1

An introduction to Taiwan market

by using Markov model• models • examples • the kernel density estimation

2

股票

價格

ρ ＋ (x)

ρ － (x)

ρ ＋ (x1)

ρ － (x1)

ρ ＋ (x2)

ρ － (x2)

第一天

收盤價

第二天

收盤價

第三天

收盤價

紅線 :St

黑線 :Xt

3

Markov model {Xn}

)}(or )(:0inf{

where

),,(),(

)(),( 1

xxSt

dyxpdyxp

dySPdyxp

t

dc

x

4

St ：表示沒有上下限的股票價格且它的 generator 是

xxb

xxL

)( )(2

12

22 ＋σ

5

股票價格

第一天收盤價

)(x

)(x

開盤價 x

Xt

St

6

股票價格

)(x

)(x

開盤價 xXt St

第一天收盤價

7

)(x

)(x

開盤價 xSt≡Xt

第一天收盤價

股票價格

8

Examples

• Binary model (Cox-Ross-Rubinstein)

.)()(

)())(,(

,)()(

)())(,(

xx

xxxxp

xx

xxxxp

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Binary model

• {Xn} is a martingale

• If the state space of {Xn} is R+,then lim Xn =0.

• If the state space of {Xn} is R,then {Xn} is recurrent.

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European call

)(1|1

),())(()())(()(

Then

.)()(Let

nnNnN

nNxnN

n XfTrFXfErC

xxfxxfxTf

Kxxf

11

.ˆ clear that isIt .)1(ˆ fact,In

)(ˆ)(ˆ)(ˆ

))(ˆ,(,)(ˆ)(ˆ

)(ˆ))(ˆ,(

by ˆ Define

,0for )1()1()(ˆ),1()1()(ˆLet

))(,(,))(,(

by define ),,0(Let

,0for )1()(,)1()(let ,0,0Fix

1

11

xXEXrX

xx

xxxxp

xx

xxxxp

X

xaxrxbxrx

ab

rbxxp

ab

arxxp

Xbr

xxaxxbxba

xnn

n

n

n

Example

12

nN

j

jnNjjnNjnN

nnNxnN

n

KbaxppjjnN

nNr

n,xc

Xn,cFKXErC

n

xxaxxbx

ba

0

)(

)(

)1()1()1(!)!(

)!(1

)(

where

),ˆ( toequal is |)ˆ()1(

at time callEuropean of price Then the

,0for )1()(,)1()(

let ,0,0Fix

European call

ab

rbp

13

.for

),,0(for (x)

)(1Esup)(1E(x)

.,),0(:0inf and 1

Let

.1 ofsolution a is 1 Suppose

,0for )(,)(

xx

1

xKx

xxK

KXrKXr

XnK

rqepe

xxexxex

n

tt

Optimal stopping time

14

Kernel density estimation

a.e. )( )()(1

)(

i.i.d. ,, Suppose

1

21

nxFxXIn

xF

XXn

iin

a.e. )( )()(1

)(1

nxFh

XxK

nhxF

n

i n

i

nn

15

).1,()(

)()(),(

,),(),(

by defined is

).1,()(

1,for )(,)(

1,for )(,)(

)1,0(,,1,1 Assume

.0)(limLet

)(

2

x

x

xd

c

n

x

PxQ

dyxQdyxp

dyyxqdyxp

X

PxQ

xcxxdxx

xaxxbxx

dacb

xmx

Positive recurrent Markov chain

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Steps of the kernel density estimation

• estimate q(x,y) such that pc(x,y)=q(x,y)dy

• estimate Q+(x) such that pd(x,dy)=Q+(x)δρ+(dy)

• estimate Q-(x) such that pd(x,dy)=Q-(x)δρ-(dy)

• define estimatorsnn ˆ,ˆ

17

5

1

1

,)(1

)(ˆ

Let

nh

h

XxK

nhx

n

n

i n

i

nn

Kernel density estimator

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)( if )(

,)(),( if ),(),(

,),(1

)(ˆ

Let1

0

xyxQ

xxyyxqyxk

yXkn

yn

iin

Another density estimator

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20

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