[PDF] Une approche mathématique de linvestissement boursier





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Une approche mathématique de linvestissement boursier

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Preferimos realizar uma busca por produções a. N investir na formação continuada que lhe proporcione ampliar o seu repertório.

45.4

45.745.8

80 9010070shares shares shares shares?????? ???? ??????? ?? ?????? ???????

U U

T(v) =TX

i=0vipi+pTT X i=0v iTX i=0jvipij bYn+1=bY1?Xn+12CX1+bY2?Xn+12CX2? bY???

Y=X+; N(0;2)

b =argmin(jjYXjj22) b

Ridge=argmin(jjYXjj22+Ridgejjjj22)

b

Lasso=argmin(jjYXjj22+Lassojjjj1)

b

EN=argmin(jjYXjj22+EN1jjjj1+EN2jjjj22)

M M M L L L C C C M M

0buyL0sellC0buyC0sellM0buyM0sellL1buyL1sellC1buyC1sellM1buyM1selljO????? ????? ????? ????? ???? ???? ???? ???? ???? ???? ???? ????

jO1????? ????? ???? ????? ????? ?????

0buyL0sellC0buyC0sellM0buyM0sellL1buyL1sellC1buyC1sellM1buyM1selljL0buy? ? ? ? ? ? ? ? ? ? ? ?

jL0sell? ? ? ? ? ? ? ? ? ? ? ? jC0buy? ? ? ? ? ? ? ? ? ? ? ? jC0sell? ? ? ? ? ? ? ? ? ? ? ? jM0buy? ? ? ?????? ? ??? jM0sell? ? ? ? ?????? ? ?? jL1buy? ? ? ? ? ? ? ??? ?? jL1sell? ? ? ? ? ? ? ? ?? ??? jC1buy? ? ? ? ? ??? ? ? ? ? jC1sell? ? ? ? ? ? ??? ? ? ? jM1buy? ? ? ? ? ?? ?? ? ? ? Cr h(i;j) =Correlation(Ni(t+h+)Ni(t+);Nj(t+h)Nj(t)) Cr m(t) =m+MX n=1X T i1buyfM0buy;L1buy;M1buy;M1sellg L

1sellfM0sell;L1sell;M1buy;M1sellg

C

1buyfL1buyg

C

1sellfL1sellg

M

1buyfM0buy;M1buyg

M E ?v? ? ?????? ??????? 45.4

45.745.8

U

1=v0(p1p0)jv0jp0jv1v0jp1

U

1=v0p0v1p1+v1p1+v0p1jv0p0j jv1p1j

U

T(v) =TX

i=0vipi+pTT X i=0v iTX i=0jvipij ??????T?PT i=0vi= 0? J (v) =TX i=0(v+ipaski+vipbidi) +TX i=0(v+ipaskivipbidi) PT J (v) =TX i=0K1X j=0(vj+ ipaskij+vj ipbidij) +TX i=0K1X j=0(vj+ ipaskijvj ipbidij) ?bidQijvj ?0vj+ PT1 ?v+i=PK1 j=0vj+ ?vi=PK1 j=0vj ?cTx ?Gxh ?cT=pask0(1 +);:::;paskT1(1 +);pbid0(1);:::;pbidT1(1) ?xT=v+T;vT ?G=2 6

666664I O

I O O I OI L L LL3 7

777775

?x2R2T ?G2R6TR2T ?cTx ?Gxh ?Ax=o ?cT=pask0(1 +);:::;paskT1(1 +);pbid0(1);:::;pbidT1(1);o ?xT=v+T;vT;vT ?G=2 6

666664I O O

I O O O I O OI O O O I O OI3 7

777775

?A= [I;I;] ?=2 6

66666641 0 00

11 0??????

0

00 113

7

7777775

?x2R3T ?G2R6TR3T ?A2RTR3T

O(T2)??O(T)?

U

ClosePriceOpenPriceOpenPrice

??? ??????? ???? ?? ???? ??? ?????? ????? ??? ???? ??? ????? ?? ??? ????? ???? ?? ????t??? 1+Q2? T1+Q1 +Q2 t)? ?? ??? ????t? tdt)? ??? ?????? ??????? ??? ??? ?? ??? Liq bid=P5 ask)? ???????EMA(d;X)?? ?????d?? ?????? ??EMA(d;X)t0=Xt0??? ???t1i? EMA(d;X)ti=!Xti+ (1!)EMA(d;X)ti1? ?????!=min(1;titi1d

EMA?????? ???

?????? ???N(E;pn ??????? ??? ??(Epn ) = (0:62) = 26:5%? ?? ???? ????? ???? ??? ??? ?? ? ????? ??????? C M i;j

T=card(f(Xtn2CXi;Ytn2CYj)g)

Y tn=Ln(S tn+1S ????? ???? ? ??????? ???? ??50:30%? ?????? ?????P(A=A) = 48:02%?P(B=A) = 51:98%? ??? ????? ????50%?????P(B=A)???P(A=B)??? ?????? ????50%? C c c Y1=( +1??XT2CX1

1??XT2CX2

c Y2=(

E(YjY2CY2\X2CX1)??XT2CX1

E(YjY2CY1\X2CX2)??XT2CX2

c Y3=(

E(YjX2CX1)??XT2CX1

E(YjX2CX2)??XT2CX2

1?Q 2???Q 3? ????k= 4? ????? ??? ???????G1;G2;G3???G4?? ??? ??????? ??? ??? ????? ?? ??? ??????? ?G2+G32 ???G3+G42

Y=X+; N(0;2)

XiX i(Xi)?? ????? ??Xi?? ?? ???????b??? ?? ?????? ?? b =argmin(J=jjYXjj22) J = 2tXX2tXY 2J = 2tXX

E(bjX) = (tXX)1tXE(YjX) = (tXX)1tXX=

V ar(bjX) = (tXX)1tXV ar(YjX)X(tXX)1= (tXX)1tX2IX(tXX)1=2(tXX)1 E(jjbjj22jX) =E(tYX(tXX)2tXYjX)) =Trace(X(tXX)2tX2I) +jjjj22=2Trace((tXX)1) +jjjj22 MSE(b) =E(jjbjj22jX) =E(jjbjj22jX) jjjj22=2Trace((tXX)1) =2X1 i ????? ??? ??????? ??? ???? ??????? ????? ???()i?????? ??? ????? ?????? ??tXX? ?????? bY=Xb?? ???? ?? ??????? ??? ??????? ?? ???? ?? ??????? ??

1:000 2:000

3:000 5:999

x y =4:000

11:999

=>x y =2:000 1:000 ???? ?????? ? ????? ?????? ?? ??? ??????A?

1:001 2:000

3:000 5:999

x y =4:000

11:999

=>x y =0:400 2:200 ???? ?????? ? ????? ?????? ?? ??? ??????B?

1:000 2:000

3:000 5:999

x y =4:001

11:999

=>x y =3:999 4:000 ? ??K(A) =jjAjj2jjA1jj2? ?????jj:jj2??????? ???l2????? ?????? ??? ? ??????X??jjXjj2=pt XX??? ??? ? ??????A??jjAjj2= maxjjXjj26=0jjAXjj2jjXjj2? ??? ???AX=B???A(X+X) =B+B????jjXjj2jjXjj2K(A)jjBjj2jjBjj2 ???AX=B???(A+A)(X+X) =B????jjXjj2jjX+Xjj2K(A)jjAjj2jjAjj2 ??? ???A2Rp;p?X2Rp????jjXjj26= 0? =>jjAXjj2 jjAjj2jjXjj2??? ??? ???A2Rp;p?B2Rp;p?

jjABjj2= maxjjXjj26=0jjABXjj2jjXjj2= maxjjBXjj26=0jjABXjj2jjBXjj2jjBXjj2jjXjj2maxjjYjj26=0jjAYjj2jjYjj2maxjjXjj26=0jjBXjj2jjXjj2=jjAjj2jjBjj2

=>jjABjj2 jjAjj2jjBjj2??? ???A;B;X???? ????AX=B??? ???A(X+X) =B+B??? ???? ??? ??? ???X=A1B??? ????? ???jjXjj2=jjA1Bjj2 jjA1jj2jjBjj2 ???? ???jjBjj2=jjAXjj??? ????? ???jjBjj2 jjAjj2jjXjj2??? ???? ??? ??? ????jjXjj2jjBjj2 jjA1jj2jjBjj2jjAjj2jjXjj2 jjXjj2jjXjj2K(A)jjBjj2jjBjj2 ???A;B;X???? ????AX=B??? ???(A+A)(X+X) =B??? ???? ??? ??? ????X=A1A(X+X)? ????? ??? ??? ??? ???????jjXjj2 jjA1jj2jjAjj2jjX+Xjj2 jjXjj2jjX+Xjj2K(A)jjAjj2jjAjj2 c =argmin(jjYXjj22+jjjj22) b

R=argmin(jjYXjj22+Rjjjj22)

b

R= (tXX+RI)1tXY=Zb where Z= (I+R(tXX)1)1=W1

E(bRjX) =E(ZRbjX) =Z

V ar(bRjX) =V ar(ZbjX) =2Z(tXX)1tZ

MSE(bR) =E(t(Zb)(Zb)jX) =E(ttZZjX)2tZ+t

=Trace(tZZ2(tXX)1) +ttZZ2tZ+t (tXX)1=Z1I R

IZ= (IZ)WW1= (WI)W1=R(tXX)1W1=R(WtXX)1=R(tXX+RI)1

MSE(bR) =Trace(2Z

RZ)Trace(2Z

RZ2) +t(IZ)2

=2Xi(i+R)2+2Rt(tXX+RI)2 ?????? ??K(tXX+RI) =max+R min+Rmax b

EN=argmin(jjYXjj22+EN1jjjj1+EN2jjjj22)

M M M L L L C C C M M buyLsellLC buyCsellCM buyMsellMO 0L1LC 0C1CM 0M1MO

1jO1C1jO1M1jO1O

O L

1jO1???? ????? ???? ???? ???? ???? ?????

C M

0buyL0sellC0buyC0sellM0buyM0sellL1buyL1sellC1buyC1sellM1buyM1selljL0buy????? ???? ????? ????? ???? ???? ???? ???? ???? ???? ???? ????

jL0sell???? ????? ????? ????? ???? ???? ???? ???? ???? ???? ???? ???? jC0buy????? ????? ????? ???? ???? ???? ???? ???? ???? ???? ???? ???? jC0sell????? ????? ???? ????? ???? ???? ???? ???? ???? ???? ???? ???? jM0buy????? ???? ???? ???? ????? ???? ???? ???? ???? ???? ????? ???? jM0sell???? ????? ????? ???? ???? ????? ???? ???? ???? ???? ???? ????? jL1buy????? ???? ???? ????? ???? ???? ???? ???? ???? ???? ???? ????? jL1sell???? ????? ????? ???? ???? ???? ???? ???? ???? ???? ????? ???? jC1buy????? ????? ????? ???? ???? ???? ???? ???? ???? ???? ???? ???? jC1sell????? ????? ???? ????? ???? ???? ???? ???? ???? ???? ???? ???? jM1buy????? ???? ???? ????? ???? ???? ????? ???? ???? ???? ???? ???? jM1sell????? ????? ????? ???? ???? ???? ???? ???? ???? ???? ???? ???? jO????? ????? ????? ????? ???? ???? ???? ???? ???? ???? ???? ????

0buyL0sellC0buyC0sellM0buyM0sellL1buyL1sellC1buyC1sellM1buyM1selljL0buy? ? ? ? ? ? ? ? ? ? ? ?

jL0sell? ? ? ? ? ? ? ? ? ? ? ? jC0buy? ? ? ? ? ? ? ? ? ? ? ? jC0sell? ? ? ? ? ? ? ? ? ? ? ? jM0buy? ? ? ?????? ? ??? jM0sell? ? ? ? ?????? ? ?? jL1buy? ? ? ? ? ? ? ??? ?? jL1sell? ? ? ? ? ? ? ? ?? ??? jC1buy? ? ? ? ? ??? ? ? ? ? jC1sell? ? ? ? ? ? ??? ? ? ? jM1buy? ? ? ? ? ?? ?? ? ? ? L C M L C M ??????fM0buy;M0sellg? L C M

0buyL0sellC0buyC0sellM0buyM0sellL1buyL1sellC1buyC1sellM1buyM1selljL0buy????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ?????

jL0sell????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ????? jC0buy????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ????? jC0sell????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ????? ????? jM0buy????? ????? ????? ????? ???? ???? ???? ???? ????? ????? ???? ????? jM0sell????? ????? ????? ????? ???? ???? ???? ???? ????? ????? ????? ???? jL1buy????? ????? ????? ????? ???? ???? ???? ???? ????? ????? ????? ???? jL1sell????? ????? ????? ????? ???? ???? ???? ???? ????? ????? ???? ???? jC1buy????? ????? ????? ????? ????? ????? ???? ???? ????? ????? ????? ????? jC1sell????? ????? ????? ????? ????? ????? ???? ???? ????? ????? ????? ????? jM1buy????? ????? ????? ????? ???? ???? ???? ???? ????? ????? ???? ????? jM1sell????? ????? ????? ????? ???? ???? ???? ???? ????? ????? ????? ???? L

0buyL0sellC0buyC0sellM0buyM0sellL1buyL1sellC1buyC1sellM1buyM1selljL0buy???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ?????

jL0sell???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jC0buy???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jC0sell???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jM0buy???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jM0sell???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jL1buy???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jL1sell???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jC1buy???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jC1sell???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jM1buy???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? jM1sell???? ???? ???? ???? ????? ????? ????? ????? ????? ????? ????? ????? C h(i;j) =1h

Cov(Ni(t+h+)Ni(t+);Nj(t+h)Nj(t))

Cr h(i;j) =Correlation(Ni(t+h+)Ni(t+);Nj(t+h)Nj(t)) N t=X 1i? fTitg T i=TiTi1 (t) = limh#01h

E[N(t+h)N(t)jFt]

F N m(t) =X 1i? fTitg?fXi=mg m(t) = limh#01h

E[Nm(t+h)Nm(t)jFt]

?? ??? ???? ?? ??? ?????Tnk??????? ???kth??????? ???? ?? ?? ????? ?? ????n? ??????? ??8m2 f1;:::;Mg? m(t) =m+MX n=1 mnZ t 0 mn(ts)dNn(s) m(t) =m+MX n=1X T i1emn(t2s)dNn(s) M X m=1 m(t)(t) 0< tT

0(t) =(t)MX

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