1. 金融商品設計與評價hw11
計財系大三 林奕全

2. Outline 1. Down-and-in put 2.closed-form for other barrier option
(以 Up-and-in/out call為例)
-解析解
-MC (Up-and-in/out call)
-MC(BlsEurCall)

3. AssetPaths1 function SPaths = AssetPaths1( s0,r,sigma,T,NSteps,NRepl )
SPaths=zeros(NRepl,1+NSteps);
SPaths(:,1)=s0;
dt=T/NSteps;
nudt=(r-0.5*sigma^2)*dt;
sidt=sigma*sqrt(dt);
for i=1:NRepl
for j=1:NSteps
SPaths(i,j+1)= SPaths(i,j)*exp(nudt+sidt*randn);
end

4. DOPutMC(Down-and-in put)
function [P,CI,NCrossed ] =DOPutMC( s0,X,r,T,sigma,sb,NSteps,NRepl)
Payoff=zeros(NRepl,1);
NCrossed=0;
for i=1:NRepl
Path=AssetPaths1(s0,r,sigma,T,NSteps,1);
crossed=any(Path<=sb);

5. if crossed==0
Payoff(i)=0;
NCrossed=NCrossed+1;
else
Payoff(i)=max(0,X-Path(NSteps+1));
end
[P,aux,CI]=normfit(exp(-r*T)*Payoff)

6. s0=50;
X=50;
r=0.1;
T=2/12;
sigma=0.4;
sb=40;
NSteps=60;
NRepl=50000;
randn('seed',0);
[D0PutMC,CI,NCrossed]=DOPutMC(s0,X,r,T,sigma,sb,NSteps,NRe pl);
D0PutMC CI
NCrossed

7. BlsMcEurPut function[Price,CI]=BlsMcEurPut(s0,X,r,T,sigma,NRepl)
nuT=(r-0.5*sigma^2)*T;
siT=sigma*sqrt(T);
DiscPayoff=exp(-r*T)*max(X- s0*exp(nuT+siT*randn(NRepl,1)),0);
[Price,Varprice,CI]=normfit(DiscPayoff);

8. s0=50;
X=50;
r=0.1;
T=2/12;
sigma=0.4;
NRepl1=50000;
randn('seed',0);
[MC50000,CI50000]=BlsMcEurPut(s0,X,r,T,sigma,NRepl)

9. Price CI
NCrossed
Down-and-out Put (MC)
1.3600
[1.3303,1.3808]
7392
Down-and-in Put (MC)
1.4912
[1.4574,1.5251]
42608
Total
2.8512
50000
MCEurput
2.8296
[2.7941,2.8652]
blsprice
2.8304

10. UICall formula

11. UICall function [UICallprice, UOCallprice]=UICall (s0,X,r,T,sigma,sb )
lamda=(r/(sigma^2))+1/2;
y1=(log(sb/s0)+lamda*(sigma^2)*T)/(sigma*sqrt(T));
x1=(log(s0/sb)+lamda*(sigma^2)*T)/(sigma*sqrt(T));
y=(log(sb^2/(s0*X))+lamda*(sigma^2)*T)/(sigma*sqrt(T));

12. UICallprice=max(0,s0. normcdf(x1)-X. exp(- r. T). normcdf(x1-sigma
UICallprice=max(0,s0*normcdf(x1)-X*exp(- r*T)*normcdf(x1-sigma*sqrt(T))- s0*((sb/s0)^(2*lamda))*(normcdf(-y)-normcdf(- y1))+X*exp(-r*T)*((sb/s0)^(2*lamda-2))*(normcdf(- y+sigma*sqrt(T))-normcdf(-y1+sigma*sqrt(T))));
UOCallprice=blsprice(s0,X,r,T,sigma)-UICallprice;
end

13. s0=50;
X=50;
r=0.1;
T=2/12;
sigma=0.4;
sb=70;
NStep=60;
NRepl=50000;
sb1=sb*exp(0.5826*0.4*sqrt(1/12/30));
[UICallprice, UOCallprice]=UICall (s0,X,r,T,sigma,sb1 )

14. UICallprice =
0.7103
UOCallprice =
2.9465

15. UICallMC
function [P,CI,NCrossed ] =UICallMC( s0,X,r,T,sigma,sb,NSteps,NRepl)
Payoff=zeros(NRepl,1);
NCrossed=0;
for i=1:NRepl
Path=AssetPaths1(s0,r,sigma,T,NSteps,1);
crossed=any(Path>=sb);
if crossed==0
Payoff(i)=0;
NCrossed=NCrossed+1;
else
Payoff(i)=max(0,Path(NSteps+1)-X);
end
[P,aux,CI]=normfit(exp(-r*T)*Payoff)

16. s0=50;
X=50;
r=0.1;
T=2/12;
sigma=0.4;
sb=70;
NRepl=50000;
NSteps=60;
randn('seed',0);
[P,CI,NCrossed ] =UICallMC( s0,X,r,T,sigma,sb,NSteps,NRepl)

17. UOCallMC
function [P,CI,NCrossed ] =UOCallMC( s0,X,r,T,sigma,sb,NSteps,NRepl)
Payoff=zeros(NRepl,1);
NCrossed=0;
for i=1:NRepl
Path=AssetPaths1(s0,r,sigma,T,NSteps,1);
crossed=any(Path>=sb);
if crossed==0
Payoff(i)=max(0,Path(NSteps+1)-X);
else
Payoff(i)=0;
NCrossed=NCrossed+1;
end
[P,aux,CI]=normfit(exp(-r*T)*Payoff)

18. s0=50;
X=50;
r=0.1;
T=2/12;
sigma=0.4;
sb=70;
NRepl=50000;
NSteps=60;
randn('seed',0);
[P,CI,NCrossed ] =UOCallMC( s0,X,r,T,sigma,sb,NSteps,NRepl)

19. BlsMcEurCall( function[Price,CI]=BlsMcEurCall(s0,X,r,T,sigma,NRepl)
nuT=(r-0.5*sigma^2)*T;
siT=sigma*sqrt(T);
DiscPayoff=exp(- r*T)*max(s0*exp(nuT+siT*randn(NRepl,1))-X,0);
[Price,Varprice,CI]=normfit(DiscPayoff);

20. Price CI
NCrossed
Up-and-in Call (MC)
0.7034
[ ]
48318
Closed form
0.7103
Up-and-out Call (MC)
2.9359
[2.8976,2.9742]
1694
2.9465
Total
3.6393
50000
MCEurCall
3.6781
[3.6295,3.7267]
blsprice
3.6568