HW 11100071021

Close Formfunction [ cdi, cdo, pdi, pdo ] = BOD_cf ( S0,K,r,q,T,sigma,Sb )

ssqT = sigma*sqrt(T);

lamda = ( r-q+0.5*sigma^2 ) / sigma^2 ;

y = ( log( Sb^2 / (S0*K) ) / ssqT ) + lamda*ssqT;

x1 = ( log(S0/Sb) / ssqT ) + lamda*ssqT;

y1 = ( log(Sb/S0) / ssqT ) + lamda*ssqT;

[c , p] = blsprice(S0,K,r,T,sigma,q);

if Sb <= K

cdi = S0*exp(-q*T)*( (Sb/S0)^(2*lamda) )*normcdf(y) - K*exp(-r*T)*( (Sb/S0)^(2*lamda-2) )*normcdf(y-ssqT);

cdo = c - cdi;

pdi = -S0*exp(-q*T)*normcdf(-x1) + K*exp(-r*T)*normcdf(-x1+ssqT) + S0*exp(-q*T)*( (Sb/S0)^(2*lamda) )*( normcdf(y) - normcdf(y1) ) - K*exp(-r*T)*( (Sb/S0)^(2*lamda-2) )*( normcdf(y-ssqT) - normcdf(y1-ssqT) );

pdo = p - pdi;

else

cdo = S0*exp(-q*T)*normcdf(x1) - K*exp(-r*T)*normcdf(x1-ssqT) - S0*exp(-q*T)*( (Sb/S0)^(2*lamda) )*normcdf(y1)

+ K*exp(-r*T)*( (Sb/S0)^(2*lamda-2) )*normcdf(y1-ssqT);

cdi = c - cdo;

pdo = 0;

pdi = p;

end

end

Close Form adjust to discretefunction [ cdi, cdo, pdi, pdo ] = BOD_cf_d ( S0,K,r,q,T,sigma,Sb,m )

Sb = Sb*exp(-0.5826*sigma*sqrt(T/m));

……

Monte Carlofunction [ c , p ] = BO_MC ( S0,K,r,T,sigma,Sb,ud,io,NRepl,NStep,rmode )

% rmode : 0(default)=>randn 1=>Halton

% ud : 0=>up 1=>down

% io : 0=>in 1=>out

for i = 1 : NRepl

MCS = AssetPaths(S0,r,sigma,T,NStep,1,rmode);

crossed_u = any(MCS >= Sb);

crossed_d = any(MCS <= Sb);

pc(i) = max( 0, MCS(NStep+1) - K );

pp(i) = max( 0, K - MCS(NStep+1) );

if ( (ud==0 && io==0) && crossed_u==0 ) || ( (ud==0 && io==1) && crossed_u==1 ) || ( (ud==1 && io==0) && crossed_d==0 ) || ( (ud==1 && io==1) &&

crossed_d==1 )

pc(i) = 0;

pp(i) = 0;

end

end

[c var ci] = normfit( exp(-r*T)*pc );

[p var ci] = normfit( exp(-r*T)*pp );

end

DemoS0 = 50; K = 50; r = 0.1; T = 2/12; sigma = 0.4; Sb = 40;

NRepl = 50000;

NStep = 60;

[ cdi, cdo, pdi, pdo ] = BOD_cf ( S0,K,r,0,T,sigma,Sb )

CF = ones(1,1000) * pdi;

rand('seed',0);

randn('seed',0);

figure;

for i = 1:600

[a, b, c, d] = BOD_cf_d ( S0,K,r,0,T,sigma,Sb,i );

Descrete(i) = c;

end

plot(1:600,Descrete,1:600,CF(1:600))

Up and In Call

1~10000 1~600

figure;

MC_R = zeros(1,100);

for i = 1:100

[a ,MC_R(i)] = BO_MC ( S0,K,r,T,sigma,Sb,1,0,i*100,NStep,0 );

x = i

end

plot(1:100,MC_R,1:100,CF(1:100));

title('MC with randn');

figure;

for i = 1:600

[a, b, c, d] = BOD_cf_d ( S0,K,r,0,T,sigma,Sb,i );

Descrete(i) = c;

end

plot(1:600,Descrete,1:600,CF(1:600))