The volatility surface generated by an option

Heston: simulation and calibration

Heston: simulation and calibration

Heston

For my assignment project in the Derivatives MSc course I chose to focus on the Heston Model. I did it using Matlab.

The Black and Scholes Model has stochastic returns. Heston models prices as also having stochastic volatility.

My assignment project addressed the behaviour of an option, both in a “B&S world” and in a “Heston world”, showing differences between the two such as kurtosis and the smile effect. I simulated stock trajectories both with MonteCarlo (it’s two correlated stochastic processes – returns and volatility) and using the analytic formula. With the analytic formula I plotted the volatility surface of such an option. I also calibrated the parameters of the analytic formula on market data, choosing an appropriate optimization function.

The files of my project are:

I paste the .pdf paper and all the code below (unfortunately I lose the format):

Heston
Avvio.m:
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%---1) MONTECARLO: SIMULIAMO nTrials TRAIETTORIE DI PREZZO in HESTON e in
% BLACK & SCHOLES ---%%% Prezzo, volatilità, curtosi

clc;
clear all;
warning off;
close all;

nTrials = 200;
strike = 15;
inizioPrezzi = 5;
numPrezzi = 1;

speed = 3;
nPeriods = 500;
scadenza = 50;
mu = 0.02;
deltaTime=1/nPeriods;
volavola = 3;
level = 0.9;
startstatevola = 1.3;
call = 1;
r = 0.01;
correlation = 0.5;

startState = [15; startstatevola];
% B&S: usa vola di lp di Heston (con X)
[PrezziBeS, VolaBeS, TempiBeS] = generaPrezziBeS(mu, level, startState, nPeriods, nTrials, deltaTime);
% Heston
[PrezziHeston, VolaHeston, tempiHeston] = generaPrezziHeston(mu, speed, level, volavola, correlation, startState, nPeriods, nTrials, deltaTime);
volaHeston = mean(VolaHeston, 1); % vettore con le volatilità medie nelle diverse scadenze

% Plotta prezzi e volatilità dei due
figure()
subplot(2,2,1);
hold on
for i = 1:nTrials
plot(TempiBeS, PrezziBeS(i,:));%prezzo
title('B&S: PREZZI');
end
subplot(2,2,3);
hold on;
for i = 2:nTrials
plot(TempiBeS, VolaBeS(i,:)./PrezziBeS(i,:));%prezzo
title('B&S: sigma');
end
subplot(2,2,2);
hold on
for i = 1:nTrials
plot(tempiHeston, PrezziHeston(i,:));%prezzo
title('HESTON: PREZZI');
end
subplot(2,2,4);
hold on;
for i = 2:nTrials
plot(tempiHeston, VolaHeston(i,:));
title('Heston: sigma');
end
% MOSTRA LA CURTOSI
figure();
lineWidth = 2;
subplot(1,2,1);
hold on;
returns = tick2ret(PrezziBeS'); % Calcola rendimenti delle traiettorie generate con Heston
muHist = mean(mean(returns));
sigmaHist = sqrt(var(returns(:)));
curtosi = kurtosis(returns(:));
histogram(real(returns),'Normalization','pdf');
y = -0.15:0.001:0.15;
f = exp(-(y-muHist).^2./(2*sigmaHist^2))./(sigmaHist*sqrt(2*pi));
plot(y,f,'LineWidth',lineWidth) % Disegna la normale
title(strcat('Black & Scholes, curtosi: ', num2str(curtosi)));

subplot(1,2,2);
hold on;
returns = tick2ret(real(PrezziHeston'));
muHist = mean(mean(returns));
sigmaHist = sqrt(var(returns(:)));
curtosi = kurtosis(returns(:));
histogram(real(returns),'Normalization','pdf');
y = -0.15:0.001:0.15;
f = exp(-(y-muHist).^2./(2*sigmaHist^2))./(sigmaHist*sqrt(2*pi));
plot(y,f,'LineWidth',lineWidth)
title(strcat('Heston, curtosi: ', num2str(curtosi)));

%% 2) MONTECARLO: SMILE (prezzo fisso-strike varia) HESTON

clc;
clear all;
% close all;
rng default;

%parametri da modificare
nTrials = 400
nSteps = 100;
prezzo = 15;
inizioStrike = 10;
numStrike = 10;

%parametri da lasciare
speed = 0.2;
nPeriods = 5;
deltaTime=1/nPeriods;

ignoraVola =true;

mu = 0.02;
volavola = 1.83;
levola = 0.02; %volatilità di partenza e di lp
level = levola;
startstatevola = levola;

r = 0.01; % tasso istantaneo di rendimento risk free
correlation = -0.5;

Vola3Scadenze = zeros(numStrike,3);
scadenza1 = 2;
scadenza2 = 3;
scadenza3 = 4;
prezziScad = zeros(numStrike,1);
rng default;
startState = [prezzo; startstatevola];

for u = 1:numStrike
[PrezziHeston, tempiHeston] = generaPrezziHeston2(mu, speed, level, volavola, correlation, startState, nPeriods, nTrials, deltaTime, ignoraVola, nSteps);

if (inizioStrike+u)>prezzo
call = 1;
callbls = true;
else
call = 0;
callbls = false;
end

% interrompi(u)
payoffsHeston = optionPrices2(call, inizioStrike+u, r, 1/nPeriods, tempiHeston, PrezziHeston);
mediePayoffs = payoffsHeston;

volatility = blsimpv(prezzo, inizioStrike+u, r, scadenza1/nPeriods, mediePayoffs(scadenza1),200, 0,[],callbls);
vola3Scadenze(u,1) = volatility;
volatility = blsimpv(prezzo, inizioStrike+u, r, scadenza2/nPeriods, mediePayoffs(scadenza2),200, 0,[],callbls);
vola3Scadenze(u,2) = volatility;
prezziScad(u) = mediePayoffs(scadenza2);
volatility = blsimpv(prezzo, inizioStrike+u, r, scadenza3/nPeriods, mediePayoffs(scadenza3),200, 0,[],callbls);
vola3Scadenze(u,3) = volatility;
end
% beep on;
% beep;
figure();
scatter(inizioStrike+1:numStrike+inizioStrike, prezziScad);
title(strcat('Prezzi opzioni put - call, HESTON, So= ', num2str(prezzo), 'maturity = ', num2str(scadenza2/nPeriods*250), ' giorni'));
xlabel('Strike');
ylabel('Prezzo');

figure();
hold on;
scatter((inizioStrike+1:1:prezzo), vola3Scadenze(1:prezzo-inizioStrike,1),'r','x');
scatter((prezzo+1:1:numStrike+inizioStrike), vola3Scadenze(prezzo-inizioStrike+1:numStrike,1),'b','x');
a = vola3Scadenze(1:numStrike,1)';
ind = inizioStrike+1:numStrike+inizioStrike;
k = ~isnan(a);
p = polyfit(ind(k),a(k),2);
refcurve(p);
axis([10 20 0 0.8]);
title(strcat('Moneyness - implied volatility, HESTON, So= ', num2str(prezzo), ' maturity= ', num2str(scadenza1/nPeriods*250), ' giorni'));
ylabel('Implied volatility');
xlabel('Strike');
legend(strcat('Y=(', num2str(p(1)),')X^2 +(', num2str(p(2)),')X+(', num2str(p(3)),')'));

figure();
hold on;
scatter((inizioStrike+1:1:prezzo), vola3Scadenze(1:prezzo-inizioStrike,2),'r','x');
scatter((prezzo+1:1:numStrike+inizioStrike), vola3Scadenze(prezzo-inizioStrike+1:numStrike,2),'b','x');
a = vola3Scadenze(1:numStrike,2)';
ind = inizioStrike+1:numStrike+inizioStrike;
k = ~isnan(a);
p = polyfit(ind(k),a(k),2);
refcurve(p);
axis([10 20 0 0.8]);
title(strcat('Moneyness - implied volatility, HESTON, So= ', num2str(prezzo), ' maturity= ', num2str(scadenza2/nPeriods*250), ' giorni'));
ylabel('Implied volatility');
xlabel('Strike');
legend(strcat('Y=(', num2str(p(1)),')X^2 +(', num2str(p(2)),')X+(', num2str(p(3)),')'));

figure();
hold on;
scatter((inizioStrike+1:1:prezzo), vola3Scadenze(1:prezzo-inizioStrike,3),'r','x');
scatter((prezzo+1:1:numStrike+inizioStrike), vola3Scadenze(prezzo-inizioStrike+1:numStrike,3),'b','x');
a = vola3Scadenze(1:numStrike,3)';
ind = inizioStrike+1:numStrike+inizioStrike;
k = ~isnan(a);
p = polyfit(ind(k),a(k),2);
refcurve(p);
axis([10 20 0 0.8]);
title(strcat('Moneyness - implied volatility, HESTON, So= ', num2str(prezzo), ' maturity= ', num2str(scadenza3/nPeriods*250), ' giorni'));
ylabel('Implied volatility');
xlabel('Strike');
legend(strcat('Y=(', num2str(p(1)),')X^2 +(', num2str(p(2)),')X+(', num2str(p(3)),')'));

%% MONTECARLO: SMILE (B&S)

clc;
clear all;
% close all;
rng default;

%parametri da modificare
nTrials = 1000;

prezzo = 15;
inizioStrike = 10;
numStrike = 10;

%parametri da lasciare
speed = 0.2;
nPeriods = 5;
deltaTime=1/nPeriods;
nSteps = 100;
ignoraVola =true;

mu = 0.02;
volavola = 1.83;
levola = 0.5;
level = levola;
startstatevola = levola;
r = 0.01;
correlation = -0.5;

Vola3Scadenze = zeros(numStrike,3);
scadenza1 = 2;
scadenza2 = 3;
scadenza3 = 4;
prezziScad = zeros(numStrike,1);
rng default;
startState = [prezzo; startstatevola];

for u = 1:numStrike
[PrezziBeS, VolaBeS, tempiBeS] = generaPrezziBeS2(mu, level, startState, nPeriods, nTrials, deltaTime, ignoraVola, nSteps);
clear VolaBeS;
if (inizioStrike+u)>prezzo
call = 1;
callbls = true;
else
call = 0;
callbls = false;
end
u
payoffsBeS = optionPrices(call, inizioStrike+u, r, 1/nPeriods, tempiBeS, PrezziBeS);
mediePayoffs = payoffsBeS;

volatility = blsimpv(prezzo, inizioStrike+u, r, scadenza1/nPeriods, mediePayoffs(scadenza1),200, 0,[],callbls);
vola3Scadenze(u,1) = volatility;

volatility = blsimpv(prezzo, inizioStrike+u, r, scadenza2/nPeriods, mediePayoffs(scadenza2),200, 0,[],callbls);
vola3Scadenze(u,2) = volatility;
prezziScad(u) = mediePayoffs(scadenza2);

volatility = blsimpv(prezzo, inizioStrike+u, r, scadenza3/nPeriods, mediePayoffs(scadenza3),200, 0,[],callbls);
vola3Scadenze(u,3) = volatility;
end
% beep on;
% beep;
figure();
scatter(inizioStrike+1:numStrike+inizioStrike, prezziScad);
title(strcat('Prezzi opzioni put - call, B&S, So= ', num2str(prezzo), ' maturity= ', num2str(scadenza2/nPeriods*250), ' giorni'));
xlabel('Strike');
ylabel('Prezzo');

figure();
hold on;
scatter((inizioStrike+1:1:prezzo), vola3Scadenze(1:prezzo-inizioStrike,1),'r','x');
scatter((prezzo+1:1:numStrike+inizioStrike), vola3Scadenze(prezzo-inizioStrike+1:numStrike,1),'b','x');
a = vola3Scadenze(1:numStrike,1)';
ind = inizioStrike+1:numStrike+inizioStrike;
k = ~isnan(a);
p = polyfit(ind(k),a(k),1);
refcurve(p);
axis([10 20 0 0.8]);
title(strcat('Moneyness - implied volatility, B&S, So= ', num2str(prezzo), 'maturity= ', num2str(scadenza1/nPeriods*250), ' giorni'));
xlabel('Implied volatility');
ylabel('Strike');

figure();
hold on;
scatter((inizioStrike+1:1:prezzo), vola3Scadenze(1:prezzo-inizioStrike,2),'r','x');
scatter((prezzo+1:1:numStrike+inizioStrike), vola3Scadenze(prezzo-inizioStrike+1:numStrike,2),'b','x');
a = vola3Scadenze(1:numStrike,2)';
ind = inizioStrike+1:numStrike+inizioStrike;
k = ~isnan(a);
p = polyfit(ind(k),a(k),1);
refcurve(p);
axis([10 20 0 0.8]);
title(strcat('Moneyness - implied volatility, B&S, So= ', num2str(prezzo), ' maturity= ', num2str(scadenza2/nPeriods*250), ' giorni'));
xlabel('Implied volatility');
ylabel('Strike');

figure();
hold on;
scatter((inizioStrike+1:1:prezzo), vola3Scadenze(1:prezzo-inizioStrike,3),'r','x');
scatter((prezzo+1:1:numStrike+inizioStrike), vola3Scadenze(prezzo-inizioStrike+1:numStrike,3),'b','x');
a = vola3Scadenze(1:numStrike,3)';
ind = inizioStrike+1:numStrike+inizioStrike;
k = ~isnan(a);
p = polyfit(ind(k),a(k),1);
refcurve(p);
axis([10 20 0 0.8]);
title(strcat('Moneyness - implied volatility, B&S, So= ', num2str(prezzo), ' maturity= ', num2str(scadenza3/nPeriods*250), ' giorni'));
xlabel('Implied volatility');
ylabel('Strike');
% beep;

%% 3) - FORMULA ANALITICA - volatility surface
clear all
% close all;
s0=1500;
vola0 = 1;
level = 0.1;
volavola = 5;
mu = 0.05;
speed = 2;
correlation = -0.3;
r = 0.03;

inizioK = 500;
fineK = 2500;
nintervalliK = 40;
intervalliK = ceil(fineK-inizioK)/nintervalliK;

k = inizioK:intervalliK:fineK;
k = k/s0;

ttm = 2:-0.04:0.1;

[X, Y] = meshgrid(ttm, k);
Z = zeros(length(k),length(ttm));
for i=1:length(k)
for j=1:length(ttm)
OptionPrice = hestonAnalitica(s0/s0, vola0, speed, level, volavola,...
correlation, r, ttm(j), k(i));
Z(i, j) = blsimpv(s0/s0, k(i), r, ttm(j), max(OptionPrice,0),100);
end
i
end

Y=-Y;
X=X*250;
figure();
mesh(X, Y, Z);
xlabel('Time to Maturity: giorni');
ylabel('Moneyness: Strike/prezzo');
zlabel('Implied volatility');
title('VOLATILITY SURFACE');

%% 4) CALIBRAZIONE
%shuffledArray = orderedArray(randperm(size(orderedArray,1)),:);
clear all; close all;
%variabili da prendere nella funzione di costo
global strikes; global maturity; global price; global r;

data = load('c_data2.csv');
% data = data(randperm(size(data,1)),:);
% for u = 5:-1:0
u = 0;
numData1 = u*380+1;
numData2 = numData1+30;

% numData1 = 1;
% numData2 = 30;
% numData = length(data(:,1));
strikes = data(numData1:numData2,1);
maturity = data(numData1:numData2,2); %frazioni di anno fra 3/9/2017 e scadenza
bid = data(numData1:numData2,3);
ask = data(numData1:numData2,4);
price = data(numData1:numData2,5);
% x0:
% s0, x1:vola0, x2:speed, x3:level, x4:volavola,x5:correlation, x6:r, ttm, k
% x0 = [1;0.5;0.5;0.5;-0.5; 0.01]
x0 = [0.5;0.5;0.5;0.5;-0.5]
r = 0.01;
startprice = 2476.55;

%price = hestonAnalitica(startprice, x0(1),x0(2),x0(3),x0(4),x0(5),x0(6), ttm, k)

%c = costFunction(x0)

lb = [0;0;0;0;-1]
ub = [1;1;10;10;1]
x = lsqnonlin(@costFunction,x0,lb,ub)

n = length(price)
predicted_price = zeros(n,1);

for i=1:n
predicted_price(i) = hestonAnalitica(startprice, x(1),x(2),x(3),x(4),x(5),r, maturity(i), strikes(i));
end
%[bid,price, predicted_price,ask]
subplot(3,2,u+1);
scatter(price,predicted_price,'filled');
hold on;
% confronto predetti/bisettrice
plot([min(price),max(price)],[min(price),max(price)]);
title(strcat(num2str(min(strikes)),' - ',num2str(max(strikes))))
xlabel('predetti')
ylabel('effettivi')
%%per far respirare il pc
% if mod(u,2)==1
% beep();
% disp('premere un tasto');
% pause();
% end;
% end

%%

inizioK = 1500;
fineK = 3000;
nintervalliK = 40;
intervalliK = ceil(fineK-inizioK)/nintervalliK;
s0 = startprice;

k = inizioK:intervalliK:fineK;
k = k/s0;

ttm = 1:-0.02:0.05;

[X, Y] = meshgrid(ttm, k);
Z = zeros(length(k),length(ttm));
for i=1:length(k)
for j=1:length(ttm)
OptionPrice = hestonAnalitica(s0/s0, x(1), x(2), x(3), x(4), x(5), r, ttm(j), k(i));
Z(i, j) = blsimpv(s0/s0, k(i), r, ttm(j), max(OptionPrice,0),100);
end
i
end

Y=-Y;
X=X*250;
figure();
mesh(X, Y, Z);
xlabel('Time to Maturity: giorni');
ylabel('Moneyness: Strike/prezzo');
zlabel('Implied volatility');
title('VOLATILITY SURFACE');
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function [pay] = optionPrices(call, strike, r, deltaTime, tempi, prezziGenerati)
%HESTONOPTIONPRICE Summary of this function goes here

attualizzazione = exp(-tempi*r);
payoffs = prezziGenerati-strike;
if call==1
payoffs = max(payoffs, 0);
else
payoffs = max (-payoffs, 0);
end
% Attualizza i payoff
pay = mean(payoffs);
pay = pay*diag(attualizzazione);

end
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function [ prezzo ] = hestonAnalitica(s0, vola0, speed, level, volavola,...
correlation, r, ttm, k )
%HESTONANALITICA Summary of this function goes here
% Detailed explanation goes here

integrando1 = @(s,s0, vola0, level, speed, volavola, r, correlation, ttm, k) real(exp(-i.*s*log(k)).*...
funzCarHeston(s0, vola0, level, speed, volavola, r, correlation, ttm, s-i)./...
(i*s.*funzCarHeston(s0, vola0, level, speed, volavola, r, correlation, ttm, -i)));
integrando2 = @(s,s0, vola0, level, speed, volavola, r, correlation, ttm, k) real(exp(-i.*s*log(k)).*...
funzCarHeston(s0, vola0, level, speed, volavola, r, correlation, ttm, s)./...
(i*s));

integrale1 = integral(@(s)integrando1(s,s0, vola0, level, speed, volavola, r, correlation, ttm, k),0,100);
integrale2 = real(integral(@(s)integrando2(s,s0, vola0, level, speed, volavola, r, correlation, ttm, k),0,100));

p1 = 0.5+integrale1/pi;
p2 = 0.5+integrale2/pi;

prezzo = s0*p1-k*p2*exp(-r*ttm);
end
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function [prezziGenerati, volaGenerati, tempi] = generaPrezziBeS(mu, level, startState, nPeriods, nTrials, deltaTime)

% GENERAPREZZIB&S Summary of this function goes here
% Genera nTrials traiettorie secondo B&S, su nPeriods scadenze.

F = @(t, X) mu*X;
G = @(t, X) X*level;
besSde = sde(F, G, 'StartState', startState(1));
[traiettorie, tempi, z] = simByEuler(besSde, nPeriods, 'nTrials', nTrials, 'DeltaTime', deltaTime);
traiettorie = permute(traiettorie, [3, 1, 2]);
prezziGenerati = traiettorie(:, :, 1);
volaGenerati = G(tempi, prezziGenerati);

end
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function costo = costFunction(x)

global strikes; global maturity; global price; global r;
%COSTFUNCTION Summary of this function goes here
% Detailed explanation goes here

startprice = 2476.55;
% k = 1925;
% ttm = 0.0383561643835616;
% actualprice = 551.15;

n = length(strikes);
cost = zeros(n,1);
for i=1:n
prezzoHest = hestonAnalitica(startprice, x(1),x(2),x(3),x(4),x(5),r, maturity(i), strikes(i));
%prezzoHest = hestonAnalitica(startprice, x(1),x(2),(x(4)+x(3)^2)/(2*x(2)),x(4),x(5),x(6), maturity(i), strikes(i));
cost(i) = (prezzoHest-price(i)).^2;
end
% costo = sum(cost)/n;
costo = cost;
end
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function [ funzCarHeston ] = funzCarHeston(s0, vola0, level, speed, volavola, r, correlation, ttm, s)
%FUNZCARHESTON Summary of this function goes here
% Detailed explanation goes here

alpha = -s.*s/2 - i*s/2;
beta = alpha - correlation*volavola*i*s;
gamma = volavola*volavola/2;
h = sqrt(beta.*beta - 4*alpha*gamma);
rplus = (beta + h)/volavola/volavola;
rminus = (beta - h)/volavola/volavola;
g=rminus./rplus;

C = spee

function [prezziGenerati, tempi] = generaPrezziHeston(mu, speed, level, volavola, correlation, startState, nPeriods, nTrials, deltaTime, ignoraVola, nSteps)
% GENERAPREZZIHESTON Summary of this function goes here
% Genera nTrials traiettorie secondo il processo di Heston, su nPeriods scadenze.

hestonSde = heston(mu, speed, level, volavola,...
'StartState', startState, 'Correlation', correlation);
[traiettorie, tempi] = simByEuler(hestonSde, nPeriods,...
'nTrials', nTrials, 'DeltaTime', deltaTime, 'nSteps', nSteps);
prezziGenerati = squeeze(traiettorie(:,1,:))';
end

d*(rminus*ttm-(2/volavola^2).*log((1-g.*exp(-h*ttm))./(1-g)));
D = rminus.*(1-exp(-h*ttm))./(1-g.*exp(-h*ttm));

funzCarHeston = exp(C*level+D*vola0+i*s*log(s0*exp(r*ttm)));

end

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