Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations

Code 1

MATLAB code.

 clear all; close all; clc; clf;

K = 100; L = 2.5*K; T = 1/365; sigma = 0.3; r = 0.03;

cash = 100; Nx = L+1; x = linspace(0,L,Nx); h = x()-x();

payoff(1:Nx) = 0.0; payoff(x >= K) = cash;

%% Price %%

d1 = (log(x/K) + (r+sigma2/2)*T)/(sigma*sqrt(T));

d2 = d1 - (sigma*sqrt(T));

exact = cash*exp(-r*T).*normcdf(d1);

figure(); hold on; grid on;

plot(x,payoff,^,k-^,,x,exact,^,r-^,); axis([0 L -5 1.05*cash]);

%% Delta %%

del = cash*exp(-r*T).*normpdf(d1)./(sigma*sqrt(T)*x);

figure(); hold on; grid on;

plot(del,^,r-^,,^,LineWidth^,,1); axis([0 L -5 1.1*max(del)]);

%% Gamma %%

gam = -cash*exp(-r*T) * (d1.*normpdf(d2))./((sigma*x).2*T);

figure(); hold on; grid on; plot(gam,^,r-^,,^,LineWidth^,,1);

axis([0 L -1.1*max(gam) 1.1*max(gam)]);

%% theta %%

 the = cash*exp(-r*T)*(r*normcdf(d2) + normpdf(d2)...

  .* (d1/(2*T) - r/(sigma*sqrt(T))));

 figure(); hold on; grid on; plot(the,^,r-^,,^,LineWidth^,,1);

 axis([0 L -1.1*max(the) 1.1*max(the)]);

 %% Vega %%

 veg = -cash*exp(-r*T)*(normpdf(d2)).*d1/sigma;

 figure(); hold on; grid on; plot(veg,^,r-^,,^,LineWidth^,,1);

 axis([0 L -1.1*max(veg) 1.1*max(veg)]);

 %% Rho %%

 rho = cash*exp(-r*T)*((-T*normcdf(d2)) + sqrt(T)/sigma*(normpdf(d2)));

 figure(); hold on; grid on; plot(rho,^,r-^,,^,LineWidth^,,1);

 axis([0 L 3*min(rho) 1.1*max(rho)]);