Stock Price Process and Statistical Mechanics (Mathematical Economics)

Stock

Price Process

and

Statistical

Mechanics

日本大学. 文理学部 黒田 耕嗣 (Koji Kuroda)

Dept. of Math., College ofHunanities and Sciences,

Nihon University

Abstract

Amodel ofan auction for astock market isintroduced to derive adiscrete timestock

price process. Aprobability distribution of trader’s positions configuration is given by

aGibbs distribution in which trading strategies of traders and an effect from market

sentiments are contained. To derive ascaling limit the method of statistical mechanics

is applied to thisprocess , and avolatility function and adrift function

are

described in

terms ofpolymer weight functions.

Keywords: Auction$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l},\mathrm{G}\mathrm{i}\mathrm{b}\mathrm{b}\mathrm{s}$ distribution, Stockpriceprocess, Polymer expansion

1Introduction

To describe astock priceprocess manystochasticprocessessuch

as

BlackSholesmodel[l],

ICR model [2] and Vasicek model [9] are investigated, particularly in an option price

theory.

In thisarticle

we

introduce

an

auction model for astock marketcomposedof$\mathrm{N}$ traders

and asingle stock. We derive adiscrete time stochastic process describing astock price

process from this auction model, in which each trader

can

trade unit number of stocks at

数理解析研究所講究録 1337 巻 2003 年 191-204

each time $t\in\Lambda_{n}=\{1,2, \cdots, n\}$

.

It is also assumedthat each trader

can

take

one

ofthe

following three positions, buying position, selling position and neutral position at each

time $t\in\Lambda_{n}=\{1,2, \cdots, n\}$.

Probability distribution for trader’s positions is given by aGibbs measure, in which

tradingstrategies of traders and

an

effect from market sentiments

are

contained. Gibbs

measures

isintroducedbyDobrushin [4], LanfordandRuelle [8] toinvestigateastatistical

mechanics from mathematicalpoint of view. It is characterized byan interaction energy.

In

our

model this interactionenergy is determined from trading strategies of traders and

market sentiments.

Applyingamethod ofpolymer expansion developed in atheoryof statis tical

mechan-ics $[3],[6]$

we

derive acontinuous time stock price

process,

$S_{t}=S_{0} \exp\{\int_{0}^{t}\sigma(s)dB(s)+\int_{0}^{t}\mu(s)ds\}$

by scaling the discretetime process, where $B(t)$ is astandard Brownian motion.

The function $\sigma(s)$ is called avolatility function which describes the strength of the

variation of the stock price, and the function$\mu(s)$ is calledadrift functionwhichdescribes

the trend of the stock price process.

We describe these functions in terms of polymer weight functions defined by the

in-teraction

energies which

determined

by trading strategies

and

market

sentiments.

2Model

As stated in the introduction each trader

can

take

one

of the following three positions,

buying positiondenoted$\mathrm{b}\mathrm{y}+$, selling positiondenotedby -and neutral position denoted

by 0.

Denote apositions configuration of$\mathrm{N}$ traders at time$t\in\Lambda_{n}$ by

$\omega_{t}=(\omega_{t}(1), \omega_{t}(2),$$\cdots,\omega_{t}(N))$

asequence of positions configuration from time 0to $n-1$ by

$\omega$$=(\omega_{0}, \cdots,\omega_{n-1})$

.

and the totality of$\omega$ by $\Omega_{n}$

.

For any $\omega_{t}$ the number of traders with buying position, selling position, and neutral

position are denoted by$\omega_{t}^{+}$, $\omega_{t}^{-}$ and $\omega_{t}^{0}$ respectively. The number of market participants

is given by $\omega_{t}^{+}+\omega_{t}^{-}$

.

We introduce anotion of modified number of market participants

by

$|\omega_{t}|=\{$

$\omega_{t}^{+}-\omega_{t}^{-}-d_{0}$ if$\omega_{t}$ is active

0otherwise

where $d_{0}$ is apositive constant.

We say$\omega_{t}$(or time $t$) is active if $|\omega_{t}|>0$ and static if $|\omega_{t}|=0$

.

We also put

$<\omega_{1}>=\{$

($v_{t}^{+}-\omega_{t}^{-}-d_{0}$ if$\omega_{t}^{+}-\omega_{t}^{-}>d_{0}$

0if $|\omega_{t}^{+}-\omega_{t}^{-}|\leq d_{0}$

$-(\omega_{t}^{-}\omega_{t}^{+}-d_{0})$ if$\omega_{t}^{-}-\omega_{t}^{+}>d_{0}$

If $<\{v_{t-1}>>0(<0)$, then the number of traders with buying (selling) position is

greater than the number of traders withselling(buying) positionandthestock pricegoes

(down) at time $t$

.

Let $S_{t}$ be astock price at time $t$

.

When $\omega$ is given

we

define the

stock priceprocess from $\omega$ by

$\frac{S_{t}}{S_{t-1}}=e^{\alpha<\mathrm{I}v_{t-1}>}$ .

It is easily

seen

that

$S_{t}=S_{0} \exp\{\sum_{u=0}^{t-1}<\omega_{u}>\}$.

Gibbs

measure

We introduceaGibbs

measure

as

aprobability distribution of positions configuration

$\omega$

.

First

we

define

an

interaction

energy

$H(\omega)$ by

$H( \omega)=\beta_{1}\sum_{t=0}^{n-1}|\omega_{t}|^{2}+\beta_{2}\sum_{t=0}^{n-1}\Phi(\omega_{t}|\omega_{t-a}, \cdots,\omega_{t-1})$

$- \beta_{3}\sum_{t=0}^{n-1}c(t)f_{1}(|\omega|_{t,a})|\omega_{t}|-\beta_{4}\sum_{t=0}^{n-1}d(t)f_{2}(<\omega>_{t,a})<\omega_{t}>$

where $\beta_{1}$,$\beta_{2}$,$\beta_{3}$, and $\beta_{4}$ are positive numbers which control the strength of interactions

and $c(t)$ and $d(t)$

are

positive functions defined on $\Lambda_{n}$.

Nowwe define aGibbs distribution of$\omega$ by

$P_{n}( \omega)=\frac{1}{Z_{n}}\exp\{-H(\omega)\}$,

where $Z_{n}$ is anormalizationconstant called apartition function.

For this Gibbs distribution , the probability $P(\omega)$ is high(low) if the energy $H(\omega)$ is

low(high).

The first term of $H(\omega)$ is aterm controling the number of market participants. The

second term corresponds to trading strategies of$N$ traders and it is assumed that

(A-1) $\Phi(\omega_{t}|\omega_{t-a}, \cdots, \omega_{t-1})=0$ if$\omega_{t}$ is static

(A-2) $\Phi(\overline{\omega}_{t}|\overline{\omega}_{t-a}, \cdots,\overline{\omega}_{t-1})=\Phi(\omega_{t}|\omega_{t-a}, \cdots,\omega_{t-1})$

(A-3) $\Phi(\omega_{t}|\omega_{t-a}, \cdots,\omega_{t-1})>-c|\omega_{t}|$

where $\overline{\omega}_{t}$ is areflectionimage of$\omega_{t}$ defined by $\overline{\omega}_{t}=-\omega_{t}$

.

Traders lookat ahistory $(\omega_{t-a}, \cdots, \omega_{t-1})$ ofpositions configurations and determine

a

present position$\omega_{t}$

.

The third term describes

an

effect on the market from atotal number of market

participants from time $t-a$ to $t-1$ and $|\omega|_{t,a}$is given by $|\omega|_{t,a}=|\omega_{t-a}|+\cdots+|\omega_{t-1}|$

.

If$f_{1}(|\omega|_{t,a})>0$, this term worksfor increasing anumber ofmarket participants.

We

assume

that $f_{1}(x)\leq c_{1}x$ $(x>0)$ for

some

$c_{1}>0$

.

The firth term describes

an

effect

on

the market from achenge of stock price from

time $t-a$ to $t-1$ and $<\omega>_{t,a}$ is given by $<\omega>_{t,a}=<\omega_{t-a}>+\cdots+<\omega_{t-1}>$. We

assume

that

(1) $f_{2}(x)\leq c_{2}|x|$

(2) $f_{2}(0)=0$

(3) $f_{2}(x)+\mathrm{f}2(\mathrm{x})\geq 0$ for all $x\in \mathrm{R}$

(4) $f_{2}(x)+f_{2}(-x)>\epsilon_{1}$ unless $|x-b_{3}|<\delta_{0}$ or $|x+b_{3}|<\delta_{0}$ or $|x|<\delta_{0}$ for some

$\epsilon_{1}>0$,_{$b_{3}>0$},$\delta_{0}>0$

.

If $f_{2}(<\omega>_{t,a})>0$, this term works for increasing anumber oftraders with buying

positions

3Method

of

cluster expansion

The method of polymer expansion has been developed by mainly Gallavotti[6], and Del

GrOssO[3], for rigorous investigationsof phase transitions in lattice spin systems. (See also

Phister[5] for anice summary of this method.)

In this section

we

summarizeamethod of polymer expansionto state

our

results.

For agiven configuration $\omega$

we

denote by $\{t_{1}, \cdots, t_{k}\}$ aset of all active times. De

compose aset

$\alpha(t_{1})\cup\alpha(t_{2})\cup\cdots\cup\alpha(t_{k})$

into aset of connected components $W_{1}$,$\cdots$ ,$W_{m}$, where $\alpha(t_{i})=\{t_{i}-a, \cdots, t_{:}-1\}$

.

We

callacouple$\xi^{i}$of$W_{\mathrm{i}}=\{t_{f}, \cdots, t_{q}\}$ and the configuration $\{\omega_{t_{f}}, \cdots,\omega_{t_{q}}\}$

on

$W_{\dot{1}}$acluster’

and denote it by

$\xi^{\dot{1}}$ _$=$ $(\begin{array}{lll}\omega_{f} \cdots \omega_{q}f \cdots q\end{array})$

For any cluster $\xi^{:}$ given in (3.1) put

$p(\xi^{\dot{1}})=\{f, \cdots, q\}$

.

When aconfiguration $\omega\in\Omega_{n}$ is given , afamilyofclusters $\{\xi^{1}, \cdots, \xi^{m}\}$ is determined

uniquely, but this correspondence is not one-t0-0ne.

Todefine probability distributionof$\{\xi^{1}, \cdots,\xi^{m}\}$wefirst introduceaset$A(\xi^{1}, \cdots,\xi^{m})$

by

$A(\xi^{1}, \cdots,\xi^{m})=$

{

$\omega\in\Omega_{n};\{\xi^{1}$,$\cdots$ ,$\xi^{m}\}$ is obtained from $\omega$

as

aset of clusters

}.

Then aprobability distribution $P(\xi^{1}, \cdots, \xi^{m})$ for aset ofclusters is defined by

$P( \xi^{1}, \cdots , \xi^{m})=\sum_{\omega\in A(\xi^{1},\cdots t^{m})}P(\omega)$.

It follows from the definition of interaction energies that the

occurrence

of clusters

becomes independent and

$P( \xi^{1}, \cdots,\xi^{m})=\prod_{\dot{\iota}=1}^{m}\mathcal{W}(\xi^{i})$

$,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathcal{W}(\xi)$ is given by

$\mathcal{W}(\xi)=e^{-|\mathrm{p}(\xi)|\log m\mathrm{o}}\prod_{t\in\hat{p}(\xi)}\exp\{-\beta_{1}|\xi_{t}|^{2}-\beta_{2}\Phi(\xi_{t}|\xi_{t-a}, \cdots, \xi_{t-1})$

$+\beta_{3}c(t)f_{1}(|\xi|_{t,a})|\xi_{t}|+\beta_{4}d(t)f_{2}(<\xi>_{t,a})<\xi_{t}>\}$

Algebraic formalism of cluster expansion

Let $\not\subset$ be aset of all clusters in $\mathrm{Z}$, and 2be aset of mappings given by

$\mathfrak{U}=\{A : C arrow \mathrm{N};|A|<\infty\}$,

where

$|A|= \sum_{\xi\in \mathrm{C}}A(\xi)$

.

Each $A\in \mathfrak{U}$ stands for aconfiguraion of finite number of clusters with multiplicity.

Furthermore,

we

define afunctional space $L$ by

$L$ $=$

{

$\varphi:\mathfrak{U}arrow \mathrm{R};\sup_{|A|=n}|\varphi(A)<\infty$for any $n$

}.

For any $\varphi_{1}$,$\varphi_{2}\in L$ define aproduct $\varphi_{1}*\varphi_{2}$ by

$\varphi_{1}*\varphi_{2}(A)=\sum_{A_{1}+A_{2}=A}\frac{A!}{A_{1}!}A_{2}!\varphi_{1}(A_{1})\varphi_{2}(A_{2})$

where the

sum runs over

all ordered pairs ($A_{1}$,A2) such that $A_{1}+A_{2}=A$

.

It is easily

seen

that

$\varphi_{1}*(\varphi_{2}*\varphi_{3})=(\varphi_{1}*\varphi_{2})*\varphi_{3}$

.

This functional space $L$becomes acommutative algebra with aunit element 1given

by

$1(A)=\{$1if

$A=\emptyset$

0otherwise.

Wedefine subspaces

_!and

$L_{1}$ given by

a

$=\{\varphi\in \mathrm{i}2;\varphi(\emptyset)=0\}$, $L_{1}=\{\varphi\in L; \varphi(\emptyset)=1\}$,

and amapping$\mathrm{E}\mathrm{x}\mathrm{p}:h$ $arrow L_{1}$ by

$\mathrm{E}\mathrm{x}\mathrm{p}\varphi(A)=\sum_{k=0}^{\infty}\frac{1}{k!}\varphi*\cdots*\varphi(A)$

.

As

an

inverse mapping of$\mathrm{E}\mathrm{x}\mathrm{p}$ we define amapping$\mathrm{L}\mathrm{o}\mathrm{g}:L_{1}arrow \mathcal{L}_{0}$ by

${\rm Log} \varphi(A)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k!}\varphi_{0}*\cdots\varphi_{0}(A)$

where

$\varphi_{0}=\varphi-1$

.

Then

we

have the followig relation between two mappings

${\rm Log}(\mathrm{E}\mathrm{x}\mathrm{p}\varphi)=\varphi$ for any $\varphi\in L_{0}$

$\mathrm{E}\mathrm{x}\mathrm{p}({\rm Log}\varphi)=\varphi$ for any $\varphi\in L_{1}$.

We say $\chi\in L$ is multiplicative if

$\chi(A_{1}+A_{2})=\mathrm{x}(\mathrm{A}\mathrm{i})\cdot\chi(A_{2})\mathrm{f}\mathrm{o}\mathrm{r}$any$A_{1}$,$A_{2}\in \mathfrak{U}$.

The following Lemma is afundamental Lemma for cluster expansion.

Lemma 3.2([3])

If$\chi$ is multiplicative and

$\sum_{A\in \mathfrak{U}}\frac{{\rm Log}\varphi(A)\chi(A)}{A!}<\infty$

then

$\sum_{A\in \mathfrak{U}}\frac{|\varphi(A)\chi(A)|}{A!}<\infty$

and

$\sum_{A\in\alpha}\frac{\varphi(A)\chi(A)}{A!}=\exp\{\sum_{A\in \mathfrak{U}}\frac{{\rm Log}\varphi(A)\chi(A)}{A!}\}$

.

Application of cluster expansion to the probability distribution of clusters

Now

we

applythe method of cluster expansion to the probability distribution

$P(\xi_{1}, \cdots,\xi_{m})$ and derive ascaling limit of$W_{t}(\cdot)$

.

First

we

define functionals $\phi_{0}(\xi)$,$\phi_{1}(\xi)$,$\phi_{2}(\xi)$ by

$\phi_{0}(\xi)=\exp\{-\beta_{1}|\xi|^{2}-\beta_{2}\Phi(\xi_{t}|\xi_{t-a}, \cdots,\xi_{t-1})-\log m_{0}|p(\xi)|\}$

$\phi_{1}(\xi)=\exp\{\beta_{3}\sum_{t\in \mathrm{p}(\xi)}c(t)f_{1}(|\xi|_{t,a}>)|\xi_{t}|\}$

$\phi_{2}(\xi)=\exp\{\beta_{4}\sum_{t\in p(\xi)}d(t)f_{2}(<\xi>_{t,a})<\xi_{t}>\}$

and

we

define $\phi_{0}(A),\phi_{1}(A)$,$\phi_{2}(A)$,$\alpha(A)$ by

$\phi_{0}(A)=\prod_{\xi\in \mathrm{C}}\phi_{0}(\xi)^{A(\xi)}$ $\phi_{1}(A)=\prod_{\xi\in \mathrm{C}}\phi_{1}(\xi)^{A(\xi)}$ $\phi_{2}(A)=\prod_{\xi\in \mathrm{C}}\phi_{1}(\xi)^{A(\xi)}$

$\alpha(A)=\{$1if

$A!=1,p(\xi^{i})\cap p(\xi^{j})=\emptyset$ for all $\xi^{\dot{\iota}}\neq\xi^{j}\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}A$

0otherwise.

Also we put

$<A>= \sum_{\xi\in C}<\xi>A(\xi)$

$f_{1}(A)= \beta_{3}\sum_{\xi\in C}\sum_{t\in \mathrm{p}(\xi)}c(t)f_{1}(|\xi_{0}|_{t,a})|\xi_{t}|A(\xi)$

$f_{2}(A)= \beta_{4}\sum_{\xi\in C}\sum_{t\in \mathrm{p}(\xi)}d(t)f_{2}(<\xi>_{t,a})<\xi_{t}>A(\xi)$,

and define arefletion image$\overline{A}$ of_$A$ by_{$\overline{A}(\xi)=A(\overline{\xi})$}.

The following Lemma is obtained in the similar way developed in $\mathrm{I}$

.

Lemma 3.3

For any $\beta_{2}>0$ and $\beta_{3}>0$ we have

$\sum_{\xi\in \mathrm{C}_{j}O\in p(\xi)}W(\xi)<1$

for sufficiently large $\beta_{1}>0$

.

Put

$\beta_{1}=\inf\{\beta_{1}>0;\sum_{\xi\ni 0}\exp\{-\beta_{1}|\xi|^{2}-c_{3}|\xi|)+\log m_{0}|p(\xi)|\}<1\}<\infty$

.

For any $A\in A$

we

call $A$ apolymer if

$\cup\xi Ap(\xi)$

isconnected.

Lemma 3.4

$\alpha^{T}(A)=0$ unless $A$is apolymer.

Lemma 3.5

(1) If$\beta_{1}>\beta_{0}$, then wehave

$\sum_{A\ni 0}\frac{\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)}{A!}|\alpha^{T}(A)|\geq g(\beta_{1})$

where $\alpha^{T}(A)=\log\alpha(A)$ and $g(\beta_{1})arrow 0$

as

$narrow\infty$.

(2) For any $0<c<1$

we

have

$\sum_{A\ni 0,|A|^{2}\geq k}\frac{\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)}{A!}|\alpha^{T}(A)|\geq g((1-c)\beta_{1})e^{-\mathrm{c}\beta_{1}k}$

if $(1-c)\beta_{1}>h$, where

$|A|^{2}= \sum|\xi|^{2}A(\xi)$

.

$\xi$

As $\phi_{0}(A)$,$\phi_{1}(A)$ and $\phi_{2}(A)$ are multiplicative, by applying the method of cluster

ex-pansion wehave

$Z_{n}= \exp\{\sum_{A\subset\Lambda_{n}}\frac{\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)\alpha^{T}(A)}{A!}\}$

.

4Main

Result

We define ascaled

process

$W_{t}^{(n)}(\cdot)$ by

$W_{t}^{(n)}( \cdot)=\frac{1}{\sqrt{n}}W_{[nt]}(\cdot)$

.

Decompose $\Lambda_{n}$ into aset of intervals with length $n^{\alpha}$:

$\Lambda_{n}=B_{n}(1)\cup\cdots\cup B_{n}(n^{1-\alpha})$

where $B_{n}(k)=((k-1)n^{\alpha}, kn^{\alpha}]$.

Furthermore

we

assume

that

$c(t)=h( \frac{k}{n^{1-\alpha}})$, $d(t)= \frac{1}{\sqrt{n}}g(\frac{k}{n^{1-\alpha}})$ $(t\in B_{n}(k))$

for continuous functions $f(x)$ and $g(x)$,

and put

$h$ $= \inf\{\beta_{1}>h;\sum_{i(A)=0}|A|^{4}e^{-(\beta_{1}|A|^{2}-c_{3}|A|)_{\frac{|\alpha^{T}(A)|}{A!}<\infty}}\}$.

Theorem 4.1 If $\beta_{1}>\beta\alpha$

), then afinite dimensional distribution of $W_{t}^{(n)}(\cdot)$ converges to the

corre-spond finite dimensional distribution of

$\int_{0}^{t}\sigma(s)dB(s)+\int_{0}^{t}\mu(s)ds$

where

$\sigma^{2}(s)=\sum_{:(A)=0}<A>^{2}e^{\beta_{3}h(\epsilon)f_{1}(A)_{\frac{\phi_{0}(A)\alpha^{T}(A)}{A!}}}$

$\mu(s)=\sum_{:(A)=0}g(s)e^{\beta_{3}h(\epsilon)f_{1}(A)}f_{2}(A)<A>\frac{\phi_{0}(A)\alpha^{T}(A)}{A!}$

Outline ofthe proof ofTheorem 4.1

We state

an

outline of the proof ofTheorem 4.1. (See [7] for detail.) The main tool

for the proof is the method of polymer expansion developed in the previous section.

Applying Lemma 3.2 for $\varphi(A)=\exp\{iz\frac{1}{\sqrt{n}}W_{[nt]}(A)\}\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)\alpha(A)$ we have

the following description of acharacteristic function $\varphi_{t}^{(n)}(z)$ of

one

dimensional

distribu-tionof $W_{t}^{(n)}$,

$\varphi_{t}^{(n)}(z)=E[\exp\{iz\frac{1}{\sqrt{n}}\sum_{u\leq[nt]}<\xi_{u}>\}]$

$= \exp\{\sum_{A\subset\Lambda_{n}}(e^{\dot{\iota}z\frac{1}{\mathrm{v}^{\acute{n}}}W_{[nt]}(A)}-1)\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)\frac{\alpha^{T}(A)}{A!}\}$

.

Using the Taylor’s expansion

we

have

$\sum_{A\subset\Lambda_{n}}(e^{\dot{|}z\frac{1}{\sqrt{n}}W_{[nt]}(A)}-1)\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)\frac{\alpha^{T}(A)}{A!}\}$ $=izI_{1}(n)- \frac{1}{2}z^{2}I_{2}(n)-\frac{iz^{3}}{6}I_{3}(n)$, where $I_{1}(n)= \frac{1}{\sqrt{n}}\sum_{A\subset\Lambda_{n}}W_{[nt]}(A)\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)\frac{\alpha^{T}(A)}{A!}$ I2(n) $= \frac{1}{n}\sum_{A\subset\Lambda_{n}}W1^{nt}](A)^{2}\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)\frac{\alpha^{T}(A)}{A!}$ $I_{3}(n)= \frac{1}{n\sqrt{n}}\sum_{A\subset\Lambda_{\hslash}}W_{[nt]}(A)^{3}e^{z\theta}.\cdot\neq_{n}W_{[nt]}(A)\phi_{0}(A)\phi_{1}(A)\phi_{2}(A)\frac{\alpha^{T}(A)}{A!}$

for

some

$\theta_{1}\in(0,1)$

.

Employing the method of poymer expansion developed in the theory of statistical

mechanics we have thefollowing results.

When $\beta_{1}>\mathrm{f}\mathrm{f}\mathrm{i}$ we provethat

$\lim_{narrow\infty}I_{1}(n)=\int_{0}^{t}\mu(s)ds$

$\lim_{narrow\infty}I_{2}(n)=\int_{0}^{t}\sigma^{2}(s)ds$

$\lim_{narrow\infty}I_{3}(n)=0$

It follows from this propositionthat

$\varphi_{t}^{(n)}(z)$ $arrow\exp\{iz\int_{0}^{t}\mu(s)ds-\frac{1}{2}z^{2}\int_{0}^{t}\sigma^{2}(s)ds\}$

Hence the

one

dimensional distribution of $W^{(n)}(t)$ converges to the corresponding

distribution of

$\int_{0}^{t}\mu(s)ds+\int_{0}^{t}\sigma(s)dB(s)$

.

Acharacteristic function of afinite dimensional distribution $\varphi_{t_{1},\cdots,t_{k}}$$(z_{1}, \cdots, z_{k})$ of

$W^{(n)}(t)$ is given by

$\varphi_{t_{1,\prime}t_{k}}\ldots(z_{1}, \cdots, z_{k})=E[\exp\{\frac{1}{\sqrt{n}}\sum_{k=1}^{m}z_{k}W_{[nt_{k}]}\}]$

.

Using the result in

one

dimensional

case

we have

$\varphi_{t_{1,\prime}t_{k}}\ldots(z_{1}, \cdots, z_{k})arrow\exp\{i\sum_{j=1}^{k}z_{j}\int_{0}^{t_{j}}\mu(s)ds-\frac{1}{2}\sum_{p=1}^{k}\sum_{q=1}^{k}z_{\mathrm{p}}z_{q}\int_{0}^{t_{\mathrm{p}}\wedge t_{q}}\sigma^{2}(s)ds\}$

as

$narrow\infty$

.

This implies that the finite dimensional distribution of $W^{(n)}(t)$ converges to

the corresponding distribution of

$\int_{0}^{t}\mu(s)ds+\int_{0}^{t}\sigma(s)dB(s)$

.

Remark

on

trading volume

Inthis auction model atrading volume$v_{t}$ at time $t$ is given by

$v_{t}={\rm Min}\{\omega_{t}^{+},\omega_{t}^{-}\}$

.

Let

us

remark that trading volume is not always

zero

when$t$isstatic andaprobability

distributionof trading volume$v_{t}$for static$t$is uniformdistribution. So, the expectation

value of$v_{t}$ for static$t$ is given by

$e(v)= \frac{e(v)}{n_{0}}$

$e(v)=. \sum_{\omega_{t}\cdot \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}v(\omega_{t})$ and $n_{0}=. \sum_{\omega_{\ell}.\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}1$.

We denote by$V[nt]$ atotal trading volume traded in time interval _{$[0, [nt]]$} and considera

asymptoticbehavior of$E[V_{[nt]}]$ and $V[V_{[nt]}]$,where $E[V_{[nt]}]$ and $V[V_{[nt]}]$

are an

expectation

and avarianceofVjnt], respectively.

Using the method of polymer expansion

we

describe acharacteristic

function

ofVjnt]

andobtain the formulas of$E[V_{[nt]}]$ and $V[V_{[nt]}]$ in terms of polymer weight functions.

Prom these formulas and using the

same

method obtaining the scaling limit of the

characteristicfunction for afinitedimensional distribution $W[nt]$

we

have

$\lim_{narrow\infty}\frac{1}{n}E[V_{[nt]}]=\int_{0}^{t}(\sum_{i(A)=O}v(A)e^{h(x)f_{1}(A)}\frac{\phi_{0}(A)\alpha^{T}(A)}{A!}dx$

$\lim_{narrow\infty}\frac{1}{n}V[V_{[nt]}]=t(\frac{e(v^{2})}{n_{0}}-(\frac{e(v)}{n_{0}})^{2})$

$-( \frac{e(v^{2})}{n_{0}}-(\frac{e(v)}{n_{0}})^{2})\int_{0}^{t}\sum_{i(A)=O}|p(A)|e^{h(x)f1(A)}\frac{\phi_{0}(A)\alpha^{T}(A)}{A!}dx$

$+ \int_{0}^{t}\sum_{:(A)=O}(v(A)-\frac{e(v)}{n_{0}}||p(A)|)^{2}e^{h(x)f_{1}(A)}\frac{\phi_{0}(A)\alpha^{T}(A)}{A!}dx$

where

$v(A)= \sum_{\xi}\sum_{u\in p(\xi)}v(\xi_{u})A(\xi)$

$e(v^{2})=. \sum_{\{v_{t}\cdot \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}}v(\omega_{t})^{2}$

Refferences

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Political Economy, 81,

637-654

[2] $\mathrm{C}\mathrm{o}\mathrm{x},\mathrm{J}.\mathrm{C}$ , $\mathrm{I}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{l},\mathrm{J}.\mathrm{E}.$, and $\mathrm{R}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{S}.\mathrm{A}.(1985)$: “A theory of the term structure of

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