金融商品設計與評價 - PowerPoint Presentation

Slide1

金融商品設計與評價hw14

計財系大三 林奕全

Slide2

Outline

1. range forward

-

出口商

(long

美元賣權

)

-

進口商

(long

美元買權

)

2.Sallie Mae

3.finite difference in European Call

-Explicit approach

-Implicit approach

Slide3

流程:以出口商為例

Step1:

給定

s0,X,r,rf,T,sigma

由

B-S

算出賣權的權利金

,

假設為

P

Step2:

令

P =a*exp(-r*T)*

normcdf

(-d2)-s0*exp(-

rf

*T)*

normcdf

(-d1)

解未知數

a(

以

P

為權利金的買權執行價格

)

Slide4

a=20:0.5:35;

s0=27;

X=26.7;

sigma=0.0311;

rf

=0.0304;

r=0.0565;

T=0.25;

d1=(log(s0./a)+(r-rf+sigma^2./2).*T)/(sigma.*sqrt(T));

d2=d1-sigma.*

sqrt

(T);

[K]=s0.*exp(-rf.*T).*

normcdf

(d1)-a.*exp(-r.*T).*

normcdf

(d2);

[ C,P ] =

Curoption

( s0,X,r,rf,T,sigma);

figure

plot(

a,P

-K,'-');

title('Solution');

xlabel

('a')

ylabel

('P-K')

legend('value')

Slide5

Slide6

Curoption

function [ C,P ] =

Curoption

( s0,X,r,rf,T,sigma )

d1=(log(s0/X)+(r-rf+sigma^2/2)*T)/(sigma*sqrt(T));

d2=d1-sigma*

sqrt

(T);

C=s0*exp(-

rf

*T)*

normcdf

(d1)-X*exp(-r*T)*

normcdf

(d2);

P=X*exp(-r*T)*

normcdf

(-d2)-s0*exp(-

rf

*T)*

normcdf

(-d1);

end

Slide7

Demovalue

function [ K ] =

Demovalue

(s0,a,r,rf,T,sigma,jud)

d1=(log(s0/a)+(r-rf+sigma^2/2)*T)/(sigma*sqrt(T));

d2=d1-sigma*

sqrt

(T);

if

jud

==0

K=s0*exp(-

rf

*T)*

normcdf

(d1)-a*exp(-r*T)*

normcdf

(d2);

else

K=a*exp(-r*T)*

normcdf

(-d2)-s0*exp(-

rf

*T)*

normcdf

(-d1);

end

end

Slide8

approx

function [sol] = approx(C,P,X,s0,r,rf,T,sigma,jud )

a=X;

S=zeros(1,2000);

if

jud

==0

for

i

=2:1:2001

S(i-1)=

Demovalue

(s0,a+(i-1)*0.001,r,rf,T,sigma,jud);

end

Slide9

for

i

=2:1:2001

if (P-S(i-1))*(P-S(

i

))<0

c=P-S(i-1);

d=P-S(

i

);

sol=(c/(c-d))*(a+i*0.001)+(-d/(c-d))*(a+(i-1)*0.001);

break;

end

end

Slide10

else

P=C;

for

i

=2:1:2001

S(i-1)=

Demovalue

(s0,a-(i-1)*0.001,r,rf,T,sigma,jud);

end

Slide11

for

i

=2:1:2001

if (P- S(i-1))*(P- S(

i

))<0

c=P-S(i-1);

d=P-S(

i

);

sol=(-c/(d-c))*(a-i*0.001)+(d/(d-c))*(a-(i-1)*0.001);

break;

end

end

end

end

Slide12

Curinterval

function [

lower,upper

] =

Curinterval

(s0,X,r,rf,T,sigma,jud )

[ C,P ] =

Curoption

( s0,X,r,rf,T,sigma );

[ K ] =

Demovalue

(s0,X,r,rf,T,sigma,jud);

if

jud

==0

sol = approx(C,P,X,s0,r,rf,T,sigma,jud );

lower=X;

upper=sol;

else

sol = approx(C,P,X,s0,r,rf,T,sigma,jud);

lower=sol;

upper=X;

end

end

Slide13

出口商

s0=27;

X=26.7;

>> sigma=0.0311;

>> rf=0.0304;

>> r=0.0565;

>> jud=0;

>> T=0.25;

Slide14

lower =

26.7000

upper =

27.6658

Slide15

進口商

s0=27;

X=27.65;

>> sigma=0.0311;

>> rf=0.0304;

>> r=0.0565;

>> jud=1;

>> T=0.25;

Slide16

lower =

26.7152

upper =

27.6500

Slide17

SalliieMaeBond

function [ price ] =

SalliieMaeBond

( s0,r,T,sigma,N )

deltaT

=T/N;

strike=[131.75,131.75,129.5,127,124.5];

u=exp(sigma.*

sqrt

(

deltaT

));

d=1./u;

p=(exp(r.*deltaT)-d)/(u-d);

lattice=zeros(N+1,N+1);

for j=0:N

s=(50*((s0-124.5)/s0))*(

u^j

)*(d^(N-j));

if s<9.25

lattice(N+1,j+1)=9.25;

end

end

Slide18

for

i

=N-1:-1:0

for j=0:i

k=strike(ceil(5*(i+1)/N));

s=s0*(

u^j

)*(d^(

i

-j));

lattice(i+1,j+1)=max((50*((s-k)/s)),exp(-r*

deltaT

)*(p*lattice(i+2,j+2)+(1-p)*lattice(i+2,j+1)));

if lattice(i+1,j+1)<9.25*exp(-r*(T-(5*

i

/N)))

lattice(i+1,j+1)=9.25*exp(-r*(T-(5*i/N)));

end

end

end

price=lattice(1,1);

Slide19

N=500;

s0=10:5:500;

sigma=0.4;

r=0.02;

T=5;

Slide20

Slide21

EuCallExpl1

function [

price,matval,vetS,vetT

] =EuCallExpl1 (s0,X,r,T,sigma,Smax,dS,dt )

M=round(

Smax

/

dS

);

dS

=

Smax

/M;

N=round(T/

dt

);

dt

=T/N;

matval

=zeros(M+1,N+1);

vetS

=

linspace

(0,Smax,M+1);

vetj

=0:N;

veti

=0:M;

vetT

=

linspace

(0,T,N+1);

Slide22

matval

(:,N+1)=max(vetS-X,0);

matval

(1,:)=0;

matval(M+1,:)=(vetS(M+1)-X).*exp(-r*dt*(N-vetj));

a=0.5*

dt

*(sigma^2*

veti

-r).*

veti

;

b=1-dt*(sigma^2*veti.^2+r);

c=0.5*

dt

*(sigma^2*

veti+r

).*

veti

;

Slide23

for j=N:-1:1

for

i

=2:M

matval(i,j)=a(i)*matval(i-1,j+1)+b(i)*matval(i,j+1)+c(i)*matval(i+1,j+1);

end

end

idown

=floor(s0/

dS

);

iup

=ceil(s0/

dS

);

Slide24

if

idown

==

iup

price=

matval

(idown+1,1);

else

price=

matval

(idown+1,1)+(s0-(idown+1)*

dS

)*(

matval

(iup+1,1)-

matval

(iup,1))/

dS

;

end

Slide25

s0=50;

X=50;

r=0.1;

T=5/12;

sigma=0.3;

>>

Smax

=100;

>>

dS

=2;or

dS

=1.2

>>

dT

=5/1200;

Slide26

[Expds1,matval,vetS,vetT] =EuCallExpl1 (s0,X,r,T,sigma,Smax,dS,dT);

mesh(

vetT,vetS,matval

);

xlabel

('Time');

>>

ylabel

('Stock price');

>> Title('European Call

Option,Explicit

Method');

Slide27

Slide28

Slide29

EuCallImpl1(Confused)

function [

price,matval,vetS,vetT

] =EuCallImpl1 (s0,X,r,T,sigma,Smax,dS,dt )

M=round(

Smax

/

dS

);

dS

=

Smax

/M;

N=round(T/

dt

);

dt

=T/N;

matval

=zeros(M+1,N+1);

vetS

=

linspace

(0,Smax,M+1);

vetj

=0:N;

veti

=0:M;

Slide30

vetT

=

linspace

(0,T,N+1);

matval

(:,N+1)=max(vetS-X,0);

matval

(1,:)=0;

matval(M+1,:)=(Smax-X)*exp(-r*dt*(N:-1:0));

a=0.5*(r*

dt

*veti-sigma^2*

dt

*(veti.^2));

b=1+sigma^2*

dt

*(veti.^2)+r*

dt

;

c=(-0.5)*(r*dt*veti+sigma^2*dt*(veti.^2));

coeff=diag(a(3:M),-1)+diag(b(2:M))+diag(c(2:M-1),1);

[L,U]=

lu

(

coeff

);

aux=zeros(M-1,1);

Slide31

for j=N:-1:1

aux(1)=-a(2)*

matval

(1,j);

matval

(2:M,j)=U\(L\(

matval

(2:M,j+1)+aux));

end

idown

=floor(s0/

dS

);

iup

=ceil(s0/

dS

);

if

idown

==

iup

price=

matval

(idown+1,1);

else

price=

matval

(idown+1,1)+(s0-idown*

dS

)*(

matval

(idown+2,1)-

matval

(idown+1,1))/

dS

;

end

Slide32

s0=50;

X=50;

r=0.1;

T=5/12;

sigma=0.3;

>>

Smax

=100;

>>

dS

=2;or

dS

=1.2

>>

dT

=5/1200;

Slide33

[Expds1,matval,vetS,vetT] =EuCallImpl1 (s0,X,r,T,sigma,Smax,dS,dT);

mesh(

vetT,vetS,matval

);

xlabel

('Time');

>>

ylabel

('Stock price');

>> Title('European Call

Option,Implicit

Method');

Slide34

Slide35