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The Stationary, Continuous time, Discrete Space Model with Polya Trees for Micro-data Analysis and Option
Pricing
August 25, 2004
Masaru Hashimoto Peter Lenk
University of Michigan Business School
Ann Arbor, MI 48109-1234
The Stationary, Continuous time, Discrete Space Model with Polya Trees for Micro-data Analysis and Option Pricing
Abstract
The stationary, continuous time, discrete space (SCD) model is developed, and this paper highlights some aspects of the SCD model. The SCD model, a class of stationary Markov processes, is useful to investigate phenomenon in financial micro data where information about time in seconds and a trading price for each transaction is available. The SCD model has two components; one component for the inter arrival time, the time of two adjacent transactions, and one component for the transition of a stock price to a new price. The representation theorem for Markov processes assures that the inter arrival time follows the exponential distribution, and the sequence of the stock prices is a Markov chain with some transition matrix. This paper suggests the model using hierarchical Bayes (HB) model via Markov Chain Monte Carlo (MCMC) in the exponential distribution. For the second component, a Polya tree, one of the Bayesian nonparametric distributions, estimates the transition probability. The Polya tree estimation is particularly useful for micro data because it induces smoothing of the micro data or the “thin data” where there is few or no observation for some of the large number of states. This paper suggests that a one-dimensional or two-dimensional Polya tree is used, depending on the assumption that the transition probabilities are independent of stock prices. The SCD model is so general that it can simulate changes in volatility or jumps which commonly appear in financial data. The paper presents some results on empirical study of the SCD model. First, Monte Carlo simulation generates sample paths for Intel stock prices under this model. Then, the SCD model is applied for option pricing, and the result is compared with prices derived by some existing models such as the Black-Scholes and Binomial tree models. Bayesian Nonparametric, Polya Tree, Hierarchical Bayes Model, Markov Chain Monte Carlo, Microstructure, Option Pricing
2
1 Introduction
We develop a model which is useful to investigate phenomenon in financial micro
data, in particular, equity data. The micro data such as TAQ (Trade and Quotes) record
all the transactions. The data include time (in seconds) of each transaction, the stock
price for each transaction, the trading volume, and so on. The model is called stationary,
continuous time, discrete space model, which is a class of stationary Markov processes.
The stationary, continuous time, discrete space model accounts for the price discreteness
and the continuous trading time with random intervals of the transactions in the micro
data. This model has two components; (i) a component which models the time between
two adjacent transactions, inter arrival time. (ii) a component which models how the
stock price move for the next time period.
A theorem for Markov processes assures that the inter arrival time in the first
component has the exponential distribution. Modeling the exponential distribution for
the inter arrival time by using hierarchical Bayes model is described in this paper.
For the second component, the transition probabilities of the stock price
movements are estimated by Polya trees, one of the Bayesian nonparametric
distributions. In the micro data, there are a large number of possible discrete values for
stock prices, states, in the model. Therefore, there is often a few or no observation for
some states, which gives us a problem in estimation. Bayesian nonparametic distribution
particularly works well to make these “thin data” (relative to the number of state)
smoother. Another advantage is that the transition probabilities estimated by Polya trees
can capture some of characteristics in financial data such as show assymmetricity and
excessive kurtosis.
3
As an application, we explore option pricing by the stationary, continuous time,
discrete space model and compare the performance of this model with some of the
existing models. The stationary, continuous time, discrete space model with Polya tree is
motivated by the binomial tree method introduced by Cox, Ross, and Rubinstein (1979).
The structure of the stationary continuous time discrete space model with Polya tree is
similar to that in the binomial tree model. However, there are two major differences.
First, while the stock price at a certain time simply moves either upward with probability
p or downward with probability 1-p for the next time period in the binomial model, the
stock price in the new model could move to various prices for the next time period. Here
the transition probability is estimated by Polya trees with historical data as mentioned
above. Note that this model allows us to have only discrete prices just like the quoted
prices in the market are discrete. Second, while each time step is fixed in the binomial
tree model, it is random in Polya tree model. This random interval, i.e. inter arrival time,
is drawn from the exponential distribution. These differences would lead us to improve
the performance in stock price modeling and option pricing.
The Black and Scholes (1973) formula is one of the few analytical formulas that
allow pricing of the derivative securities. However, there are some restrictive
assumptions in the derivation of the Black and Scholes formula. Some of the
assumptions are (1) volatility is constant; (2) the stock price has a lognormal distribution.
The removal of these assumptions for the stationary, continuous time, discrete space
model would allow us to correct the bias in option pricing cased by the stronger
assumptions in the Black-Scholes model. The new model also captures some of the
characteristics in the movement of underlying assets such as stochastic volatility and
4
jumps. The stationary, continuous time, discrete space model is very general in a sense
that other models such as Black-Scholes model, stochastic volatility model, and jump
model can be considered as subsets of this model.
This paper is organized as follows. Section 2 starts with description of Polya Tree.
The extension of the one-dimensional Polya Tree to two-dimensional bivariate Poly tree
is also explained in the section. In Section 3, we introduce the stationary continuous time
discrete model with Ploya tree. Section 4 shows some results when we apply the
stationary continuous time discrete model to stochastic modeling of stocks and option
pricing. INTEL stock price data for January 1998 retrieved from TAQ database is used
in this section. Section 5 concludes with future direction of the research.
2 Polya Tree
Freedman (1963) introduced the earliest priors for nonparametric problems. He
introduced tail-free and Dirichlet random measures. Dubins and Freedman (1965),
Fabius (1964), and Ferguson (1973, 1974) formalized and explored the notion of a
Dirichlet process. The Bayesian nonparametric literature has grown rapidly since the
work of Ferguson (1973).
The Polya tree distribution as a generalization of the Dirichlet distribution has
received much attention recently. Polya trees can be used to define probability
distributions on any space of interest Ω. The Polya trees rely on a binary partitioning of
the space Ω. We must define a partition of Ω, Π = , and a set of nonnegative real
numbers A =
τB
τγ , associated with each element of Π. We can then recursively define
the partitions by letting , be the set of level 1 partitions which exclusively splits 0B 1B
5
Ω. We define , the ‘offspring’ of , so that , , , and denote the
sets at level 2, and so on. In general a parent has ‘children’ and , where
∩ = ∅ and ∪ = . A good analogy in understanding this concept is that
of a particle cascading through the tree. The particle starts in Ω and moves into with
probability , or into with probability C = 1 -C . In general, on entering , the
particle can move to either or with probabilityC and C = 1- C , respectively.
For Polya trees, these probabilities are random quantities which follow Beta distributions,
such that C ∼ Beta(
00B
B
01B
0τ
B
0τ
0B
B
1
00B
0τ
01B 10B
0τ
1τ
11B
1τ
0τ
τ B B
0τB 1τB 1τB
1
B
τB
0
0B
τB0C
0τ
0
τ 1τB
γ , 1τγ ) and = 1- where 1τC 0τC 0τγ and 1τγ are both nonnegative.
The last stage of defining Polya tree distribution is defining the A = τγ . The formal
definition is given by Lavine (1992).
0γ 1γ 00γ 0 00 C10
0τ 0τγ 1τγ 1
m
τC 0τC ,
ττ ...1
−j 01τmB ττ ...1
0 )
−C1
1(
∏=
m
j j:1τ∏
=
m
j;1 τ−j...1 ττ,... 1
DEFINITION 2.1 (Lavine, 1992) (Univariate) Polya Tree Prior.
A random probability measure P on Ω is said to have a Polya tree distribution, or a
Polya tree prior, with parameter (Π, A), written P ∼ PT(Π, A), if there exist non-negative
numbers A = ( , , ,…) and random variables C = (C ,C , …) such that the
following hold:
(i) all the random variables in C are independent,
(ii) for every τ, C ∼ Beta( , ) and = 1-
(iii) for every m = 1,2,… and every τ = ,
P( ) = , == j
C0
1τ
6
where the first terms, that is, for j = 1, are interpreted as C and 1 -C . 0 0
In practice, we cannot work with an infinite number of levels, thus we use a
partially specified Polya tree distribution (Lavine, 1992), where the partitioning occurs up
to a fixed level M. The feasibility of the partially specified Polya tree distribution is
discussed by Lavine (1994). For the prior parameter, we often choose at
level m for some c>0. Ferguson (1974) shows that P is absolutely continuous with
probability 1 by choosing . The Dirichlet process arises when
2cmcm ==τγ
2mcm = mmcc
2= . The
random variables Θ are said to be a sample from P if, given P, they are i.i.d. with
distribution P. An important fact about Polya trees is that they are conjugate. If P has a
Polya tree distribution and | P ~ P, then P |
,...
Θ
, 21 Θ
Θ has a Polya tree distribution (Ferguson,
1974, Mauldin, Sudderth, and Williams, 1992). We can easily update a Polya tree after
observing Θ ; for every ε such that i iΘ ∈ , add 1 to τB τγ . We then call the new
parameters A | Θ .
Univariate Polya tree priors presented is generalized to bivariate Polya tree priors
as follows. Note that The Polya tree prior in ℜ was formally defined and developed by
Paddock (1999).
k
DEFINITION 2.2 Bivariate Polya Tree Prior.
A random probability measure P on Ω × Ω have a bivariate Polya tree distribution, or a
bivariate Polya tree prior, with parameter (Π, A), written P ~ BPT(Π, A), if there exist
non-negative numbers A = ( , , , , , , ,…) and random
variables C = ( , , ,…) such that the following hold:
00γ
01γ
1000C ,
10γ
1010
11γ ,00
00γ 0001γ 01
00γ 0101γ
00C ,00
00C 0010C C
7
(i) all the random variables in C are independent,
(ii) for every τ and υ, (C ,C ,C ) ~ and 00υτ
01υτ
10υτ ),,,|,,( 1
110
01
00
10
01
003
υτ
υτ
υτ
υτ
υτ
υτ
υτ γγγγCCCDi
= 1 – (C +C +C ) 11υτC 0
0υτ
01υτ
10υτ
(i) for every m = 1,2,…, every mτττ ...1= , and every mυυυ ...,1= ,
P( )=
where the first terms, that is, for j=1, are interpreted as C ,C ,C ,and
m
mB υυ
ττ......
1
1
=
−
−
j
j
jC
1,
......
11
11υ
υυττ
++−
∏∏∏∏===========
−
−
−
−
−
−
−
−
−
−
m
j
m
j
m
j
m
j jj
j
j
j
j
j
j
jjj
j
j
jj
j
jCCCCC
1,1:1
1...0...
0...1...
0...0...
0:1
10
0,1:1
0...1...
0,0:1
0...0... )](1[ 11
11
11
11
11
11
11
11
11
11υτ
υυττ
υυττ
υυττ
τυτ
υυττ
υτ
υυττ
00
01
10
1-(C + + ). 00
01C 1
0C
Similar to one-dimensional Polya trees defined in Chapter III, we must define a
partition of any two dimensional space of interest Ω × Ω, Π = , and a set of non-
negative real numbers A = , associated with each element of Π. Then we define the
partitions by letting be the set of level 1 partitions which exclusively
splits Ω × Ω (Figure 2.1). We define as the ‘offspring’ of .
Thus, , , , , , , , , , , , , , , ,
and denote the sets at level 2 (Figure 2.2).
υτB
1101B
υτγ
10, B
0010B
11
01
00 ,, BBB
0101B 00
11B
0101
0100
0001
0000 ,,, BBBB
0110B 01
11B 1000B 10
01B
00B
1110B00
00B
1111
0001B 01
00B 1100B 10
10B 1011B
B
In general, a parent has ‘children’ , , , and , where ∩
∩ ∩ = ∅ and ∪ ∪ ∪ = . Similar to the analogy of a
particle in one-dimensional Polya tree distribution, on entering , the particle can move
to , , , or with probability C , C , C , and , respectively.
υτB
00υτB
00υτB
11υτB
00υ
01υτB
υτB
01υ
10υτB
10υ
11υτB
υτ
11υτ
00υτB
01υτB
B
10υτB
υτB
11υτB
10υτB
01υτB 1
0υτB
B
C00υτ
01
11υτB τ τ τ
8
1
0B 11B
0
0B 01B
Figure 2.1. Partitions at level 1
11
00B 1101B 11
10B 1111B
10
00B 1100B 10
10B 1011B
01
00B 0101B 01
10B 0111B
00
00B 0001B 00
10B 0011B
Figure 2.2. Partition at level 2
Schervish (1995) generalizes theorems for conditions for which Polya trees will
yield absolutely continuous distributions with probability one.
Theorem 2.1 (Schervish,1995)
Let P be a random probability measure which follows a Polya tree prior. Suppose each
element of partition each element of Π has a positive measure with respect to a measure
ν and for every k, ∈ k
kB υυ
ττ......
1
1 kπ (the set of partition element at level k in a tree). Let
be such that for all k and all , E[ ]=ν( )/ν( ). If k
kC υυ
ττ......
1
1
k
kB υυ
ττ......
1
1
k
kC υυ
ττ......
1
1
k
kB υυ
ττ......
1
1
11
11
......
−
−
k
kB υυ
ττ
∞<∑∞
= ∈12...
...
......
])[(][
sup1
1
1
1
m B m
m
m
m
m CECVar
ννττ
ννττ
π
then with probability 1, P is absolutely continuous with respect to Lebesgue measure.
9
Paddock (1999) applies this theorem to Polya trees in . She verifies that
at level m for some constant c>0 satisfies the condition of the above
theorem.
kℜ
2cmcm ==υτγ
The posterior bivariate Polya tree distribution can be obtained in the same way as
the posterior univariate Polya tree distribution is obtained. Suppose we observe Θ . For
every τ and υ such that Θ ∈ , we add 1 to . We then call the new parameters A|
i
iυτB υ
τγ Θ .
For n observations, let Θ = ( nθθθ ...,, ,21 ). Then A|Θ is given by , where
is the number of observations in . Thus, the posterior predictive distribution,
P(
υτ
υτ
υτ γγ n+=
υτn υ
τB
1+nθ ∈ | ) is, for υτB Θ mυυυ ...1= and mτττ ...1= ,
P( 1+nθ ∈ | ) = υτB Θ
11
11
11
11
11
11
11
11
11
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
21
21
21
21
......
1...1...
1...0...
0...1...
0...0...
......
......
11
10
01
00
11
01
−
−
−
−
−
−
−
−
−
−++++
+•••
++++
+•
+ m
m
m
m
m
m
m
m
m
m
m
m
m
m
nn
nn
n υυττ
υυττ
υυττ
υυττ
υυττ
υυττ
υυττ
υτ
υτ
υτ
υτ
υτ
υυττ
υυττ
γγγγ
γ
γγγγ
γγ
γ 1
1
+
nυτ
γ
1
1
10+
+υτ
γ00 + γ
Paddock (1999) derives the marginal distribution of multivariate Polya tree priors.
Theorem 2.2 (Paddock,1999) Marginal Distributions of Polya Trees
Suppose P is Polya tree prior on , with parameters (Π, A), i.e. P ~ BPT(Π, A). Then
the j-dimensional marginal distribution of P is a Polya tree.
kℜ
As an example in , suppose P is a Polya tree. The partitions at level 1 are as in
Figure 2.3. Then the parameters at level 1 follow a Dirichlet distribution,
2ℜ
10
( C , ) ~ and C = 1 – ( ). If we
would like to find the one-dimensional marginal distribution of P with respect to the
horizontal axis of the square, we must sum over the parameters of the cells along the
vertical axis to find the parameters of the marginal distribution. Thus, the marginal
distribution of C would be ( ) ~ and C = 1 - C . That is,
( ) ~ and = 1 - C .
00C ,
00C
01
10C
Beta
),,,|,,( 11
10
01
00
10
01
003 γγγγCCCDi
00C |( 0
0001 γCDi
), 11
10
01 γγγ ++ 0
1C 00
11
, 10γ
00C + 0
1C +
01
10C
)11
01 γγ ++ 0
0
( 00γ
3 Stationary, Continuous Time, Discrete Space Model
This section describes the structure of stationary, continuous time, discrete space
model for financial micro data. The structure of the model is in the class of structure for
all stationary Markov processes.
We denote X as a Markov process with state space E and transition function ( ).
We assume that the state space E is discrete. The discrete space fits well the fact that
transaction prices are always quoted in discrete units in equities. We also assume that
t→X(t,ω) is right continuous for almost all T
tP
0 ∈Ω. We denote as the state (i.e. the
stock price or the stock price change in the equity application) after n changes in the
state. We let T the instant of nth transition for the process X and let the inter arrival
time which is time between t and the instant of the next change of the state. Then we
define
nS
n tA
0T =0; T = + , n1+n nTnTA ∈ Ν
nS =X(T ), nn ∈ Ν .
11
The path t→X(t,ω) can be described as follows: The process starts with time
(ω) = 0. At time 0, the process is in some state =X(0, ω). In terms of the equity
application, it can be stated that this initial stock price at time 0 is . Then the process
remains in this state for some positive time, and at time T (ω) it jumps to a new state or a
new stock price =X(T (ω), ω). Here, the inter arrival time is (ω) = T (ω) -T (ω).
The process remains in the new state for (ω). Then at some time T (ω) = T (ω) +
(ω), it jumps to a new state or a new stock price, =X(T (ω), ω); and so on. A
realization of this process is seen in Figure 3.1.
0T
1TA
0S
0S
0TA
1
1S 1 1 0
1TA 2 1
2S 2
)(ωtS
0 ω) ω) (ω) ω) (ω) time t
Figure 3.1. A realization of a Markov Process
0A 1T (1TA ( 2T
2TA ( 3T
12
; n∈nS Ν is a Markov chain, and the inter arrival time has an exponential
distribution with the parameter depending on . This result is derived form the
following theorem.
nTA
nS
Theorem 3.1
For any n∈ Ν , j∈E, and u∈ , we have +ℜ
uinnTn ejiQTTSSuAjXP
n
)(001 ),(,...;,...,|, λ−
+ =>=
if occurs. Here, Q(i , j)≥0, Q(i, i)=0, and iX n = ∑j
jiQ ),( =1.
The proof of this theorem is found in Çinlar (1975). This result yields the
following two corollaries.
Corollary 3.1
The sequences ,…of successive states visited form a Markov chain with the
transition matrix Q.
0S , 1S
Proof of Corollary 3.1
Letting u=0 in Theorem 3.1 yields the result. ڤ
In Q matrix, each row represents the current state or the current stock price while
each column represents the state or the stock price for the next time period. In section
3.3, we discuss the estimation procedure of the transition matrix Q by using Polya trees.
Corollary 3.2
13
For any n ∈ Ν , i E, and uni,...0 ∈ nu,...,1 ∈ +ℜ , we have
,,...|,..., 110011 nnnnnTT iSiSiSuAuAPn
===>> −− = nn uiui ee )()( 110 −−− ⋅⋅⋅ λλ
Proof of Corollary 3.2
From theorem 3.1, we have
uinnT ejSiSuAP
n
)(1 ),| λ−+ ===>
independent of j. by iteration, we can get the result in Corollary 3.2. ڤ
This corollary gives us the following two statements: (i) the inter arrival times
between transactions are conditionally independent of each other given the successive
states being visited. (ii) Each such inter arrival time has an exponential distribution with
the parameter dependent on the state visited.
For Markov processes, Çinlar (1975) provides the theorem which relates the
transition functions to the data Q(i, j) and λ(i). The proof is shown in Çinlar
(1975).
),( jiPt
Theorem D.1
For any i, j ∈E the function t→ is differentiable and the derivative is continuous.
At t=0 the derivative is
),( jiPt
≠=−
=jiifjiQijiifi
jiA),()(
)(),(
λλ
For arbitrary t≥0,
14
∑ ∑∈ ∈
==Ek Ek
ttt jkAkiPjkPkiAjiPdtd ),(),(),(),(),(
Here, A is called the generator of process X. This process says that if the
transition function ( ) is known, then we can determine Q and λ through A, the
derivative of at t=0. Conversely, if Q and λ are known, then ( ) can be computed by
solving the infinite system of differential equations:
tP
tP tP
tt APPdtd
= (Kolmogorov’s backward equation)
or
APPdtd
tt = (Kolmogorov’s forward equation)
If ( ) were a numerical-valued function, the solution of either would have been . In
general, we get a similar result with the interpretation of e ,
tP Ate
At
∑∞
=
=0 !n
nn
tA Ante .
3.1 Hierarchical Bayes Model
In financial micro data, it is often difficult to estimate the parameter of the
exponential distribution if we assume that the parameter is dependent on the state or the
stock price. This is because there is few or no observation for some states in such micro
data. To ease the difficulty, we suggest hierarchical Bayes (HB) model to estimate the
parameter in the exponential inter arrival time. In HB model, each parameter jλ for the
states for j =1,…,J is assumed to have a gamma distribution with parameters α and β.
Then, α and β are assumed to have an exponential distribution and a gamma distribution,
jS
15
respectively. It can be seen that the parameters jλ ’s for j = 1,…,J are linked through
parameters α and β. This structure of the HB model would provide us reasonable
estimates of jλ for states with even few observations.
α
λα )
β0 )
0e−
(, iα
,...)1+
iJ
)1( +λ
Data ]j
jn
More formally, the following three prior distributions are assumed:
(i) jλ has a gamma distribution with a probability density function,
[ jλ | α, β ] = jejβλαβ −−
Γ1
(.
(ii) β has a gamma distribution with a probability density function,
[β | ] = 00 , sr β000
10
(sr
r
er
S −−
Γ.
(iii) α has an exponential distribution with a probability density function,
[α | u ] = u . 0α0u
Markov Chain Monte Carlo (MCMC) procedure generates the sequence
for i = 1,2,…,M, where M is the number of iterations. Here,
is generated from [ ], a full conditional for
)())()(1 ,,...., ii
Ji βλλ
)1( +ijλ Dataii
j ,,| )()( βαλ jλ . is generated
from [ ], a full conditional for β. is generated from
[ ], a full conditional for α. These full conditionals can be
operated on their proportions:
)1( +iβ
Datai ,)(α
Data
iJ
i ,| )1((1 λλβ +
ii ,,...| )1()1(1
++ βλα
)1( +iα
,
[ ]j ,,| βαλ ∝ ][][][,,|[ jData λβαβαλ
∝ ∏=
−−−j
jijj
n
ij
tj ee
1
1 βαλ λλ
∝ ),|(1∑=
++i
ijjj tnGamma βαλ
16
[ ] DataJ ,,,...| αλλβ ∏=
∝J
jj
1
]][,|[ ββαλ
∝ ∏=
−−−−J
j
srj ee j
1
11 00][ ββλα βλ
∝ ) ∑=
++J
jjsJrGamma
100 ,|( λαβ
[ ] DataJ ,,,...| 1 βλλα ∏=
∝J
jj
1
]][,|[ αβαλ
α
α
α
βλ0
)(
)(1 u
J
J
jj
e−=
Γ
⋅∏∝
∝ ))((ln 0
1)(u
J
J
jj
e−⋅⋅
−∏
Γ =
βλα
α
In order to generate the sequence of , one can use a Metropolis-Hastings
algorithm. Once we generate the sequence of , we can analyze the outputs such as the
distribution of
)(iα
(λ )ij
jλ , the posterior mean of jλ , and the posterior standard deviation of jλ .
Appendix A provides the summary for the Metropolis-Hastings algorithm as well as the
MCMC algorithm.
3.2 Transition Matrix by Polya Trees
This section provides the discussion about estimation of the transition matrix Q in
Theorem 3.1 by Polya trees. Financial micro data are often “thin data” relative to the
number of states. That is, only small number of observations or even no observation can
be found for some states. In particular, elements away from the diagonal in the transition
matrix often have few or no observation because a stock price is more likely to jump to
17
the price which is close to the current price for the next time period. One of the
advantages of using Polya tree estimation is that it can make the “thin data” with bumps
in the probability distribution much smoother. Polya tree estimation is one of the
Bayesian nonparametric estimations which can do the smoothing. Furthermore, since
most financial data show assymmetricity and excessive kurtosis, by assuming that the
transition matrix has the Polya tree distributions (unlike assuming that the process is a
Brownian motion), one can capture these characteristics of the financial data.
How we construct the transition matrix Q by Polya trees is described below. First
we consider the case that the states are the stock prices. We use the bivariate Polya trees
defined in section 2 for this case. For a simple case, we assume that there are only two
states. In Figure 2.2, corresponds to Q(1,1), corresponds to Q(1,2),
corresponds to Q(2,1) and corresponds to Q(2,2). In practice, we typically
assume that the length of each partition is the same or smaller than a tick for stock prices.
This is because we have the sufficient number of levels for the bivariate Polya tree so that
it can provide us reasonable estimates. We then count the number of observations for
each cell in data. For the above two states example, if we observe that the stock price
changes from to , we add the number of observations in Q(1,2) element by 1. By
combining the number of observations for each cell with the prior distribution, we can
update the Polya tree and can obtain the posterior predictive distribution, which is the
estimate of Q. We note that the stock price always change in the transition and the
diagonal element, Q(i, i), must be set to be 0.
1100B 11
01B
1110B 11
11B
1S 2S
We next consider the case that the states are stock prices, that is, the transition of
the stock prices is independent of the stock prices. In this case, we can use the univariate
18
Polya tree which gives us a one-dimensional Polya tree vector. Each partition in a
univariate Polya tree corresponds to a price change. Again, by using the updating as
described in section 2, we can obtain the posterior predictive distribution for the
univariate Polya tree.
How we choose the prior parameter in Polya trees is an issue. As presented in
section 2, we often choose at level m for some c. We will discuss this
issue with INTEL stock data in section 4.
2cmcm ==τγ
4 Example
In section 3, stationary, continuous time, discrete space model with Polya tree was
defined. The empirical study for the model is conducted by using INTEL stock price data
for January and February of 1998 retrieved from TAQ database. The database recorded
all the transactions occurred during the time period. We can find the time (in seconds) of
the transactions along with the transaction prices and the trading volumes. This section
provides the results of this initial empirical study.
4.1 Parameter λ from Hierarchical Bayes Model via MCMC
The sequence for i = 1,…,2000 is generated by Markov
Chain Monte Carlo Method. Table 4.1 (for the case that states are stock prices) and
Table 4.2 (for the case that states are the price changes) provide the posterior means and
standard deviations of
)()()()(1 ,,,...., iii
Ji βαλλ
jλ , α, and β. In MCMC algorithm, we set that the number of
iterations, M, is 2000 and the transitory period, B, is 1000. So, when we calculate the
above statistics, we throw the first 1000 iterations away. Also, in the MCMC algorithm,
we use the prior parameter values, =1, =2, and u =1. For Metropolis-Hastings 0r 0s 0
19
algorithm, we set =0.1 for the first case and =0.25 for the second case. During the
period, the highest price and the lowest price were $22.125 and $18.125, respectively.
For the case that states are stock prices, J=64 is used with S =18.125 and S =22.125
while for the case that states are price changes, J=32 is used with =-1 and S =1. In
both cases, S - =1/16, which is the tick size for this stock. Appendix B shows the
sequences of some of the parameters, for i=1,…,2000.
2σ
nS
2σ
)( ,iJ α
λ ,64
1 64
321S
1+n
)()()(1 ,,...., iii βλλ
βαλ ,,....,1
λ
Table 4.1. Statistics for Parameters in HB Model (States = Stock
Prices)
j Price ($) Posterior Mean for j Posterior SD for jλ
1 18 2/16 0.5010 0.2715 2 18 3/16 0.6588 0.6101 3 18 4/16 0.6537 0.2407 4 18 5/16 1.1415 0.7474 5 18 6/16 0.5382 0.1784 6 18 7/16 0.7405 0.6233 7 18 8/16 1.0094 0.3772 8 18 9/16 0.7380 0.4841 9 18 10/16 0.6140 0.2313
10 18 11/16 0.6675 0.5777 11 18 12/16 1.0165 0.3708 12 18 13/16 0.6859 0.5547 13 18 14/16 0.4664 0.2485 14 18 15/16 0.6779 0.5452 15 19 1.2085 0.4281 16 19 1/16 0.7064 0.6183 17 19 2/16 0.2790 0.0773 18 19 3/16 0.3238 0.1416 19 19 4/16 1.0837 0.2676 20 19 5/16 0.9763 0.5213 21 19 6/16 0.2582 0.0751 22 19 7/16 0.5630 0.2387 23 19 8/16 1.4472 0.3017 24 19 9/16 0.1730 0.0793 25 19 10/16 0.5619 0.2077 26 19 11/16 0.7274 0.6210 27 19 12/16 0.3691 0.0899 28 19 13/16 0.1844 0.0699 29 19 14/16 0.5345 0.1475 30 19 15/16 0.6715 0.5688 31 20 0.3314 0.0734
20
32 20 1/16 1.2768 0.4782 33 20 2/16 0.5538 0.2253 34 20 3/16 1.2478 0.6687 35 20 4/16 0.4228 0.0877 36 20 5/16 1.0116 0.3571 37 20 6/16 0.2213 0.0390 38 20 7/16 0.1986 0.0523 39 20 8/16 0.2874 0.0444 40 20 9/16 0.4340 0.1569 41 20 10/16 0.2678 0.0639 42 20 11/16 0.2756 0.1103 43 20 12/16 0.2650 0.0529 44 20 13/16 0.1346 0.0749 45 20 14/16 0.5015 0.1528 46 20 15/16 0.8093 0.3804 47 21 2.1200 0.6457 48 21 1/16 0.6725 0.6052 49 21 2/16 0.1944 0.0448 50 21 3/16 0.3276 0.1503 51 21 4/16 0.1839 0.0345 52 21 5/16 0.4747 0.2575 53 21 6/16 0.3325 0.0689 54 21 7/16 0.6886 0.5814 55 21 8/16 0.3249 0.0931 56 21 9/16 0.9879 0.4807 57 21 10/16 0.6963 0.5855 58 21 11/16 0.6784 0.5705 59 21 12/16 0.4156 0.5705 60 21 13/16 0.6877 0.5868 61 21 14/16 1.5737 0.6274 62 21 15/16 0.6972 0.5967 63 22 1.5669 0.7803 64 22 1/16 0.5869 0.3832
Posterior Mean for 0α Posterior SD for 0α Posterior Mean for 0β Posterior SD for 0β
1.5609 0.3115 2.3974 0.6440
21
Table 4.2. Statistics for Parameters in HB Model (States = Stock
Price Changes)
βαλλ ,,,...., 641
j Price Change ($) Posterior Mean for jλ Posterior SD for jλ
1 -1 0.4165 0.4375 2 -15/16 0.4419 0.4736 3 -14./16 0.4489 0.4807 4 -13/16 0.4290 0.4576 5 -12/16 0.2491 0.1421 6 -11/16 0.4363 0.4519 7 -10/16 1.3055 0.7662 8 -9/16 0.4571 0.4991 9 -8/16 0.6466 0.2918
10 -7/16 0.4426 0.4666 11 -6/16 0.3995 0.0905 12 -5/16 0.2036 0.1019 13 -4/16 0.3535 0.0482 14 -3/16 0.4034 0.1104 15 -2/16 0.2752 0.0271 16 -1/16 0.2735 0.0377 17 1/16 0.1967 0.0289 18 2/16 0.2163 0.0227 19 3/16 0.2915 0.0681 20 4/16 0.2335 0.0336 21 5/16 0.4828 0.2711 22 6/16 0.1698 0.0378 23 7/16 0.1697 0.0395 24 8/16 0.3921 0.1881 25 9/16 0.1101 0.0623 26 10/16 0.1580 0.0929 27 11/16 0.1598 0.0957 28 12/16 0.4232 0.4321 29 13/16 0.4462 0.3695 30 14/16 0.3965 0.4492 31 15/16 0.4337 0.4534 32 1 0.4365 0.4780
Posterior Mean for 0α Posterior SD for 0α Posterior Mean for 0β Posterior SD for 0β
1.2988 0.4595 2.9208 1.1721
22
4.2 Transition Matrix Q
We apply the procedure of constructing Q matrix by bivariate Polya trees outlined
in section 3 for the INTEL stock data from January 22, 1998 until February 4, 1998. We
let S =18 and =26. We set that the prior parameter at level m. We set c, d,
and the number of levels as following: c=0.01, 1, 100; d=2; # of levels=6. The 3-D plots
of Q are shown in Appendix C. The increase in c tends to make the contour of Q
smoother. Also, in these plots, it can clearly be seen that the elements closer to the
diagonal of the Q matrix tend to have higher probabilities.
1 JS dm cmc =
Table 4.3 provides the transition probability for each state when the states are the
price changes. These transition probabilities are calculated from univariate Polya tree as
described in section 3. In the univariate Polya tree, we let the number of levels 5 with 32
states. We let = -1 and =15/16=0.9375 with - =1/16 for all n=1,…J-1. The
prior parameter in the Polya tree is set to be = cm at each level m. We calculate the
transition probabilities for c=0.1, 1, and 10. The plots are shown in Figure 4.1, where we
can see that some states, namely even sixteenth, have higher transition probabilities. We
also see that as c gets larger, the plot of the transition probabilities get smoother.
1S 32S 1+nS
2
nS
mc
23
Table 4.3.
Prior: c=0.1 Prior: c=1 Prior: c=10
States Price Changes
Transition Prob
Transition Prob
Transition Prob
1 -1 0.0004 0.0018 0.0086 2 -0.9375 0.0004 0.0018 0.0086 3 -0.875 0.0004 0.0018 0.0086 4 -0.8125 0.0004 0.0018 0.0086 5 -0.75 0.0027 0.0028 0.0091 6 -0.6875 0.0015 0.0026 0.009 7 -0.625 0.0038 0.003 0.0092 8 -0.5625 0.0017 0.0027 0.0091 9 -0.5 0.0077 0.0122 0.0292
10 -0.4375 0.003 0.0106 0.0287 11 -0.375 0.0338 0.0256 0.0329 12 -0.3125 0.0091 0.0166 0.031 13 -0.25 0.1079 0.0955 0.0734 14 -0.1875 0.0282 0.0453 0.0635 15 -0.125 0.2051 0.1814 0.1026 16 -0.0625 0.117 0.1175 0.0897 17 0 0 0 0 18 0.0625 0.0865 0.0935 0.1001 19 0.125 0.1719 0.1398 0.0724 20 0.1875 0.0363 0.0521 0.0575 21 0.25 0.0926 0.0688 0.0457 22 0.3125 0.0084 0.0258 0.0387 23 0.375 0.0368 0.0325 0.0375 24 0.4375 0.0043 0.0185 0.0348 25 0.5 0.0254 0.0131 0.0132 26 0.5625 0.0065 0.0088 0.0126 27 0.625 0.0014 0.0053 0.0116 28 0.6875 0.0014 0.0053 0.0116 29 0.75 0.0025 0.0038 0.0107 30 0.8125 0.0014 0.0035 0.0106 31 0.875 0.0009 0.0032 0.0105 32 0.9375 0.0009 0.0032 0.0105
24
Figure 4.1. Transition Probability by Univariate Polya Trees for INTEL data
Transition Probability for Price Change, Prior c=0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
-1.5 -1 -0.5 0 0.5 1 1.5
Price Change
Post
erio
r Pro
babi
lity
Transition Probability for Price Changes, Prior c=1
-0.05
0
0.05
0.1
0.15
0.2
-1.5 -1 -0.5 0 0.5 1 1.5
Price Change
Post
erio
t Pro
babi
lity
Transition Probability for Price Changes, Prior c=10
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
-1.5 -1 -0.5 0 0.5 1 1.5
Price Change
Post
erio
r Pro
babi
lity
25
4.3 Monte Carlo Simulation and Option Pricing
Sample paths for INTEL stock prices are generated by Monte Carlo simulation.
The INTEL stock data from January 22 until February 4 retrieved from TAQ database are
used to estimate the parameters in our model and create the sample paths. We assume that
the process is followed by stationary, continuous time, discrete model with univariate
Polya tree. So, we assume that the transition probabilities are independent of the stock
prices, and the states in the model are assumed to be the stock price changes. The inter
arrival times between two adjacent transactions are drawn from the one parameter model
of the exponential distribution. More specifically, the inter arrival times are generated by
the random number generator for the exponential distribution with =0.2509. This value
for is the reciprocal of the mean inter arrival time during this period, which is the
maximum likelihood estimate of the exponential distribution. We note that these inter
arrival times generated by the simulation are random. Once the inter arrival time is
drawn and the next transaction time is determined, the price change (and therefore the
new stock price for the next time period) is drawn from the transition probabilities
generated by the univariate Polya tree. We set that the number of levels in the Polya tree
is six. The prior parameter is = at each level m with c=0.01, 0.1, 1, and 10. In
this simulation, we suppose that the initial stock price is $22. Once the stock price for the
next period is determined, we again generate the inter arrival time from the exponential
distribution and fix the next transaction time. We repeat this process until the summation
of the inter arrival times hit T=390 minutes. For each value of the prior parameter c, 100
sample paths are generated. Then, 100 terminal stock prices at time T are created for
each value of c. Means and variances of the terminal stock prices for c=0.01, 0.1, 1, and
λ
λ
mc 2cm
26
10 are shown in Table 4.4. We see that the variance gets larger as c gets larger. Some of
the sample paths are provided in Figure 4.2. These sample paths show us that two
distinct characteristics different from sample paths created by assuming that the process
is a Brownian motion. First, the sample paths by the stationary, continuous time, discrete
space model appear to show changes in volatility throughout the time period. Second, the
paths created by this model have the characteristics that the stock price occasionally
exhibit jumps.
Table 4.4.
Terminal Stock Price at T Mean Variance
c=0.01 25.1566 2.5973 c=0.1 24.9615 2.8257 c=1 24.9476 6.2185
c=10 23.6263 24.5438
27
Figure 4.2. Sample Paths for INTEL Stock Prices
28
Figure 4.6. Sample Paths for INTEL Stock Prices (Continued)
In order to find the European option prices, we adjust the terminal stock prices
generated under the real world to the terminal stock prices under the risk neutral world.
From the terminal prices created above, the expected value of the implied growth rate tµ
can be found. Then by using ErE tt +=][µ [Risk Premium], we find the expected risk
premium. Here, we assume that the risk-free rate is r =0.0524 which is the Treasury Bill
rate in January, 1998. For our INTEL example, E[Risk Premium] for the model with
c=0.01, 0.1,1, and 10 are 0.1433, 0.1354, 0.1359, and 0.0743. We then subtract the
excess prices added by the expected risk premium from the terminal stock prices under
the real world. With these adjusted terminal prices, European call and put options with
strike prices K=18, 19, 20, 21, 22, 23, 24, 25, 26 and 27 are priced. First, we find the
expected payoff of European call options, E[max( -K), 0] and the expected payoff of
European put options, E[max(K- ), 0]. Then, the payoffs are discounted with the risk-
free rate. The results are found in Table 4.5 and Table 4.6. Option Prices by stationary,
continuous time, discrete model are compared with those by Black-Scholes Formula and
t
TS
TS
29
Binomial tree model with 30 steps. In Black-Scholes formula, the volatility is assumed to
be equal to the historical volatility during the period, = 0.4068. In case of European
call (put) options, for the higher (low) strike prices, the option prices derived by
stationary, continuous time, discrete space model get higher than the prices calculated by
Black-Scholes and Binomial tree. The fact that the stationary, continuous time, discrete
space model allows the paths to have jumps would be one explanation of this result.
2σ
Table 4.5. European Call Option Prices by Three Models
(In the table, SCD model stands for stationary, continuous time, discrete model)
Strike Price Black –Scholes
Binomial Tree
SCD model with c=0.01
SCD model with c=0.1
SCD model with c=1
SCD model with c=10
18 4.0037 4.004 4.0065 4.0058 4.0450 4.5970 19 3.0039 3.004 3.0245 3.0439 3.1327 3.7579 20 2.0066 2.006 2.0791 2.1230 2.3124 3.0051 21 1.0566 1.056 1.2679 1.2950 1.5778 2.3661 22 0.3549 0.352 0.6368 0.6641 0.9859 1.8530 23 0.0619 0.060 0.2728 0.2797 0.5813 1.4431 24 0.0050 0.004 0.0853 0.0909 0.3163 1.0990 25 0.0002 0.000 0.0051 0.0111 0.1760 0.8126 26 0.0000 0.000 0.0000 0.0000 0.0832 0.5796 27 0.0000 0.000 0.0000 0.0000 0.0255 0.3831
Table 4.6. European Put Option Prices by Three Models
(In the table, SCD model stands for stationary, continuous time, discrete model)
Strike Price Black -Scholes
Binomial Tree
SCD model with c=0.01
SCD model with c=0.1
SCD model with c=1
SCD model with c=10
18 0.0000 0.000 0.0028 0.0021 0.0413 0.5932 19 0.0000 0.000 0.0206 0.0400 0.1286 0.7539 20 0.0025 0.002 0.0749 0.1189 0.3082 1.0009 21 0.0522 0.052 0.2636 0.2906 0.5735 1.3617 22 0.3503 0.347 0.6322 0.6596 0.9814 1.8485 23 1.0571 1.055 1.2680 1.2749 1.5765 2.4384 24 2.0001 2.000 2.0803 2.0859 2.3113 3.0935 25 2.9950 2.995 2.9999 3.0058 3.1708 3.8074 26 3.9946 3.995 3.9946 3.9946 4.0778 4.5742 27 4.9948 4.995 4.9944 4.9944 5.0199 5.3775
30
How we choose the prior parameter is an important issue. In the previous section,
we price the European call and put options by the stationary continuous time model with
Polya trees with 4 different prior parameters c=0.01, c=0.1, c=1, and c=10. This section
is devoted to discuss the procedure of combining the option prices by the different prior
parameters.
First, we suppose that there is the finite number of c’s for the prior parameters that
we would like to use. With the finite number of c’s, the expected value of option is
calculated by the total probability formula,
]|[]|[][ DatacPcOptionEOptionEc
⋅= ∑ ,
where
∑ ⋅⋅
=
c
cPcDataPcPcDataPDatacP
][]|[][]|[]|[ and ∑
c
cP ][ =1.
P[c] is considered as a weight for each parameter. If we do not have any prior
knowledge, it would be reasonable for us to assign the equal weight for each parameter c.
]|[ DatacP is equal to:
]|)([1
1
1cBPE
m
m
mall
n∏⋅⋅⋅=
⋅⋅⋅⋅⋅⋅
τττττ
ττ = ][][][1
1
1
21
21
21
1
11 ∏∏∏
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
m
m
mall
n
all
n
all
n CECECEττ
ττττ
τττ
ττττττ
Because these capital C’s have beta distributions by the definition of Polya tree, we have:
At level 1,
)12()1()1(
)1()1()12(][
102
12
02
22
2
1010
nncncnc
cccCCE nn
++⋅Γ+⋅Γ⋅+⋅Γ
⋅⋅Γ⋅⋅Γ
⋅Γ= ,
at level 2,
)22()2()2(
)2()2()22(][
01002
012
002
22
2
01000100
nncncnc
cccCCE nn
++⋅Γ+⋅Γ⋅+⋅Γ
⋅⋅Γ⋅⋅Γ
⋅Γ=
31
and so on. Here the prior parameter is c = cm at each level m. We calculate
] for all the partitions in the Polya tree and multiply these expected
values in order to find . Operating on the logarithm scale would provide us
numerical stability:
m2
11
1
01
1 10[ j
j
j
j
nn CCE ττττ
ττττ⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅ ⋅
]|[ DatacP
∑ ⋅⋅
= −
−
c
MAXcDataP
MAXcDataP
cPecPeDatacP
)()(]|[ ])|[log(
])|[log(
,
where MAX = max . Table 4.7 shows with various c’s
ranging form 0.001 to 10. We note that ≈0 for c < 0.01 and c>0.1.
))|(log( cDataPc
]|[ DatacP
]|[ DatacP
Table 4.7. P[C|Data]
C P(c) P(c|Data) 0.001 0.0588 0
0.0025 0.0588 0 0.005 0.0588 0
0.0075 0.0588 0 0.01 0.0588 0.0004 0.025 0.0588 0.5405 0.05 0.0588 0.4425 0.075 0.0588 0.0163 0.1 0.0588 0.0002
0.25 0.0588 0 0.5 0.0588 0
0.75 0.0588 0 1 0.0588 0
2.5 0.0588 0 5 0.0588 0
7.5 0.0588 0 10 0.0588 0
Table 4.8 provides with 4 possible c’s, c = 0.01, 0.1, 1, and 10. Four
different possible combinations of the weights P[c] are given. With these combinations
]|[ DatacP
32
of weights, the option prices are revised by combining the prices for the 4 possible prior
parameter c’s. Table 4.9 and Table 4.10 summarize the result.
Table 4.8. P[C|Data]
(1) Weight 1
C P[c] P[c|Data] 0.01 0.25 0.6457 0.1 0.25 0.3543 1 0.25 0 10 0.25 0
(2) Weight 2
C P[c] P[c|Data] 0.01 0.25 0.4767 0.1 0.5 0.5233 1 0.125 0 10 0.125 0
(3) Weight 3
C P[c] P[c|Data] 0.01 0.5 0.7847 0.1 0.25 0.2153 1 0.125 0 10 0.125 0
(4) Weight 4
C P[c] P[c|Data] 0.01 0 0 0.1 0 0 1 0.5 1 10 0.5 0
33
Table 4.9 European Call Option Prices for Combined SCD models
(In the table, SCD model stands for stationary, continuous time, discrete model)
Strike Price Black -Scholes
Binomial Tree
Weight 1: Combined
SCD model
Weight 2: Combined
SCD model
Weight 3: Combined
SCD model
Weight 4: Combined
SCD model 18 4.0037 4.004 4.0063 4.0061 4.0064 4.0450 19 3.0039 3.004 3.0314 3.0347 3.0287 3.1327 20 2.0066 2.006 2.0947 2.1021 2.0885 2.3124 21 1.0566 1.056 1.2775 1.2821 1.2737 1.5778 22 0.3549 0.352 0.6465 0.6511 0.6427 0.9859 23 0.0619 0.060 0.2753 0.2764 0.2743 0.5813 24 0.0050 0.004 0.0873 0.0882 0.0865 0.3163 25 0.0002 0.000 0.0072 0.0082 0.0064 0.1760 26 0.0000 0.000 0.0000 0.0000 0.0000 0.0832 27 0.0000 0.000 0.0000 0.0000 0.0000 0.0255
Table 4.10. European Put Option Prices for Combined SCD models
(In the table, SCD model stands for stationary, continuous time, discrete model)
Strike Price Black -Scholes
Binomial Tree
Weight 1: Combined
SCD model
Weight 2: Combined
SCD model
Weight 3: Combined
SCD model
Weight 4: Combined
SCD model 18 0.0000 0.000 0.0025 0.0024 0.0026 0.0413 19 0.0000 0.000 0.0274 0.0307 0.0247 0.1286 20 0.0025 0.002 0.0905 0.0979 0.0844 0.3082 21 0.0522 0.052 0.2731 0.2777 0.2694 0.5735 22 0.3503 0.347 0.6419 0.6465 0.6381 0.9814 23 1.0571 1.055 1.2705 1.2716 1.2695 1.5765 24 2.0001 2.000 2.0823 2.0832 2.0815 2.3113 25 2.9950 2.995 3.0020 3.0030 3.0012 3.1708 26 3.9946 3.995 3.9946 3.9946 3.9946 4.0778 27 4.9948 4.995 4.9944 4.9944 4.9944 5.0199
34
5 Discussion
This paper illustrates the stationary, continuous time, discrete space (SCD) model.
The SCD model, a class of stationary Markov processes, is useful to investigate
phenomenon in financial micro data. The paper highlights the usefulness of the Polya
tree estimation in financial micro data. It induces smoothing of the micro data or the
“thin data” where there is few or no observation for some of the large number of states.
The SCD model is so general that it can simulate changes in volatility, jumps, excess
kurtosis, skewness which commonly appear in financial data. Finally, the paper presents
an example by using Intel tick by tick data. Option prices derived by SCD model and
some existing models such as the Black-Scholes and Binomial tree models are compared.
We will state further research in this chapter. The result shows that the option prices
derived by SCD model tends to overestimate the prices by standard Black-Sholes model
when the strike price is away from at-the-money.
For future direction of this research, we would extend the example to some other
data than INTEL stock data and other time periods than January, 1998 will be
investigated. In section 4, the sample paths are generated by assuming that the process is
followed by stationary, continuous time, discrete space model with univariate Polya tree
and one common parameter in the exponential distribution of inter arrival time. We will
use the model with bivariate Polya tree and the model with hierarchical Bayes (HB)
model in the exponential inter arrival time to create sample paths and price European
options.
35
6 Appendices
A.1 Markov Chain Monte Carlo (MCMC) Algorithm
The following is the MCMC algorithm which generates the sequences of for i = 1,2,…,M, where M is the number of iterations: )()()()(
1 ,,,...., iiiJ
i βαλλ(1) Initialize the prior parameters, , , and . 0r 0s 0u(2) Initialize MCMC parameters, M (# of iterations) and B(transitory period). (3) Set the number of states J. (4) Define Matrix for , where i=1,…,M. )()()()(
1 ,,,...., iiiJ
i βαλλ(5) Read in the data, , , and for j=1,…,J. jn jt ijt
(6) Initialize the starting values for . βαλλ ,,,....,1 J
For example, . 1,1,1 )()()( === iiij andβαλ
(7) For i =1 to M, generate , , and from [ ], )(ijλ
)(iβ )(iα Dataiiij ,,| )1()1()( −− βαλ
[ ], and [ ]. Save , , and DataiiJ
ii ,,...,| )1()()()( −αλλβ DataiiJ
ii ,,,...| )()()(1
)( βλλα )(ijλ
)(iβ
. )(iα (8) Do output analysis. For example,
Posterior Mean of λ = ∑+=−
M
Bi
ijBM 1
)(1 λ and Posterior Standard deviation of
λ = ∑+=
−−
M
Bij
ijBM 1
2)( )(1 λλ .
A.2 Metropolis-Hastings Algorithm
)(iα
|α
can be generated by using Metropolis-Hastings algorithm. As in section 3.2.2,
[ ] . Let DataJ ,,,...1 βλλ))((ln 0
1)(u
J
J
jj
e−⋅⋅
−∏
Γ∝ =
βλα
α )(απ = . ))((ln 0
1)(u
J
J
jj
e−⋅⋅
−∏
Γ =
βλα
α
(1) Initialize the value of . 2σ(2) Set , the current value of α on iteration i. )(
0iαα =
(3) Generate candidate cα from g( · | 0α ), where g( · | 0α ) = Normal(log cα | log 0α , . That is, the probability distribution
function of g is
)2σ2
222
1 α
απσ
−− c
ec
0 )logα2 (log1σ .
(4) Find P = min )|()()|()(
00
0
αααπαααπ
c
cc
gg⋅⋅
,1
36
(5) Accept cα on iteration i+1 with probability P and set =)1( +iα cα . Keep 0α on iteration i+1 with probability 1-P and set = )1( +iα 0α .
B Sequence of Parameters Generated by MCMC for INTEL data
B.1 Parameters v.s. Iterations (States = Stock Prices)
37
38
B.2 Parameters v.s. Iterations (States = Price Changes)
39
40
41
C 3-D Plots for Q Matrix for INTEL Data (x axis = State i, y axis = State j, and z axis = Posterior Probability) C.1 c=0.01, d=2, and 6 levels
C.2 c=1, d=2, and 6 levels
42
C.3 c=100, d=2, and 6 levels
43
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