All The Matlab Codes

Published Date: 02 Nov 2017

All the Matlab codes which we have used to work put the results are given.

A1.1 European Call

function call_price=EuroCall(S, E, r, sigma, time)

S = 10; E = 5; time = 0.25; r = 0.06; sigma = 0.3;

time_sqrt = sqrt(time);

d1 = (log(S/E)+r*time)/(sigma*time_sqrt)+0.5*sigma*time_sqrt;

d2 = d1-(sigma*time_sqrt);

call_price = S*normcdf(d1)-E*exp(-r*time)*normcdf(d2);

A1.2 European Put

function put_price=EuroPut(S, E, r, sigma, time)

S = 10; E = 5; time = 0.25; r = 0.06; sigma = 0.3;

d1=(log(S/E)+(r+sigma^2/2)*time)/(sigma*sqrt(time));

d2=d1-sigma*sqrt(time);

put_price=E*exp(-r*time)*normcdf(-d2)-S*normcdf(-d1);

A1.3 European call with dividends

function call_price=EuroCallWithDivs(S, E, r, sigma, time, D)

S = 30; E = 20; time = 0.5; r = 0.06; sigma = 0.3; D=0.04;

time_sqrt = sqrt(time);

d1 = (log(S/E)+(r-D)*time)/(sigma*time_sqrt)+0.5*sigma*time_sqrt;

d2 = d1-(sigma*time_sqrt);

call_price=S.*exp(-D.*time).*normcdf(d1,0,1)-E.*exp(-r.*time).*normcdf(d2,0,1);

A1.4 European put with dividends

function put_price=EuroPutWithDivs(S, E, r, sigma, time, D)

S = 30; E = 20; time = 0.5; r = 0.06; sigma = 0.3; D=0.04;

d1=(log(S/E)+((r-D)+sigma^2/2)*time)/(sigma*sqrt(time));

d2=d1-sigma*sqrt(time);

put_price=-S.*exp(-D.*time).*normcdf(-d1,0,1)+E.*exp(-r.*time).*normcdf(-d2,0,1);

A2 Finite Difference Methods

A2.1 Explicit Method

function AmerPutPrice=AmerPutEM(S0, E, r, sigma, T, M, N)

T = 0.5;

S0 = 30;

E = 20;

sigma = 0.3;

r = 0.06;

N = 2430;

M = 2430;

dt = T/N;

mu = r - sigma^2/2;

dx = sigma*sqrt(3*dt);

pu = dt*(sigma^2/2/dx^2 + mu/2/dx);

pm = 1 - dt*sigma^2/dx^2 - r*dt;

pd = dt*(sigma^2/2/dx^2 - mu/2/dx);

S = zeros(2*M+1, N+1);

V = zeros(2*M+1, N+1);

I = [0:1:N];

J = [M:-1:-M]';

S = S0*exp(J.*dx);

V(:,end) = max(E - S, 0);

for j=N:-1:1

for i=2:2*M

V(i,j) = pu*V(i-1,j+1) + pm*V(i,j+1) + pd*V(i+1,j+1);

end

V(2*M+1,j) = V(2*M,j) + (S(2*M) - S(2*M+1));

V(1,j) = V(2,j);

for i=1:2*M+1

V(i,j) = max(E - S(i), V(i,j));

end

end

AmerPutPrice = V(M+1,1)

A2.2a Fully Implicit Method

function ImplicitPrice=AmerPutIM(S0, E, r, sigma, T, M, N)

T = 0.5;

S0 = 30;

E = 20;

sigma = 0.3;

r = 0.06;

PutCall = 'P';

EuroAmer = 'A';

N = 2430;

M = 2430;

dt = T/N;

mu = r - sigma^2/2;

dx = sigma*sqrt(3*dt);

pu = -dt/2*(sigma^2/dx^2 + mu/dx);

pm = 1 + dt*sigma^2/dx^2 + r*dt;

pd = -dt/2*(sigma^2/dx^2 - mu/dx);

S = zeros(2*M+1, N+1);

V = zeros(2*M+1, N+1);

J = [M:-1:-M]';

S = S0*exp(J.*dx);

clear J

if strcmp(PutCall,'P')

V(:,end) = max(E - S, 0);

else

V(:,end) = max(S - E, 0);

end

if strcmp(PutCall,'P')

lambda_L = max(0,E - S(2*M+1));

lambda_U = 0;

elseif strcmp(PutCall,'C')

lambda_L = 0;

lambda_U = max(0,S(1) - E);

end

for j=N:-1:1

C = [lambda_U; V(2:2*M,j+1); lambda_L];

V(:,j) = SolveTriangular(C,pu,pm,pd,lambda_L,lambda_U);

if strcmp(EuroAmer,'A')

for i=1:2*M+1;

switch PutCall

case 'P'

V(i,j) = max(V(i,j), E - S(i));

case 'C'

V(i,j) = max(V(i,j), S(i) - E);

end

end

end

end

clear i j

ImplicitPrice = V(M+1,1);

A2.2b Fully Implicit Method

function y = SolveTriangular(C,pu,pm,pd,lambda_L,lambda_U);

M = (length(C)-1)/2;

pmp(2*M) = pm + pd;

pp(2*M) = C(2*M) + pd*lambda_L;

for j=2*M-1:-1:2

pmp(j) = pm - pu*pd/pmp(j+1);

pp(j) = C(j) - pp(j+1)*pd/pmp(j+1);

end

y(1) = (pp(2) + pmp(2)*lambda_U)/(pu + pmp(2));

y(2) = y(1) - lambda_U;

for j=3:2*M;

y(j) = (pp(j) - pu*y(j-1))/pmp(j);

end

y(2*M+1) = y(2*M) - lambda_L;

A2.3 Crank Nicolson Method

function AmerPutPrice=AmerPutCN(S0, E, r, sigma, T, M, N)

T = 0.5;

S0 = 30;

E = 20;

sigma= 0.3;

r = 0.06;

N = 2430;

M = 2430;

dt = T/N;

mu = r- sigma^2/2;

dx = sigma*sqrt(3*dt);

pu = -1/4*dt*(sigma^2/dx^2 + mu/dx);

pm = 1 + dt*sigma^2/2/dx^2 + r*dt/2;

pd = -1/4*dt*(sigma^2/dx^2 - mu/dx);

S = zeros(2*M+1, N+1);

V = zeros(2*M+1, N+1);

J = [M:-1:-M]';

S = S0*exp(J.*dx);

V(:,end) = max(E - S, 0);

pmp = zeros(2*M+1,N+1);

pp = zeros(2*M+1,N+1);

C = zeros(2*M+1,N+1);

for j=N+1:-1:2

pmp(2*M,j) = pd+pm;

for i=2*M-1:-1:2

pmp(i,j) = pm - pd/pmp(i+1,j)*pu;

end

end

lambda_L = S(2*M+1) - S(2*M);

lambda_U = 0;

for j=N+1:-1:2;

for i=2*M:-1:2

if i==2*M

pp(i,j) = -pu*V(i-1,j) - (pm-2)*V(i,j) - pd*V(i+1,j) + pd*lambda_L;

else

pp(i,j) = -pu*V(i-1,j) - (pm-2)*V(i,j) - pd*V(i+1,j) - pd/pmp(i+1,j)*pp(i+1,j);

end

end

j=j-1;

for i=1:2*M+1

if i==1

C(i,j) = (pp(i+1,j+1) + pmp(i+1,j+1)*lambda_U) / (pmp(i+1,j+1) + pu);

V(i,j) = max(E - S(i), C(i,j));

elseif i<2*M+1

C(i,j) = (pp(i,j+1) - pu*C(i-1,j))/pmp(i,j+1);

V(i,j) = max(E - S(i), C(i,j));

else

C(i,j) = C(i-1,j) - lambda_L;

V(i,j) = max(E - S(i), C(i,j));

end

end

j=j+1;

end

V

AmerPutPrice = V(M+1,1)

A3 Binomial Method

A3.1 Binomial Method European Call

function call_price=european_call_bin(S, E, r, sigma, t, M)

S=45; E=40; r=0.1; sigma=0.25;

t=0.5; M=10;

R = exp(r*(t/M));

Rinv = 1.0/R;

u = exp(sigma*sqrt(t/M));

uu = u*u;

d = 1.0/u;

p_up = (R-d)/(u-d);

p_down = 1.0-p_up;

prices=zeros(M+1,1);

prices(1) = S*(d^M);

for ( i=2:(M+1) )

prices(i) = uu*prices(i-1);

end

call_values=zeros(M+1,1);

call_values = max(0, (prices-E));

for ( M=M:-1:1 )

for ( i=1:1:(M) )

call_values(i) = ( p_up*call_values(i+1)+p_down*call_values(i) )*Rinv;

end

end

call_price = call_values(1);

For the European Put you just have to replace max(0, (prices-E)) with max(0, (E-prices))

A3.2 Binomial Method American Call

function call_price=american_call_bin(S, E, r, sigma, t, M)

S=45; E=40; r=0.1; sigma=0.25;

t=0.5; M=10;

R = exp(r*(t/M));

Rinv = 1.0/R;

u = exp(sigma*sqrt(t/M));

d = 1/u;

p_up = (R-d)/(u-d);

p_down = 1-p_up;

prices = zeros(M+1);

prices(1) = S*(d^M);

uu = u*u;

for i=2:M+1

prices(i) = uu*prices(i-1);

end

call_values = max(0, (prices-E));

for M=M:-1:1

for i=1:M+1

call_values(i) = (p_up*call_values(i+1)+p_down*call_values(i))*Rinv;

prices(i) = d*prices(i+1);

call_values(i) = max(call_values(i),prices(i)-E);

end

end

call_price=call_values(1);

For the American Put you just have to replace max(0, (prices-E)) with max(0, (E-prices)) and also replace max(call_values(i),prices(i)-E) with max(call_values(i),E-prices(i)).

A3.3 Binomial Method for European and American Options with Dividends

function BioWoithDivs=AmerEuroDivsBin(S0, E, r, sigma, T, M, N, D)

BSCall = inline('s*exp(-D*T)*normcdf((log(s/E) + (r-D+sigma^2/2)*T)/sigma/sqrt(T)) - E*exp(-r*T)*normcdf((log(s/E) + (r-D+sigma^2/2)*T)/sigma/sqrt(T) - sigma*sqrt(T))',...

's','E','r','D','sigma','T');

BSPut = inline('E*exp(-r*T)*normcdf(-(log(s/E) + (r-D+sigma^2/2)*T)/sigma/sqrt(T) + sigma*sqrt(T)) - s*exp(-D*T)*normcdf(-(log(s/E) + (r-D+sigma^2/2)*T)/sigma/sqrt(T))',...

's','E','r','D','sigma','T');

S0 = 30;

r = 0.06;

D = 0.04;

sigma = 0.3;

N = 10;

E = 20;

T = 0.5;

PutCall = 'P';

EuroAmer = 'A';

dt = T/N;

u = exp(sigma*sqrt(dt));

d = 1/u;

p = (exp((r-D)*dt)-d)/(u-d);

S = zeros(2*N+1,N+1);

S(N+1,1) = S0;

for j=2:N+1

for i=N-j+2:2:N+j

S(i,j) = S0*u^(N+1-i);

end

end

V = zeros(2*N+1,N+1);

switch PutCall

case 'C'

V(:,N+1) = max(S(:,N+1) - E, 0);

case 'P'

V(:,N+1) = max(E - S(:,N+1), 0);

end

for j=N:-1:1

for i=N-j+2:2:N+j

switch EuroAmer

case 'A'

if strcmp(PutCall, 'C')

V(i,j) = max(S(i,j) - E, exp(-r*dt)*(p*V(i-1,j+1) + (1-p)*V(i+1,j+1)));

else

V(i,j) = max(E - S(i,j), exp(-r*dt)*(p*V(i-1,j+1) + (1-p)*V(i+1,j+1)));

end

case 'E'

V(i,j) = exp(-r*dt)*(p*V(i-1,j+1) + (1-p)*V(i+1,j+1));

end

end

end

TreePrice = V(N+1,1)

if strcmp(PutCall, 'C')

EuroPrice = BSCall(S0,E,r,D,sigma,T)

else

EuroPrice = BSPut(S0,E,r,D,sigma,T)

end

if strcmp(PutCall, 'P') & strcmp(EuroAmer, 'A')

ExercisePremium = TreePrice - EuroPrice

end

Project Plan

Weeks 3-8

During week 3 I will be given my dissertation topic and know what dissertation I will be doing. I will try and see my supervisor on a weekly basis so that I have regular feedback on my dissertation. Early during this period I will do some background research on, Options, Black-Scholes and Asset price model. To do this research I will need to look for books, which will help me for my background information and during the dissertation. During week 8, the background, project plan and Gantt chart will be due in.

Weeks 9-12

During this 4 week period I will do more research on Options, specifically American style options and look at upper and lower boundary conditions for both American and European style options, and do the write up for these topics.

Weeks 13-14

During these two weeks, which are holidays, I will start looking into detail the Binomial Model, and see what effect it has on valuing options. Also I will start doing research into Finite Difference Methods.

Weeks 15-19

I will start doing in depth research in to Finite Difference Methods. Also during these 5 weeks I will start implementing on Matlab.

Weeks 20-24

In these weeks, I will analyse the results I get from Matlab and complete my final write up. The dissertation is due in on week 25, but I???ve left an extra week for anything I???ve left out or any final checks.

Weeks 25-29

Week 29 is when the oral presentation is, so in these weeks I will prepare my presentation and rehearse for it, also go over my dissertation as I will be getting asked questions.