A Binomial Asset Pricing Model in a Categorical Setting

TL;DR

This paper develops a binomial asset pricing model within a categorical probability framework, incorporating generalized filtrations to model information loss, and explores valuation of financial claims under these conditions.

Contribution

It introduces a novel categorical approach to asset pricing with generalized filtrations, capturing information loss scenarios not addressed by traditional models.

Findings

Model accommodates agents forgetting information at specific times.

Valuations of financial claims are analyzed under non-standard filtrations.

Framework extends classical asset pricing to more realistic information dynamics.

Abstract

Adachi and Ryu introduced a category Prob of probability spaces whose objects are all probability spaces and whose arrows correspond to measurable functions satisfying an absolutely continuous requirement in [Adachi and Ryu, 2019]. In this paper, we develop a binomial asset pricing model based on Prob. We introduce generalized filtrations with which we can represent situations such as some agents forget information at some specific time. We investigate the valuations of financial claims along this type of non-standard filtrations.

Equations

$$\mathbf{Prob}(\bar{X},$$

$$\mathcal{T}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{T}$$

$$\mathcal{T}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$U:\mathbf{Prob}\to \mathbf{Meas}$$

$$\tau :=\{{\tau }_s:\mathcal{T}(s,-)\dot{\to }L\circ F{\}}_{s\in Obj(\mathcal{T})}$$

$$I{d}_s\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\tau }_{s,t}(i)=[v_s\circ (Fi{)}^{+}{]}_{{\sim }_{{\mathbb{P}}_{Ft}}}.$$

$$AP(F)\cong \prod _{t\in Obj(\mathcal{T})}L(Ft).$$

$$y_t:\mathrm{Nat}(\mathcal{T}(t,-),L\circ F)\cong (L\circ F)t.$$

$$x=\{x_t{\}}_{t\in Obj(\mathcal{T})}$$

$$x_t:=x(t)\in L(Ft).$$

$$s\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\int }_{A}ud{\mathbb{P}}_X={\int }_{f^{-1}(A)}vd{\mathbb{P}}_Y.$$

$$X$$

$$(\mathcal{E}\circ F)i(x(t))=x(s).$$

$$s\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathbb{P}}_n(\{a\}):=\prod _{i=1}^n{p}_i^{d_i}(1-p_i{)}^{1-d_i}.$$

$$\mathcal{B}({\ast }_{m,n}):=f_{m,n}^{-}:=(f_m\circ f_{m+1}\circ \cdots \circ f_{n-1}{)}^{-}$$

$$\omega \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$B_{n+1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$f_n:=f_n^{full}.$$

$$f_n:=\{\begin{array}{l}{f}_n^{drop}\text{if}n=k,\\ f_n^{full}\text{if}n\ne k.\end{array}$$

$$f_n:=\{\begin{array}{l}{f}_n^{drop}\text{if}{k}_0\le n\le T-k_1\\ f_n^{full}\text{if}0\le n<k_0\text{or}T-k_1<n\le T.\end{array}$$

$$E^{f_n^{-}}(v)(a){\mathbb{P}}_n(\{a\})=\sum _{b\in f_n^{-1}(a)}v(b){\mathbb{P}}_{n+1}(\{b\}).$$

$$f_n^{-1}(a)=(f_n^{full}{)}^{-1}(a)=\{a0,a1\}.$$

$$E^{f_n^{-}}(v)(a)$$

$$=v(a0)(1-p_{n+1})+v(a1)p_{n+1}.$$

$$B_n\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E^{f_n^{-}}({\xi }_{n+1})(a)$$

Full text

This work was supported by JSPS KAKENHI Grant Number 18K01551.

1. Introduction

Adachi and Ryu introduced the category $\mathbf{Prob}$ as an adequate candidate of the category of probability spaces with good arrows. They show the existence of the conditional expectation functor from $\mathbf{Prob}$ to $\mathbf{Set}$, which is a natural generalization of the classical notion of conditional expectation ([Adachi and Ryu, 2019]).,

In this paper, we develop a binomial asset pricing model based on the category $\mathbf{Prob}$. Generalized filtrations defined in this setting change not only $\sigma$-algebras but also probability measures and 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 not only 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 specific time. This paper investigate the valuations of financial claims along these non-standard filtrations.

First, in Section 2, we review the concept of categorical probability theory and introduce generalized filtrations and adapted processes and martingales along them. In this setting, our probability spaces are changing as time goes on. For example, we may have a bigger underlying set 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 model and investigate a few generalized filtrations and possibility of valuations along them. We also provide 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 were mainly introduced in [Adachi and Ryu, 2019] as a preparation for Section 3.

Let $\bar{X}=(X,\Sigma_{X},\mathbb{P}_{X})$, $\bar{Y}=(Y,\Sigma_{Y},\mathbb{P}_{Y})$ and $\bar{Z}=(Z,\Sigma_{Z},\mathbb{P}_{Z})$ be probability spaces throughout this paper.

Definition 2.1**.**

[Null-preserving functions [Adachi and Ryu, 2019] ] A measurable function $f:\bar{Y}\to\bar{X}$ is called null-preserving if $f^{-1}(A)\in\mathcal{N}_{Y}$ for every $A\in\mathcal{N}_{X}$, where $\mathcal{N}_{X}:=\mathbb{P}_{X}^{-1}(0)\subset\Sigma_{X}$ and $\mathcal{N}_{Y}:=\mathbb{P}_{Y}^{-1}(0)\subset\Sigma_{Y}$.

Definition 2.2**.**

[Category $\mathbf{Prob}$ [Adachi and Ryu, 2019] ]

A category $\mathbf{Prob}$ is the category whose objects are all probability spaces and the set of arrows between them are defined by

[TABLE]

where $f^{-}$ is a symbol corresponding uniquely to a function $f$.

We write $Id_{X}$ for an identity measurable function from $\bar{X}$ to $\bar{X}$, while writing $id_{X}$ for an identity function from $X$ to $X$. Therefore, the identity arrow of a $\mathbf{Prob}$-object $\bar{X}$ is $Id_{X}^{-}$.

Definition 2.3**.**

[Generalized Filtrations] Let $\mathcal{T}$ be a fixed small category which we sometimes call the time domain. A $\mathcal{T}$-filtration is a functor $F:\mathcal{T}\to\mathbf{Prob}$.

When we say filtrations in the classical setting, we keep using a same underlying set $\Omega$ throughout time. This situation can be represented by the following diagram.

[TABLE]

However, in our new setting, the filtration can change not only $\sigma$-fields but also probability measures and underlying sets as the following diagram shows.

[TABLE]

One of the implications of this generalization is that we can think possibly distorted filtrations by using adequate null-preserving function $f_{t}$.

Actually, the biggest aim of this paper is to investigate this kind of non-standard filtrations by using, as a first example, a simple binomial asset pricing model.

Let $\mathbf{Meas}$ be the category of measurable spaces that consists of all measurable spaces as its objects and all measurable functions between them as arrows.

[TABLE]

is a forgetful functor that is it maps a object $\bar{X}$ to $(X,\Sigma_{X})$ by dropping its probability measure, and an arrow $f:\bar{X}\to\bar{Y}$ to $f:(X,\Sigma_{X})\to(Y,\Sigma_{Y})$.

Later, we will investigate a modification of a given filtration $F$ to another filtration $G$ such that $U\circ F=U\circ G$, that is, the situation when they share their measurable space nature.

Before going into our concrete example, we will define adapted processes and martingales over this generalized filtrations.

Let $F$ be a fixed $\mathcal{T}$-filtration throughout this section.

Definition 2.4**.**

[$F$-Adapted Processes]

An $F$-adapted process is a collection of natural transformations

[TABLE]

For a $\mathbf{Prob}$-arrow $\varphi:\bar{X}\to\bar{Y}$, there exists a measurable function $f:Y\to X$ such that $\varphi=f^{-}$ by its definition. We write $\varphi^{+}$ for this $f$. That is, $(\varphi^{+})^{-}=\varphi$.

Now Let $\tau$ be an $F$-adapted process and $i:s\to t$ be a $\mathcal{T}$-arrow. Then, we have the following commutative diagram.

[TABLE]

For $s\in Obj(\mathcal{T})$ pick a random variable $v_{s}$ satisfying $[v_{s}]_{\sim_{\mathbb{P}_{Fs}}}=\tau_{s,s}(Id_{s})$. Then, we have

[TABLE]

That is, $\tau_{s,t}(i)$ is $(Fi)$-measurable.

Proposition 2.5**.**

Let $AP(F)$ be the set of all $F$-adapted processes. Then,

[TABLE]

Proof.

By Yoneda Lemma, we have for $t\in Obj(\mathcal{T})$,

[TABLE]

Then, $\prod_{t\in Obj(\mathcal{T})}y_{t}$ is an isomorphism denoting (2.4).

∎

For $x\in AP(F)$, we sometimes write

[TABLE]

where

[TABLE]

Remark 2.6**.**

For an arrow $i:s\to t$ in $\mathcal{T}$, in general, $Fs$ and $Ft$ are different probability spaces. So we cannot (for example) add two random variables $x_{s}\in L^{1}(Fs)$ and $x_{t}\in L^{1}(Ft)$ whose domains are $\tilde{Fs}$ and $\tilde{Ft}$.

[TABLE]

In order to import $x_{s}$ into $L^{1}(Ft)$, we take $x_{s}\circ(Fi)^{+}$ as its proxy. This fact allows us to treat $L^{1}(Ft)$ as a vector space containing all preceding random variables $x_{s}\in L^{1}(Fs)$ with $s\leq t$.

Next, we go into the definition of martingales. In order to make it possible, we need a concept of conditional expectations in the category $\mathbf{Prob}$ which was introduced in [Adachi and Ryu, 2019].

Theorem 2.7**.**

[Conditional Expectation [Adachi and Ryu, 2019]]*

Let $f^{-}:\bar{X}\to\bar{Y}$ be a $\mathbf{Prob}$-arrow. For all $v\in\mathcal{L}^{1}(\bar{Y})$ and $A\in\Sigma_{X}$, there exists $u\in\mathcal{L}^{1}(\bar{X})$ satisfying the following equation.*

[TABLE]

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 functor $\mathcal{E}:\mathbf{Prob}^{op}\to\mathbf{Set}$ as following:*

[TABLE]

We call $\mathcal{E}$ a conditional expectation functor.

Definition 2.9**.**

[$F$-Martingales]

Let $F:\mathcal{T}\to\mathbf{Prob}$ be a functor. An $F$-martingale is an $F$-adapted process $x\in AP(F)$ such that for every $\mathcal{T}$-arrow $i:s\to t$,

[TABLE]

3. A Binomial Asset Pricing Model

In this section, we introduce a binomial asset pricing model based on the category $\mathbf{Prob}$.

3.1. Filtration $\mathcal{B}$

First, we define a general scheme of our model by introducing a filtration $\mathcal{B}$.

Definition 3.1**.**

[Filtration $\mathcal{B}$]

Let $\omega$ be the category whose objects are all integers starting with [math] and for each pair of integers $m$ and $n$ with $m\leq n$ there is a unique arrow $*_{m,n}:m\to n$. That is, $\omega$ is the category corresponding to the integer set $\mathbb{N}$ with the usual total order. Let $\mathbf{p}:=\{p_{i}\}_{i=1,2,\dots}$ be an infinite sequence of real numbers $p_{i}\in[0,1]$. We define an $\omega$-filtration $\mathcal{B}:=\mathcal{B}^{\mathbf{p}}:\omega\to\mathbf{Prob}$ in the following way.

For an object $n$ of $\omega$, $\mathcal{B}n$ is a probability space $\bar{B}_{n}:=(B_{n},\Sigma_{n},\mathbb{P}_{n})$ whose components are defined as follows:

1. (1)

$B_{n}:=\{0,1\}^{n}$, the set of all binary numbers of $t$ digits, 2. (2)

$\Sigma_{n}:=2^{B_{n}}$, 3. (3)

for $a:=d_{1}d_{2}\dots d_{n}\in B_{n}$ where $d_{i}\in\{0,1\}\;(i=1,2,\dots n)$. $\mathbb{P}_{n}:\Sigma_{n}\to[0,1]$ is the probability measure defined by

[TABLE]

For integers $m$ and $n$ with $m<n$, we define

[TABLE]

where $f_{n}:=(\mathcal{B}(*_{n,n+1}))^{+}$ is a predefined null-preserving function from $B_{n+1}$ to $B_{n}$.

The filtration $\mathcal{B}$ is called non-trivial if there exists $i$ such that $0<p_{i}<1$.

Note that any function from $B_{n}$ is measurable since $\Sigma_{n}$ is a powerset of $B_{n}$.

[TABLE]

As we introduced, the functor $\mathcal{B}$ is a generalized filtration, representing a filtration over the classical binomial model, for example developed in [Shreve, 2005].

The classical version requires the terminal time horizon $T$ for determining the underlying set $\Omega:=\{0,1\}^{T}$ while our version does not require it since the time variant probability spaces can 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. (1)

$f^{full}_{n}$

[TABLE] 2. (2)

$f^{drop}_{n}$

[TABLE]

The function $f^{drop}_{n}$ can be interpreted to forget what happens at time $n$.

Note that the function $f^{full}_{n}$ is always null-preserving while $f^{drop}_{n}$ is null-preserving if and only if $p_{n}=0$.

Example 3.3**.**

[Filtrations] As we mentioned in Definition 3.1, all we need to determine the filtration is to specify $f_{n}:B_{n+1}\to B_{n}$. We have three examples of filtration $\mathcal{B}$. For $j=1,2,\dots,n$,

1. (1)

Classical filtration:

[TABLE] 2. (2)

Drop-$k$:

[TABLE] 3. (3)

Elderly person: For fixed numbers $k_{0},k_{1}\in\mathbb{N}$,

[TABLE]

Proposition 3.4**.**

For a $\mathbf{Prob}$-arrow $f^{-}_{n}:\bar{B}_{n}\to\bar{B}_{n+1}$, $v\in L^{1}(\bar{B}_{n+1})$ and $a\in B_{n}$,

[TABLE]

Especially, with the classical filtration, we have

[TABLE]

Hence

[TABLE]

Definition 3.5**.**

[$\mathcal{B}$-Adapted Process $\xi_{n}$] For $n=1,2,\dots$ define a $\mathcal{B}$-adapted process $\xi_{n}$ by

[TABLE]

Proposition 3.6**.**

For $a\in B_{n}$ with $\mathbb{P}_{n}(a)\neq 0$,

[TABLE]

where

[TABLE]

for $j=0,1$, and $\#A$ denotes the cardinality of the set $A$.

3.2. Arbitrage Strategies

Now we define two instruments tradable in our market.

Definition 3.7**.**

[Stock and Bond Processes] Let $\mu,\sigma,r\in\mathbb{R}$ be constants such that $\sigma>0$, $\mu>\sigma-1$ and $r>-1$.

1. (1)

A stock process $S_{n}:B_{n}\to\mathbb{R}$ over $\mathcal{B}$ is defined by

[TABLE]

where $\langle\rangle\in B_{0}$ is the empty sequence, and $s_{0}>0$ is a positive constant. 2. (2)

A bond process $b_{n}:B_{n}\to\mathbb{R}$ over $\mathcal{B}$ is defined by

[TABLE]

The condition $\mu>\sigma-1$ is necessary for keeping the stock price positive.

We sometimes call the triple $(\mathcal{B},S,b)$ a market But, it does not mean that the market will not contain other instruments.

Proposition 3.8**.**

For any $a\in B_{n}$,

1. (1)

E^{f_{n}^{-}}(S_{n+1})=S_{n}\big{(}(1+\mu)E^{f_{n}^{-}}(1_{B_{n+1}})+\sigma E^{f_{n}^{-}}({\xi_{n+1}})\big{)}.** 2. (2)

$E^{f_{n}^{-}}(1_{B_{n+1}})(a)=\frac{\mathbb{P}_{n+1}(f_{n}^{-1}(a))}{\mathbb{P}_{n}(a)}.$** 3. (3)

$b_{n}(a)=(1+r)^{n}.$* *

Definition 3.9**.**

[Strategies] A strategy is a sequence $(\phi,\psi)=\{(\phi_{n},\psi_{n})\}_{n=1,2,\dots}$, where

[TABLE]

Each element of the strategy $(\phi_{n},\psi_{n})$ is called a portfolio. The value $V_{n}$ of the portfolio at time $n$ is determined by:

[TABLE]

Definition 3.10**.**

[Gain Processes] A ** gain process ** of the strategy $(\phi,\psi)$ is the process $\{G^{(\phi,\psi)}_{n}\}_{n=0,1,2,\dots}$ defined by

[TABLE]

Lemma 3.11**.**

Let $n$ be an object of $\omega$ such that

[TABLE]

Then, we have

[TABLE]

Proof.

[TABLE]

∎

Definition 3.12**.**

[Arbitrage Strategies]

1. (1)

A strategy $(\phi,\psi)$ is called a $\mathcal{B}$-arbitrage strategy if \mathbb{P}_{n}\big{(}G^{(\phi,\psi)}_{n}\geq 0\big{)}=1 for every $n$, and \mathbb{P}_{n_{0}}\big{(}G^{(\phi,\psi)}_{n_{0}}>0\big{)}>0 for some $n_{0}$. 2. (2)

The market is called non-arbitrage or NA if it does not allow $\mathcal{B}$-arbitrage strategies.

Proposition 3.13**.**

If the market $(\mathcal{B},S,b)$ with a non-trivial filtration $\mathcal{B}$ is non-arbitrage, then $\lvert\mu-r\rvert<\sigma.$

Proof.

Assuming that $r\leq\mu-\sigma$ or $r\geq\mu+\sigma$, we will construct an arbitrage strategy $(\phi,\psi)$ by using the following algorithm.

⬇

for n = 0, 1, 2, ...:

observe S(n) and b(n)

if r <= mu - sigma:

    phi(n+1) > 0  # pick arbitrarily

elsif r >= mu + sigma:

    phi(n+1) < 0  # pick arbitrarily

psi(n+1) := -(S(n) / b(n)) phi(n+1)

# Then, we have S(n) phi(n+1) + b(n) psi(n+1) = 0,

# which simplifies the computation of G(n) in the following.

if n == 0:

    G(0) := 0

else:  # n > 1

    G(n) := s(n) (phi(n) * f(n-1)) + b(n) (psi(n) * f(n-1))

In the above code, ‘*’ is the function composition operator.

By Lemma 3.11, we have

[TABLE]

So we have $G^{(\phi,\psi)}_{n}\geq 0$ as long as $r\leq\mu-\sigma$ or $r\geq\mu+\sigma$.

By the way, since our filtration is non-trivial, there exists a number $n_{0}$ such that $0<p_{n_{0}}<1$. It is easy to check that

[TABLE]

which concludes that $(\phi,\psi)$ is an arbitrage strategy. ∎

3.3. Risk-Neutral Filtrations

In this subsection, we assume that $\lvert\mu-r\rvert<\sigma$.

Let us consider about the discounted stock process

[TABLE]

Definition 3.14**.**

[Risk-neutral filtrations] An $\omega$-filtration $\mathcal{C}$ is called a risk-neutral filtration having the same shape of $\mathcal{B}$ if $U\circ\mathcal{C}=U\circ\mathcal{B}$ and discounted stock process becomes a $\mathcal{C}$-martingale, that is, for any arrow of the form $*_{n,n+1}:n\to n+1$ in the category $\omega$

[TABLE]

We want to find a risk-neutral filtration $\mathcal{C}$. Here is the shape of the filtration whose detail we will determine.

Definition 3.15**.**

[Filtration $\mathcal{C}$] Let $n$ be an object of the category $\omega$.

1. (1)

$\mathbb{Q}_{n}:\Sigma_{n}\to[0,1]$ is a probability measure of $(B_{n},\Sigma_{n})$, 2. (2)

$\bar{C}_{n}:=(B_{n},\Sigma_{n},\mathbb{Q}_{n})$, 3. (3)

$g_{n}:=f_{n}$.

We define an $\omega$-filtration $\mathcal{C}$ by for $n\in Obj(\omega)$,

[TABLE]

With these notations, (3.16) is reduced to the form

[TABLE]

Theorem 3.16**.**

*A process $S_{n}^{\prime}$ is a

$\mathcal{C}$-martingale , that is, for $n\in\mathbb{N}$, $E^{g_{n}^{-}}(S^{\prime}_{n+1})=S^{\prime}_{n}$ if and only if for all $n\in\mathbb{N}$ and $a\in B_{n}$,*

[TABLE]

where for $j=0,1$

[TABLE]

and

[TABLE]

Proof.

For $a\in B_{n}$

[TABLE]

if and only if

[TABLE]

∎

In order to determine more detail of $\mathcal{C}$, we need the following condition for $\mathbb{Q}_{n}$.

Proposition 3.17**.**

The following conditions for $\mathbb{Q}_{n}$ are equivalent.

1. (1)

for all $n\in\mathbb{N}$, $a\in B_{n}$,

[TABLE] 2. (2)

for all $n\in\mathbb{N}$, $f^{full}_{n}$ is measure-preserving w.r.t. $\mathbb{Q}_{n}$, that is,

[TABLE] 3. (3)

there exists a sequence of functions $\{q_{k}:B_{k}\to[0,1]\}_{k=1,2,\dots}$ such that for all $n=1,2,\dots$ and $d_{j}=0,1$,

[TABLE]

such that for every $a\in B_{n-1}$, $q_{n}(a0)+q_{n}(a1)=1.$

In the following discussion, we assume the following assumption which is the condition (2) of Proposition 3.17.

Assumption 3.18**.**

For all $n\in\mathbb{N}$, $f^{full}_{n}$ is measure-preserving w.r.t. $\mathbb{Q}_{n}$.

By Assumption 3.18 and (3) of Proposition 3.17, we have

[TABLE]

In the rest of this subsection, we will investigate the shape of $\mathbb{Q}_{n}$ under the assumption that $S^{\prime}_{n}$ is $\mathcal{C}$-martingale.

3.3.1. Classical Filtration

First, we prepare a lemma for for the proof of the following propositions.

Lemma 3.19**.**

If $1=c_{1}x+c_{0}(1-x)$, then

[TABLE]

Proposition 3.20**.**

For a fixed $n\in\mathbb{N}$, assume that $f_{n}=f^{full}_{n}$. Then for $a\in B_{n}$ with $\mathbb{Q}_{n}(\{a\})\neq 0$, we have

[TABLE]

Note that the resulting probability depends neither on $a$ nor on $n$.

Proof.

By observing the following diagram

[TABLE]

we have

[TABLE]

By (3.19)

[TABLE]

Now since

[TABLE]

and $\mathbb{Q}_{n}(\{a\})\neq 0$, we have

[TABLE]

Hence by Lemma 3.19, we have

[TABLE]

∎

Corollary 3.21**.**

If $\mathcal{B}$ is the classical filtration, then for any $n\in\mathbb{N}$ and $a\in B_{n}$ we have

[TABLE]

where

[TABLE]

3.3.2. Drop-$k$ Filtration

Proposition 3.22**.**

For a fixed $n(=1,2,\dots)$, assume that $f_{n}=f^{drop}_{n}$. Then for $a\in B_{n-1}$ with $\mathbb{Q}_{n-1}(\{a\})\neq 0$, we have

[TABLE]

Proof.

By observing the following diagram

[TABLE]

we have

[TABLE]

By (3.19)

[TABLE]

Now since $\mathbb{Q}_{n}(\{ad_{n}\})=\mathbb{Q}_{n-1}(\{a\})q_{n}(ad_{n})$ and $\mathbb{Q}_{n-1}(\{a\})\neq 0$, we have

[TABLE]

Next, again by (3.19)

[TABLE]

By dividing both hands by $\mathbb{Q}_{n-1}(\{a\})\neq 0$,

[TABLE]

Then, since $q_{n}(a1)=0$ and $q_{n}(a0)=1$,

[TABLE]

Hence, by Lemma 3.19, we have

[TABLE]

∎

We have to check that both $f^{full}_{n}$ and $f^{drop}_{n}$ are null-preserving w.r.t. $\mathbb{Q}_{n}$.

[TABLE]

If $\mathbb{Q}_{n}(d_{1}\dots d_{n-1}1)\neq 0$, then $drop_{n}$ is null-preserving, and so is $f^{drop}_{n}$ since $f^{full}_{n}$ is measure-preserving.

Remark 3.23**.**

We have the following remarks for Figure 3.2.

1. (1)

Since the agent evaluates stock and bond along the function $f_{n}^{drop}$, she can recognise only the nodes $a0$, $a01$ and $a00$ and can not recognise the nodes $a1$, $a11$ and $a10$. We interpret these nodes $a1$, $a11$ and $a10$ as invisible. 2. (2)

The values $q_{n+1}(a11)\in[0,1]$ can be arbitrarily selected, and $q_{n+1}(a10)$ is computed by $1-q_{n+1}(a10)$. That is, the probability measure $\mathbb{Q}_{n+1}$ is not determined uniquely, so is not the risk-neutral filtration $\mathcal{C}$. 3. (3)

The probability measure $\mathbb{Q}_{n}$ is not equivalent to the original measure $\mathbb{P}_{n}$. Therefore, it is not an EMM.

Remark 3.24**.**

Let $\mathcal{C}:\omega\to\mathbf{Prob}$ be a risk-neutral filtration, and $Y:B_{T}\to\mathbb{R}$ be a payoff at time $T$. Then, for the agent who has a drop-$k$ filtration as her subjective filtration, the price of $Y$ at time $n$ with a unique arrow $i_{n}:n\to T$ is given by

[TABLE]

[TABLE]

For $a\in B_{n-2}$, you can see in Figure 3.4 that at time $n-1$ the value of $Y_{n}(a1)$ is discarded and use only the value of $Y_{n}(a0)$ for computing $Y_{n-1}(a)$.

3.3.3. Replication Strategies

Let us investigate the situation where a given strategy $(\phi,\psi)$ becomes a replication strategy of the payoff $Y$ at time $T$.

Definition 3.25**.**

[Self-Financial Strategies] A self-financial strategy is a strategy $(\phi,\psi)$ satisfying

[TABLE]

for every $n=1,2,\dots$.

For a self-financial strategy $(\phi_{n},\psi_{n})_{n=1,2,\dots}$, we have:

[TABLE]

Therefore, for $a\in B_{n}$ and $d_{n+1}\in\{0,1\}$,

[TABLE]

where

[TABLE]

Now let us assume that there exists a function $g_{n}:B_{n}\to B_{n}$ such that $f_{n}=g_{n}\circ f^{full}_{n}$ .

[TABLE]

Then $f_{n}(ad_{n+1})=g_{n}(a)$ for every $a\in B_{n}$ and $d_{n+1}\in\{0,1\}$. So the equation (3.29) becomes

[TABLE]

Hence, we have:

[TABLE]

Therefore, we can determine the appropriate strategy $(\phi_{n+1},\psi_{n+1})$ on $g_{n}(B_{n})\subset B_{n}$ by (3.32). We actually do not care the values of $(\phi_{n+1},\psi_{n+1})$ on $B_{n}\setminus g_{n}(B_{n})$.

For example, in the case of $f_{n}=f^{drop}_{n}$, the function $g_{n}:B_{n}\to B_{n}$ satisfies

[TABLE]

for all $d_{1}\dots d_{n-1}d_{n}\in B_{n}$. Looking at Figure 3.4, values in the region $B_{n}\setminus g_{n}(B_{n})$ are not necessary for computing $Y_{n-1}(a)$. Hence, determining the values of $(\phi_{n+1},\psi_{n+1})$ in $g_{n}(B_{n})$ is enough for making the practical valuation.

4. Concluding Remarks

We formulated an infinitely growing sequence of binomial probability spaces in the category $\mathbf{Prob}$. We gave some concrete (possibly distorted) filtrations. We determined the shape of the risk-neutral filtrations to the above examples. We showed the valuations of claims given at time $T$ through the distorted filtrations, and provided a replication strategy implementing the valuation.