%% Analyzing Investment Strategies with CVaR Portfolio Optimization in MATLAB - Theory
%
% Robert Taylor
% The MathWorks, Inc.
% Copyright (C) 2012 The MathWorks, Inc.
%% Covered-Call Strategy
% Generate the standard payoff diagram for covered-calls. To anchor the results, the diagram
% includes initial and expected stock prices to illustrate profitability of the covered-call
% position versus an uncovered position. Vary the parameter mu, which is the stock price drift
% parameter, to locate different payoffs. Try, for example, mu = 0.1 and mu = 0.35.
X0 = 40; % initial price of stock
r0 = 0.02; % risk free rate (annualized)
mu = 0.10; % stock price drift (annualized)
sigma = 0.24; % stock price volatility (annualized)
K = 42; % strike price of call option
TC = 64/252; % call option expiration (years)
EX = X0*exp(mu*TC);
C0 = blsprice(X0, K, r0, TC, sigma);
X = linspace(35, 45, 11)';
W0 = 100;
WU = zeros(11,1);
WC = zeros(11,1);
for i = 1:11
WU(i) = X(i);
if X(i) >= K
WC(i) = C0 + K;
else
WC(i) = C0 + X(i);
end
end
Wmax = (W0/X0)*max([WU; WC; X0]);
Wmin = (W0/X0)*min([WU; WC; X0]);
clf;
plot(X, (W0/X0)*[WU, WC]);
hold on
plot(X, (W0/X0)*X0*ones(11,1), ':r');
plot(X0*ones(2,1), [Wmax; Wmin], ':r');
plot(EX*ones(2,1), [Wmax; Wmin], ':r');
title(['\bf' sprintf('Payoff Diagram for Covered-Call Strategy (Drift %g)', mu)]);
xlabel('Stock Price at Option Expiration');
ylabel('Wealth');
text(X0, Wmin - 1, 'x_0', 'Fontsize', 8);
text(EX, Wmin - 1, 'E[X]', 'Fontsize', 8);
text(X(2), W0 - 1, 'W_0', 'FontSize', 8);
legend('Uncovered','Covered','Location','NorthWest');
hold off
%% Distribution of Covered-Call Payoffs without Re-Investment
% Generate the distribution of payoffs for covered-calls and compare with the distribution for an
% uncovered position.
X0 = 40; % initial price of stock
r0 = 0.01; % risk free rate (annualized)
mu = 0.1; % stock price drift (annualized)
sigma = 0.24; % stock price volatility (annualized)
startdate = datenum('28-Sep-2012'); % start date for investment period
enddate = datenum('21-Dec-2012'); % enddate for investment period
numdays = daysdif(startdate, enddate, 13); % number of trading days for investment period
t0 = 0; % set initial time to be 0 years
T = numdays/252; % time horizon is 1 month with T in years
K = 42; % strike price of call option
expdate = datenum('21-Dec-2012'); % expiration date of call option
TC = daysdif(startdate, expdate, 13)/252;
C0 = blsprice(X0, K, r0, TC, sigma);
numtrials = 10000; % number of trials
numsamples = 252*T*13; % number of days to time T with 13 half-hour periods per day
dt = T/numsamples; % time increment in years
% generate geometric Brownian motion process for underlying stock price
Xsde = gbm(mu, sigma, 'starttime', t0, 'startstate', X0);
[X, t] = Xsde.simulate(numsamples, 'ntrials', numtrials, 'deltatime', dt);
% set up uncovered position
WU = X(end, :)';
% set up covered position without re-investment
WC = zeros(numtrials, 1);
for i = 1:numtrials
H = 1;
assigned = false;
for j = 1:numel(t)
if j == 1
WC(i) = H*(X0 + C0);
else
WC(i) = WC(i) + H*(X(j,i) - X(j-1,i));
end
if ~assigned && X(j,i) >= K
WC(i) = WC(i) + H*(K - X(j,i));
H = 0;
assigned = true;
end
end
end
WU = (100/X0)*WU;
WC = (100/X0)*WC;
Wmin = min([min(WU), min(WC)]);
Wmax = max([max(WU), max(WC)]);
Nmax = max([max(hist(WU,20)), max(hist(WC,20))]);
clf;
subplot(2,1,1);
hist(WU, 20);
axis([Wmin,Wmax,0,1.1*Nmax]);
title('\bfDistributions of Uncovered and Covered Wealth');
xlabel('Uncovered Strategy');
subplot(2,1,2);
hist(WC, 20);
axis([Wmin,Wmax,0,1.1*Nmax]);
xlabel('Covered Strategy');
%% Maximum allowable slippage for initiating position
% Illustrate maximum allowable slippage for an initiating covered-call position such that any
% slippage above this value renders the uncovered position preferable to the covered position. An
% initiating position assumes that the position is uncovered at the start of the investment period
% and that a covered-call position is established at this time so that the call premium contributes
% to total return during the period.
% The result is a plot that displays maximum allowable slippage as a function of strike price and
% exercise probability. The red line on the plot is the maximum allowable slippage if the option
% is exercised immediately when the stock price crosses above the strike price at any time during
% the investment period.
X0 = 40; % initial stock price
T = 23/252; % duration of investment period (years)
r0 = 0.02; % risk free rate (annualized)
m = 0.1; % stock price drift (annualized)
s = 0.24; % stock price volatility (annualized)
TC = 64/252; % expiration date of call option
Knum = 21;
pnum = 20;
K = linspace(40, 45, Knum); % strike prices
p = linspace(0.05, 1, pnum); % exercise probabilities
d = NaN(Knum, pnum); % maximum slippage as function of strike and probabilities
for i = 1:Knum
XT = X0*exp(m*T);
for j = 1:pnum
C = blsprice(X0, K(i), r0, TC, s);
xi = C*(1 + r0*T)/XT;
if p(j) > xi
d(i,j) = (xi/p(j))/(1 - xi/p(j));
end
end
end
P = zeros(Knum, 1); % hitting time exercise probabilities
dP = NaN(Knum, 1); % maximum slippage for hitting time exercise probabilities
for i = 1:Knum
XT = X0*exp(m*T);
P(i) = gbm_upcrossing_prob(X0, K(i), T, m, s);
C = blsprice(X0, K(i), r0, TC, s);
xi = C*(1 + r0*T)/XT;
if P(i) > xi
dP(i) = (xi/P(i))/(1 - xi/P(i));
end
end
d = 10000*d;
dP = 10000*dP;
dmin = min(min(min(d)),min(dP));
dmin = min(0, dmin);
dmax = max(max(max(d)),max(dP));
dmax = min(2000, dmax);
clf;
surf(p, K, d, 'FaceColor', 'b', 'EdgeColor', 'b', 'FaceAlpha', 0.3);
hold on
plot3(P, K, dP, 'r', 'LineWidth', 3);
axis([min(p), max(p), min(K), max(K), dmin, dmax]);
view(60, 30);
title('\bfMaximum Allowable Slippage for Initiating Position');
xlabel('Exercise Probability');
ylabel('Strike Price');
zlabel('Maximum Slippage (Basis Points)');
hold off
%% Maximum allowable slippage for ongoing position
% Illustrate maximum allowable slippage for an ongoing covered-call position such that any
% slippage above this value renders the uncovered position preferable to the covered position. An
% ongoing position assumes that the position is covered at the start of the investment period so
% that the call premium for the position does not contribute to total return during the period.
% The result is a plot that displays maximum allowable slippage as a function of strike price and
% exercise probability. The red line on the plot is the maximum allowable slippage if the option
% is exercised immediately when the stock price crosses above the strike price at any time during
% the investment period.
X0 = 40; % initial stock price
T = 23/252; % duration of investment period (years)
r0 = 0.02; % risk free rate (annualized)
m = 0.1; % stock price drift (annualize)
s = 0.24; % stock price diffusion (annualized)
TC = 64/252; % expiration date of call option
K = linspace(40, 45, 21); % strike prices
p = linspace(0.05, 1, 20); % exercise probabilities
d = NaN(Knum, pnum); % slippage as function of strike and probabilities
for i = 1:Knum
XT = X0*exp(m*T);
for j = 1:pnum
C = blsprice(X0, K(i), r0, TC, s);
xi = C*r0*T/XT;
if p(j) > xi
d(i,j) = (xi/p(j))/(1 - xi/p(j));
end
end
end
P = zeros(Knum, 1); % hitting time exercise probabilities
dP = NaN(Knum, 1); % maximum slippage for hitting time exercise probabilities
for i = 1:Knum
XT = X0*exp(m*T);
P(i) = gbm_upcrossing_prob(X0, K(i), T, m, s);
C = blsprice(X0, K(i), r0, TC, s);
xi = C*r0*T/XT;
if P(i) > xi
dP(i) = (xi/P(i))/(1 - xi/P(i));
end
end
d = 10000*d;
dP = 10000*dP;
dmin = min(min(min(d)),min(dP));
dmin = min(0, dmin);
dmax = max(max(max(d)),max(dP));
dmax = min(20, dmax);
clf;
surf(p, K, d, 'FaceColor', 'b', 'EdgeColor', 'b', 'FaceAlpha', 0.3);
hold on
plot3(P, K, dP, 'r', 'LineWidth', 3);
axis([min(p), max(p), min(K), max(K), dmin, dmax]);
view(60, 30);
title('\bfMaximum Allowable Slippage for Ongoing Position');
xlabel('Exercise Probability');
ylabel('Strike Price');
zlabel('Maximum Slippage (Basis Points)');
hold off
