · Ω Ω Id Ω o o Ω Id Ω o o · · · Id Ω o o Generalized Filtration T F (cid:15) (cid:15) t t o o t o o · · · o o Prob ¯Ω t ¯Ω t f t ,t o o ¯Ω t f t ,t o o · · · f t ,t o o Figure 2.2.
Classical and Generalized Filtrations For further discussion about the category
Prob , see [Adachi and Ryu, 2019].
ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 5 Filtrations over a Binomial Asset Pricing Model
In this section, as a concrete example of the filtration introduced in Section 2, we look at anunusual filtration on a binomial asset pricing model.3.1.
Filtration B N . First, we define a general scheme of our model by introducing a filtration B N for an integer N . Definition 3.1 (Time Domain and Probability Space) . Let N ∈ N and s, t ∈ R be non-negativereal numbers.(1) Discrete intervals .[ s, t ] N := { n − N | n ∈ Z and s ≤ n − N ≤ t } , [ s, t ) N := { n − N | n ∈ Z and s ≤ n − N < t } , ( s, t ] N := { n − N | n ∈ Z and s < n − N ≤ t } , ( s, t ) N := { n − N | n ∈ Z and s < n − N < t } . (2) Let T N be a category whose objects are elements of [0 , ∞ ) N . For s, t ∈ [0 , ∞ ) N , T N has the unique arrow ι Ns,t from t to s if and only if t ≥ s .(3) B Nt := { , } (0 ,t ] N . (function space)(4) F Nt := 2 B Nt . (powerset)(5) Let p Ns ∈ [0 ,
1] for each s ∈ (0 , ∞ ) N . Then, a probability measure P Nt : F Nt → [0 ,
1] isdefined for every ω ∈ B Nt by P Nt ( { ω } ) := Y s ∈ (0 ,t ] N ( p Ns ) ω ( s ) (1 − p Ns ) − ω ( s ) . (6) ¯ B Nt := ( B Nt , F Nt , P Nt ) (probability space) Definition 3.2.
A filtration B N is determined by defining arrows f Ns,t below: T N B N / / Prob s ¯ B Ns t ι Ns,t O O ¯ B Nt B N ( ι Ns,t ):= f Ns,t O O The filtration B N is called non-trivial if there exists t ∈ (0 , ∞ ) N such that 0 < p t < B N , every function from B Nt to B Ns becomes a null-preserving function from ¯ B Nt to ¯ B Ns .As we introduced, the functor B N is a generalized filtration, representing a filtration over theclassical binomial model developed, for example 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 traditional Kolmogorov world. Proposition 3.3.
For a random variable X on ¯ B Nt and ω ∈ ¯ B Ns , we have E f Ns,t ( X )( ω ) P Ns ( { ω } ) = X ω ′ ∈ ( f Ns,t ) − ( ω ) X ( ω ′ ) P Nt ( { ω ′ } ) . T. ADACHI, K. NAKAJIMA AND Y. RYU
Proof.
Put A := { ω } and f s,t := f Ns,t in (2.1). Then the result is straightforward. (cid:3)
In order to see a variety of filtrations, we introduce two candidates of f Ns,t introduced inDefinition 3.2.
Definition 3.4 (Two Candidates of f Ns,t ) . Let s, t be objects of T N satisfying s < t .(1) full Ns,t ¯ B Ns ∈ ¯ B Nt ∈ full Ns,t o o ω | (0 ,s ] N ω ✤ o o (2) drop Ns,t ¯ B Ns ∈ ¯ B Nt ∈ drop Ns,t o o full Ns,t ( ω ) × (0 ,s ) N ω ✤ o o The function drop
Ns,t can be interpreted to forget what happens at time s .We can easily show the following proposition. Proposition 3.5.
For s < t < u in [0 , ∞ ] N ,(1) full Ns,t ◦ full Nt,u = full Ns,u ,(2) full
Ns,t ◦ drop Nt,u = full Ns,u ,(3) drop
Ns,t ◦ full Nt,u = drop Ns,u ,(4) drop
Ns,t ◦ drop Nt,u = drop Ns,u . Definition 3.6 (Examples of (Subjective) Filtrations) . Let s, t be any objects of T N such that s < t .(1) Classical filtration: Full N : T N → Prob is defined by
Full N ( ι Ns,t ) := full
Ns,t . (2) Dropped filtration: Drop
Nα,β : T N → Prob where α, β ∈ R are constants, is defined by Drop
Nα,β ( ι Ns,t ) := ( drop Ns,t if s = t and s ∈ [ α, β ] N , full Ns,t otherwise . A person who has a subjective filtration
Drop
Nα,β forgets the events happened during[ α, β ].Note also that the dropped filtration is well-defined by Proposition 3.5.
Definition 3.7. [ B -Adapted Process ξ Nt ] Let t ∈ [0 , + ∞ ) N . a stochastic process ξ Nt : B Nt → R is defined by ξ Nt ( ω ) := 2 ω ( t ) − ∀ ω ∈ B Nt ) . Definition 3.8.
For j = 0 , ω ∈ B Nt , I Nt ( j, ω ) := { ω ′ ∈ ( f Nt,t +2 − N ) − ( ω ) | ω ′ ( t + 2 − N ) = j } . Proposition 3.9.
For ω ∈ B Nt with P Nt ( ω ) = 0 , E f Nt,t +2 − N ( ξ Nt +2 − N )( ω ) = f Nt,t +2 − N ) − ( ω )) p Nt +2 − N − I Nt (0 , ω ) , where A denotes the cardinality of the set A . ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 7
Proof.
By Proposition 3.3, E f Nt,t +2 − N ( ξ Nt +2 − N )( ω ) = X ω ′ ∈ ( f Nt,t +2 − N ) − ( ω ) ξ Nt +2 − N ( ω ′ ) P Nt +2 − N ( ω ′ ) P Nt ( ω )= X ω ′ ∈ ( f Nt,t +2 − N ) − ( ω ) (2 ω ′ ( t + 2 − N ) − P Nt +2 − N ( ω ′ ) P Nt ( ω )= X ω ′ ∈ I Nt (1 ,ω ) P Nt +2 − N ( ω ′ ) P Nt ( ω ) − X ω ′ ∈ I Nt (0 ,ω ) P Nt +2 − N ( ω ′ ) P Nt ( ω )= X ω ′ ∈ I Nt (1 ,ω ) p Nt +2 − N − X ω ′ ∈ I Nt (0 ,ω ) (1 − p Nt +2 − N )= f Nt,t +2 − N ) − ( ω )) p Nt +2 − N − I Nt (0 , ω ) . (cid:3) Arbitrage Strategies.
Now we define two instruments tradable in our market.
Definition 3.10. [Stock and Bond Processes] Let µ, σ, r ∈ R be constants such that σ > µ > σ − r > −
1. We have the following B N -adapted processes which are two instrumentstradable in our market. Let t ∈ [0 , + ∞ ) N .(1) A stock process S Nt : B Nt → R over the filtration B N is defined by S N ( ∗ ) := s , S Nt +2 − N := ( S Nt ◦ f Nt,t +2 − N )(1 + 2 − N µ + 2 − N σξ Nt +2 − N )where ∗ ∈ B N is the unique element.(2) A bond process b Nt : B Nt → R over the filtration B N is defined by b N ( ∗ ) := 1 , b Nt +2 − N := ( b Nt ◦ f Nt,t +2 − N )(1 + 2 − N r ) . The condition µ > σ − B N , S N , b N ) a market . But, it does not mean that the marketwill not contain other instruments.The following proposition is straightforward. Proposition 3.11.
Let B Nt be a random variable on B Nt defined by B Nt ( ω ) = 1 for every ω ∈ B Nt . Then, we have for any ω ∈ B Nt ,(1) E f Nt,t +2 − N ( S Nt +2 − N ) = S Nt (cid:0) (1 + 2 − N µ ) E f Nt,t +2 − N (1 B Nt +2 − N ) + 2 − N σE f Nt,t +2 − N ξ Nt +2 − N ) (cid:1) , (2) E f Nt,t +2 − N (1 B Nt +2 − N )( ω ) = { P Nt +2 − N (( f Nt,t +2 − N ) − ( ω )) } / { P Nt ( ω ) } , (3) b Nt ( ω ) = (1 + 2 − N r ) N t . Definition 3.12. [Strategies] A strategy is a sequence ( φ, ψ ) = { ( φ t , ψ t ) } t ∈ (0 , ∞ ) N , where(3.1) φ t : B Nt − − N → R and ψ t : B Nt − − N → R . Each element of the strategy ( φ t , ψ t ) is called a portfolio . For t ∈ [0 , ∞ ) N , the value V t ofthe portfolio at time t is determined by:(3.2) V t := ( S N φ − N + b N ψ − N if t = 0 ,S Nt ( φ t ◦ f Nt − − N ,t ) + b n ( ψ t ◦ f Nt − − N ,t ) if t > . T. ADACHI, K. NAKAJIMA AND Y. RYU
Definition 3.13. [Gain Processes] A gain process of the strategy ( φ, ψ ) is the process { G ( φ,ψ ) t } t ∈ [0 , ∞ ) N defined by(3.3) G ( φ,ψ ) t := ( − ( S N φ − N + b N ψ − N ) if t = 0 , ( S Nt ( φ t ◦ f Nt − − N ,t ) + b Nt ( ψ t ◦ f Nt − − N ,t )) − ( S Nt φ t +2 − N + b Nt ψ t +2 − N ) if t > . Lemma 3.14.
Let t ∈ [0 , ∞ ) N with (3.4) S Nt φ t +2 − N + b Nt ψ n +2 − N = 0 . Then, we have (3.5) S Nt +2 − N ( φ t +2 − N ◦ f Nt,t +2 − N )+ b n +1 ( ψ n +2 − N ◦ f Nt,t +2 − N ) = (2 − N µ +2 − N σξ Nt +2 − N − − N r )(( S Nt φ t +2 − N ) ◦ f Nt,t +2 − N ) . Proof.
LHS =( S Nt ◦ f Nt,t +2 − N )(1 + 2 − N µ + 2 − N σξ Nt +2 − N )( φ t +2 − N ◦ f Nt,t +2 − N )+ ( b Nt ◦ f Nt,t +2 − N )(1 + 2 − N r )( ψ t +2 − N ◦ f Nt,t +2 − N )=(1 + 2 − N µ + 2 − N σξ Nt +2 − N )(( S Nt φ t +2 − N ) ◦ f Nt,t +2 − N ) + (1 + 2 − N r )(( b Nt ψ t +2 − N ) ◦ f Nt,t +2 − N )=(1 + 2 − N µ + 2 − N σξ Nt +2 − N )(( S Nt φ t +2 − N ) ◦ f Nt,t +2 − N ) − (1 + 2 − N r )(( S Nt φ t +2 − N ) ◦ f Nt,t +2 − N )= RHS. (cid:3)
Definition 3.15. [Arbitrage Strategies](1) A strategy ( φ, ψ ) is called a B N - arbitrage strategy if P Nt (cid:0) G ( φ,ψ ) t ≥ (cid:1) = 1 for every t ∈ [0 , ∞ ) N , and P Nt (cid:0) G ( φ,ψ ) t > (cid:1) > t ∈ [0 , ∞ ) N .(2) The market is called non-arbitrage or NA if it does not allow B N -arbitrage strategies. Proposition 3.16.
If the market ( B N , S N , b N ) with a non-trivial filtration B N is non-arbitrage,then | µ − r | < N σ. Proof.
Assuming that r ≤ µ − N σ or r ≥ µ − N σ , we will construct an arbitrage strategy( φ, ψ ) by using the following algorithm. for t = 0 , 1 , 2 , ...:t := 2^( - N ) no b s e r v e S ( t ) and b ( t )if r <= mu - 2^ N sigma :phi ( t +2^( - N )) > 0 In the above code, ‘*’ is the function composition operator.By Lemma 3.14, we have(3.6) G ( φ,ψ ) t = 2 − N ( µ + 2 N σξ Nt − r )(( S Nt − − N φ t ) ◦ f Nt − − N ,t ) . ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 9
So we have G ( φ,ψ ) t ≥ r ≤ µ − N σ or r ≥ µ + 2 N σ .By the way, since our filtration is non-trivial, there exists a number t such that 0 < p t < P Nt ( G ( φ,ψ ) t > > , which concludes that ( φ, ψ ) is an arbitrage strategy. (cid:3) Risk-Neutral Filtrations.
In this subsection, we assume that | µ − r | < N σ .Let us consider about the following discounted stock process Definition 3.17. A discount stock process ( S Nt ) ′ : B Nt → R is defined by( S Nt ) ′ := ( b Nt ) − S Nt . Definition 3.18. A risk-neutral filtration with respect to the filtration B N is a filtration C N such that U ◦ C N = U ◦ B N , where U : Prob → Meas is the forgetful functor to the category of measurable spaces, T N C N / / B N / / Prob U / / Meas and with which ( S Nt ) ′ becomes a C N -martingale, that is, E C N ( ι Ns,t ) (( S Nt ) ′ ) = ( S Ns ) ′ . In the remainder of this subsection, we will focus on proving the following theorem.
Theorem 3.19.
There exists a risk-neutral filtration with respect to the filtration
Drop
Nα,β . First, we examine what form the probability measure Q Nt : F Nt → [0 ,
1] takes when C N ( t ) =( B Nt , F Nt , Q Nt ) for a risk-neutral filtration C N , in general. Theorem 3.20.
A stochastic process ( S Nt ) ′ is a C N -martingale if and only if the followingequation holds for every t ∈ [0 , ∞ ) N and ω ∈ B Nt . Q Nt ( { ω } ) = c Q Nt +2 − N ( I Nt (1 , ω )) + c Q Nt +2 − N ( I Nt (0 , ω )) where c := 1 + 2 − N µ + 2 − N σ − N r , c := 1 + 2 − N µ − − N σ − N r . Proof.
Let ω ∈ B Nt . Then, by Proposition 3.3, we have E C N ( ι Nt,t +2 − N ) (( S Nt +2 − N ) ′ )( ω ) Q Nt ( { ω } )= X ω ′ ∈ ( f Nt,t +2 − N ) − ( ω ) ( S Nt +2 − N ) ′ ( ω ′ ) Q Nt +2 − N ( { ω ′ } )= X ω ′ ∈ ( f Nt,t +2 − N ) − ( ω ) ( b Nt +2 − N ) − ( ω ′ )( S Nt ◦ f Nt,t +2 − N ( ω ′ )(1 + 2 − N µ + 2 − N σξ Nt +2 − N ( ω ′ )) Q Nt +2 − N ( { ω ′ } )= X ω ′ ∈ ( f Nt,t +2 − N ) − ( ω ) (1 + 2 − N r ) − ( t +2 − N ) S Nt ( ω )(1 + 2 − N µ + 2 − N σξ Nt +2 − N ( ω ′ )) Q Nt +2 − N ( { ω ′ } )=( S Nt ) ′ ( ω ) X ω ′ ∈ ( f Nt,t +2 − N ) − ( ω ) − N µ + 2 − N σξ Nt +2 − N ( ω ′ )1 + 2 − N r Q Nt +2 − N ( { ω ′ } ) . Therefore, the condition ( S Nt ) ′ = E C N ( ι Nt,t +2 − N ) (( S Nt +2 − N ) ′ ) is equivalent to Q Nt ( { ω } ) = X ω ′ ∈ I Nt (1 ,ω ) − N µ + 2 − N σ − N r Q Nt +2 − N ( { ω ′ } ) + X ω ′ ∈ I Nt (0 ,ω ) − N µ − − N σ − N r Q Nt +2 − N ( { ω ′ } )= c Q Nt +2 − N ( I Nt (1 , ω )) + c Q Nt +2 − N ( I Nt (0 , ω )) . (cid:3) Definition 3.21.
For ω ∈ B Nt and d ∈ { , } , ( ωd ) ∈ B Nt +2 − N is an element satisfying( ωd )( s ) := ( ω ( s ) ( s ≤ t ) d ( s = t + 2 − N )for any s ∈ (0 , t + 2 − N ] N .Unless there is confusion, we will omit the parentheses in (( ωd ) d ) and write ωd d .In order to determine more detail of C , we need the following condition for Q Nt . Proposition 3.22.
The following conditions for Q Nt are equivalent.(1) For all t ∈ [0 , ∞ ) N and ω ∈ B Nt , (3.8) Q Nt +2 − N ( { ω , ω } ) = Q Nt ( { ω } ) . (2) For all t ∈ [0 , ∞ ) N , full Nt,t +2 − N is measure-preserving w.r.t. Q Nt , that is, (3.9) Q Nt = Q Nt +2 − N ◦ ( full Nt,t +2 − N ) − . (3) There exists a sequence of functions { q t : B Nt → [0 , } t ∈ (0 , ∞ ) N satisfying the followingconditions for every t ∈ (0 , ∞ ) N and ω ∈ B Nt ,(a) Q Nt ( { ω } ) = Q s ∈ (0 ,t ] N q s ( ω | (0 ,s ] N ) , (b) q t +2 − N ( ω
0) + q t +2 − N ( ω
1) = 1 . In the following discussion, we assume the following assumption which is the condition (3)of Proposition 3.22.
Assumption 3.23.
Suppose that there exists a sequence of functions { q t : B Nt → [0 , } t ∈ (0 , ∞ ) N satisfying the following conditions for every t ∈ (0 , ∞ ) N and ω ∈ B Nt ,(1) Q Nt ( { ω } ) = Q s ∈ (0 ,t ] N q s ( ω | (0 ,s ] N ) , (2) q t +2 − N ( ω
0) + q t +2 − N ( ω
1) = 1 . In the rest of this section, we assume Assumption 3.23, and then will determine the risk-neutral filtration C N by calculating { q t } t ∈ (0 , ∞ ) N . Lemma 3.24.
Let c and c are constants defined in Theorem 3.20. Then for any x ∈ R wehave c x + c (1 − x ) ⇐⇒ x = 12 − N − µ − rσ and − x = 12 + 2 N − µ − rσ . Proposition 3.25.
For t ∈ (0 , ∞ ) N if f Nt,t +2 − N = full Nt,t +2 − N , then for ω ∈ B Nt such that Q Nt ( { ω } ) = 0 , the following holds. q t +2 − N ( ω
1) = 12 − N − µ − rσ , q t +2 − N ( ω
0) = 12 + 2 N − µ − rσ . ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 11 ω ω ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ω B Nt ∈ B Nt +2 − N full Nt,t +2 − N o o ∈ ω ωd t +2 − N ✤ full Nt,t +2 − N o o Figure 3.1. full Nt +2 − N ,t Proof.
By observing Figure 3.1, we have( full
Nt,t +2 − N ) − ( ω ) = { ω , ω } , I Nt (1 , ω ) = { ω } , I Nt (0 , ω ) = { ω } . Then, by Theorem 3.20, Q Nt ( { ω } ) = c Q Nt +2 − N ( I Nt (1 , ω )) + c Q Nt +2 − N ( I Nt (0 , ω )) = c Q Nt +2 − N ( { ω } ) + c Q Nt +2 − N ( { ω } ) . Since Q Nt +2 − N ( { ωd t +2 − N } ) = Q Nt ( { ω } ) q t +2 − N ( ωd t +2 − N )by Assumption 3.23 and Q Nt ( { ω } ) = 0, we have1 = c q t +2 − N ( ω
1) + c q t +2 − N ( ω . Hence, by Lemma 3.24, we obtain q t +2 − N ( ω
1) = 12 − N − µ − rσ , q t +2 − N ( ω
0) = 12 + 2 N − µ − rσ . (cid:3) Note that the probability obtained in Proposition 3.25 does not depend on either ω or t . Proposition 3.26.
For t ∈ (0 , ∞ ) N , if f Nt = drop Nt,t +2 − N , then for ω ∈ B Nt − − N such that Q Nt − − N ( { ω } ) = 0 , the following holds. q t ( ω
1) = 0 ,q t ( ω
0) = 1 ,q t +2 − N ( ω
01) = 12 − N − µ − rσ ,q t +2 − N ( ω
00) = 12 + 2 N − µ − rσ . Proof.
By observing Figure 3.2, we have( drop
Nt,t +2 − N ) − ( ω
1) = ∅ ,I Nt (1 , ω
1) = I Nt (0 , ω
1) = ∅ , ( drop Nt,t +2 − N ) − ( ω
0) = { ω , ω , ω , ω } ,I Nt (1 , ω
0) = { ω , ω } ,I Nt (0 , ω
0) = { ω , ω } . ω ω ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ω ω ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ω ω ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ ω B Nt − − N ∈ B Nt full Nt − − N ,t o o ∈ B Nt +2 − N drop Nt,t +2 − N o o ∈ ω ω ✤ full Nt − − N ,t o o ωd t d t +2 − N ✤ drop Nt,t +2 − N o o Figure 3.2. drop
Nt,t +2 − N followed by full Nt +2 − N ,t Then, by Theorem 3.20, Q Nt ( { ω } ) = c Q Nt +2 − N ( I Nt (1 , ω c Q Nt +2 − N ( I Nt (0 , ω . Now, since Q Nt ( { ωd t } ) = Q Nt − − N ( { ω } ) q t ( ωd t ) by Assumption 3.23, and Q Nt − − N ( { ω } ) = 0, wehave q t ( ω
1) = 0 , q t ( ω
0) = 1 − q t ( ω
1) = 1 . Next, again by Theorem 3.20, Q Nt ( { ω } ) = c Q Nt +2 − N ( I Nt (1 , ω c Q Nt +2 − N ( I Nt (0 , ω c (cid:0) Q Nt +2 − N ( { ω } ) + Q Nt +2 − N ( { ω } ) (cid:1) + c (cid:0) Q Nt +2 − N ( { ω } ) + Q Nt +2 − N ( { ω } ) (cid:1) . By dividing both sides by Q Nt − − N ( { ω } ) = 0, we obtain q t ( ω
0) = c (cid:0) q t ( ω q t +2 − N ( ω
01) + q t ( ω q t +2 − N ( ω (cid:1) + c (cid:0) q t ( ω q t +2 − N ( ω
00) + q t ( ω q t +2 − N ( ω (cid:1) . Hence, since q t ( ω
1) = 0 and q t ( ω
0) = 1, we get1 = c q t +2 − N ( ω
01) + c q t +2 − N ( ω . Therefore, by Lemma 3.24, q t +2 − N ( ω
01) = 12 − N − µ − rσ , q t +2 − N ( ω
00) = 12 + 2 N − µ − rσ . (cid:3) Remark 3.27.
We have the following remarks for Figure 3.2,(1) Since the agent evaluates stock and bond along the function drop
Nt,t +2 − N , she can rec-ognize only the nodes ω ω
01 and ω
00 and can not recognise the nodes ω ω
11 and ω
10. We interpret these nodes ω ω
11 and ω
10 as invisible.(2) The values q t +2 − N ( ω ∈ [0 ,
1] can be arbitrarily selected, and q t +2 − N ( ω
10) is computedby 1 − q t +2 − N ( ω Q Nt +2 − N is not determined uniquely,so is not the risk-neutral filtration C N . ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 13 (3) The probability measure Q Nt is not equivalent to the original measure P Nt . Therefore, itis not an EMM. Proposition 3.28.
Both full
Nt,t +2 − N and drop Nt,t +2 − N are null-preserving with respect to Q Nt and Q Nt +2 − N .Proof. Let ω ∈ B Nt . Then, by Assumption 3.23,( Q Nt +2 − N ◦ ( full Nt,t +2 − N ) − )( ω ) = Q Nt +2 − N ( { ω , ω } ) = Q Nt ( ω ) . Hence, full
Nt,t +2 − N is null-preserving.Next, consider the case when drop Nt,t +2 − N . Then for ω ′ ∈ B Nt − − N , by Proposition 3.26, wehave Q Nt ( ω ′
1) = 0. On the other hand, we get( Q Nt +2 − N ◦ ( drop Nt,t +2 − N ) − )( ω ′
1) = Q Nt +2 − N ( ∅ ) = 0 . Therefore, drop
Nt,t +2 − N is also null-preserving. (cid:3) Theorem 3.29.
There exists a risk-neutral filtration C N for the dropped filtration Drop α,β .In this case, the probability measure Q Nt of the probability space C N ( t ) is not equivalent to theprobability measure P t of Drop α,β ( t ) . Therefore, it is not an EMM. In fact, the probabilitymeasure Q Nt is not uniquely determined. Similarly, the risk-neutral filtration C N is not uniquelydetermined.Proof. Substituting the q t obtained by Propositions 3.25 and 3.26 into Assumption 3.23, weobtain the probability measure Q Nt . On the other hand, from Proposition 3.28, the arrows full Nt,t +2 − N and drop Nt,t +2 − N are null-preserved under Q Nt . Therefore, we can say that C N is afiltration. Moreover, Q Nt clearly satisfies the necessary and sufficient conditions of Theorem3.20 from the way it is constructed. Therefore, the filtration C N is a risk-neutral filtrationwith respect to Drop α,β . By the way, in Proposition 3.26, q t +2 − N ( ω ∈ [0 ,
1] can take anyvalue. Then q t +2 − N ( ω
10) can be computed by 1 − q t +2 − N ( ω Q Nt +2 − N is not uniquely determinable. (cid:3) Valuation.
Let C N : T N → Prob be a risk-neutral filtration and Y : B NT → R be a payoffat time T . Then, the price Y t of Y at time t is given by the equation Y t := E C N ( ι Nt,T ) (( b NT ) − Y )with the unique arrow ι Nt,T : T → t .That is to say, even those who have a dropped subjective filtration can price Securities Y .However, additional consideration is needed on how these prices affect the market equilibriumprice.For ω ∈ B Nt − · − N , you can see in Figure 3.4 that at time t − − N the value of Y t ( ω
1) isdiscarded and use only the value of Y t ( ω
0) for computing Y t − − N ( ω ).3.4.1. 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 3.30. [Self-Financial Strategies] A self-financial strategy is a strategy ( φ, ψ )satisfying(3.10) S Nt φ t +2 − N + b Nt ψ t +2 − N = V t for every t ∈ (0 , ∞ ) N . ω ω ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ω ω ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ω ω ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ω ω qqqqqqqqqqqqqqqqqqqqqqqq ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ω ω ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ω ω ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ω ω ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ω B Nt − · − N B Nt − − N full t − · − N ,t − − N o o B Nt full t − − N ,t o o B Nt +2 − N drop Nt,t +2 − N o o Figure 3.3.
Filtration
Drop t − . ,t +0 . Y t +2 − N ( ω Y t ( ω ❢❢❢❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳❳❳❳ Y t +2 − N ( ω Y t − − N ( ω ) = Y t ( ω ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ Y t +2 − N ( ω Y t ( ω ❢❢❢❢❢❢❢❢❢❢❢ ❳❳❳❳❳❳❳❳❳❳❳ Y t +2 − N ( ω B Nt − − N B Nt full Nt − − N ,t o o B Nt +2 − N drop t,t +2 − N o o Figure 3.4.
Valuation through
Drop t,t
For a self-financial strategy ( φ t , ψ t ) t ∈ (0 , ∞ ) N , we have: V t +2 − N = S Nt +2 − N ( φ t +2 − N ◦ f Nt,t +2 − N ) + b Nt +2 − N ( ψ t +2 − N ◦ f Nt,t +2 − N )=( S Nt ◦ f Nt,t +2 − N )(1 + 2 − N µ + 2 − N σξ Nt +2 − N )( φ t +2 − N ◦ f Nt,t +2 − N )+ b Nt +2 − N (( b Nt ) − ( V t − S Nt φ t +2 − N ) ◦ f Nt,t +2 − N )=(1 + 2 − N µ + 2 − N σξ Nt +2 − N )(( S Nt φ t +2 − N ) ◦ f Nt,t +2 − N ) + (1 + 2 − N r )(( V t − S Nt φ t +2 − N ) ◦ f Nt,t +2 − N )=(2 − N µ − − N r + 2 − N σξ Nt +2 − N )(( S Nt φ t +2 − N ) ◦ f Nt,t +2 − N )+ (1 + 2 − N r )( V Nt ◦ f Nt,t +2 − N ) . ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 15
Therefore, for ω ∈ B Nt and d t +2 − N ∈ { , } ,(3.11) V t +2 − N ( ωd t +2 − N ) = (2 − N µ − − N r + 2 − N σ (2 d t +2 − N − S Nt ( ω t ) φ t + V ( ω t ) + (1 + 2 − N r ) V t ( ω t )where(3.12) ω t := f Nt,t +2 − N ( ωd t +2 − N ) . Now let us assume that there exists a function g t : B Nt → B Nt such that f Nt,t +2 − N = g t ◦ full t,t +2 − N . B Nt B Nt +2 − N f Nt,t +2 − N qqqqqqqqqq full t,t +2 − N & & ▼▼▼▼▼▼▼▼▼▼ B Ntg t O O Then f Nt,t +2 − N ( ωd t +2 − N ) = g t ( ω ) for every ω ∈ B Nt and d t +2 − N ∈ { , } . So the equation (3.11)becomes(3.13) V t +2 − N ( ωd t +2 − N ) = (2 − N µ − − N r +2 − N σ (2 d t +2 − N − S Nt ( g t ( ω )) φ t +2 − N ( g t ( ω ))+(1+2 − N r ) V t ( g t ( ω )) . Hence, we have: φ t +2 − N ( g t ( ω )) = V t +2 − N ( ω − V t +2 − N ( ω − N σS Nt ( g t ( ω ))(3.14) V t ( g t ( ω )) = (2 N σ − µ + r ) V t +2 − N ( ω
1) + (2 N σ − µ + r ) V t +2 − N ( ω N σ (1 + 2 − N r ) . (3.15)Therefore, we can determine the appropriate strategy ( φ t +2 − N , ψ t +2 − N ) on g t ( B Nt ) ⊂ B Nt by(3.14). We actually do not care the values of ( φ t +2 − N , ψ t +2 − N ) on B Nt \ g t ( B Nt ).For example, in the case of f Nt,t +2 − N = full t,t +2 − N , the function g t : B Nt → B Nt satisfies(3.16) g t ( ω ′ d t ) = ω ′ ω ′ ∈ B Nt − − N and d t ∈ { , } . Looking at Figure 3.4, values in the region B Nt \ g t ( B Nt )are not necessary for computing Y t − − N ( ω ). Hence, determining the values of ( φ t +2 − N , ψ t +2 − N )in g t ( B Nt ) is enough for making the practical valuation.3.5. Experienced Paths.
In this subsection, we introduce a concept of experienced pathsthat corresponds to a subjective recognition of a person’s experience.
Definition 3.31.
Let B N : T N → Prob be a filtration and t ∈ [0 , ∞ ] N .(1) Define a function e B N t : B Nt → B Nt by e B N t ( ω )( s ) := f Ns,t ( ω )( s )for ω ∈ B Nt , s ∈ (0 , t ] N and f Ns,t := B N ( ι Ns,t ).We call e B N t ( ω ) an experienced path of ω .(2) ˜ B Nt := { e B N t ( ω ) | ω ∈ B Nt } .(3) ˜ F Nt := 2 ˜ B Nt . ∗ B ∈ B ∈ ∗ B / full O O ∈ d / O O d B / full O O ∈ B / full O O ∈ d d / O O B / full O O ∈ d d / α O O d B / full O O ∈ B / drop O O ∈ d d d / O O B / drop O O ∈ d d d d / β O O d d d B / drop O O ∈ B / drop O O ∈ d d d d d d / O O B / full O O ∈ d d d d d d d / O O d d d d B full O O ∈ B full O O ∈ d d d d d d d d t O O Figure 3.5.
Experienced Paths for B N := Drop N , (4) ˜ P Nt := P Nt ◦ ( e B N t ) − . B Nt e B Nt / / ˜ B Nt [0 , F Nt P Nt o o ˜ F Nt ( e B Nt ) − o o (5) ¯˜ B Nt := ( ˜ B Nt , ˜ F Nt , ˜ P Nt ).(6) For s, t ∈ [0 , ∞ ) N ( s ≤ t ), ˜ f Ns,t : ¯˜ B Nt → ¯˜ B Ns is a function defined by˜ f Ns,t := full Ns,t | ˜ B Nt . Proposition 3.32.
A correspondence ˜ B N : T N → Prob defined by ˜ B N ( t ) := ¯˜ B Nt and ˜ B N ( ι Ns,t ) := ˜ f Ns,t is a functor, that is, a T N -filtration. Example 3.33. [Experienced Paths for B N := Drop N , ] Let B N := Drop N , and d i ∈ { , } for i ∈ N . Then, as seen in Figure 3.5, we have e B ( d d d d ) = d d d ,e B ( d d d d d d d d ) = d d d d d . ENERALIZED FILTRATIONS AND ITS APPLICATION TO BINOMIAL ASSET PRICING MODELS 17
Theorem 3.34.
The correspondence e B N : B N → ˜ B N is a natural transformation. That is, for s, t ∈ [0 , ∞ ) N ( s ≤ t ) , the following diagram commutes: T N B N e B N / / ˜ B N s ¯ B Ns e B Ns / / ¯˜ B Ns t ι Ns,t O O ¯ B Nt e B Nt / / f Ns,t O O ¯˜ B Nt ˜ f Ns,t O O Proof.
For ω ∈ B Nt and u ∈ (0 , s ] N ,˜ f Ns,t ( e B N t ( ω ))( u ) = ( full Ns,t | ˜ B Nt )( e B N t ( ω ))( u )= full Ns,t ( e B N t ( ω ))( u )= ( e B N t ( ω ) | (0 ,s ] N )( u )= e B N t ( ω )( u ) = f Nu,t ( ω )( u ) . On the other hand, e B N s ( f Ns,t ( ω ))( u ) = f Nu,s ( f Ns,t ( ω ))( u ) = f Nu,t ( ω )( u ) . (cid:3) Here is an implication of Theorem 3.34: The person who dropped her memory believes thather memory is perfect (full), while others observe that she lost her memory.Lastly, we mention the fact that in a case the given filtration is full, experienced pathscoincide with objective paths.
Proposition 3.35. If B N = Full N , then ˜ B N = B N . Concluding Remarks
In this paper, we proposed the concept of generalized filtration. It is an extended filtrationthat goes beyond the conventional framework of monotonically increasing information sequencesand allows the development of information to not only increase, but also to decrease or betwisted. It is an extended concept, just like the subjective probability measure attributedto an individual, of a subjective filtration as a history of personal information evolution. Anatural interest is to see how far conventional theories of stochastic analysis and control can bedeveloped under such generalized filtration.In this paper, as an example of an application, in addition to conventional filtration (classicalfiltration) in a binomial asset price model, we introduce a dropped filtration with loss of memoryfor a certain period of time to see whether individuals with the latter as her subjective filtrationcan in any sense price securities. This resulted in the question of whether there is a risk-neutralfiltration corresponding to this subjective filtration. We have shown the existence of such afiltration. However, the obtained risk-neutral filtration is not uniquely determined, unlike theclassical risk-neutral probability measure observed in a complete market. This means that amarket with such a generalized filtration is not complete (at least for individuals who havesuch a filtration as a subjective filtration). For other subjective filtrations not discussed in thispaper, it is possible that there may be no risk-neutral probability measure. How equilibriummarket prices are determined in such cases may be one of important themes for future research.Needless to say, the application of generalized filtrations shown in this paper is only oneexample, and many other applications are possible. As mentioned above, generalized filtrations can be used to develop conventional theories of stochastic control and stochastic differentialequations. For example, it can be used to transform a problem that is not time-consistent underclassical filtration into a time-consistent problem by twisting the filtration. The theory of filtra-tion enlargement used for credit risk calculation and insider trading analysis in finance may beable to be considered in the framework of generalized filtration [Aksamit and Jeanblanc, 2017].Furthermore, in order to study the relationship between a filtration and related risk-neutralfiltration, or filtrations defined on several different time domains, it is necessary to consider thetransformation and convergence of filtrations in a space of filtrations.
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Graduate School of Management, Tokyo Metropolitan University, 1-4-1 Marunouchi, Chiyoda-ku, Tokyo 100-0005, Japan
Email address : Takanori Adachi
College of International Management, Ritsumeikan Asia Pacific University, 1-1 Jumon-jibaru, Beppu, Oita, 874-8577 Japan
Email address : Katsushi Nakajima
Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu,Shiga, 525-8577 Japan
Email address ::