Chapter 7. Simulation Methods for VaR for Options and Bonds

x=-3:0.1:3;
plot(x,normcdf(x))
title("CDF of Normal Distribution")

import numpy as np
import matplotlib.pyplot as plt
x = np.arange(-3,3.1, step = 0.1) # Python's arange excludes the last value
plt.plot(x, stats.norm.cdf(x))
plt.title("CDF of Normal distribution")
plt.show()
plt.close()

rng default; % set seed
S=10;
rand(S,1)
randn(S,1)
trnd(4,S,1)

np.random.seed(12)     # set seed
S = 10                 # simulation size
print (np.random.uniform(size = S))
print (np.random.normal(size = S))
print (np.random.standard_t(df = 4,size = S))

yield = [5.00 5.69 6.09 6.38 6.61...
         6.79 6.94 7.07 7.19 7.30]; % yield curve
T = length(yield); 
r=0.07;                             % initial yield rate
Par=10;                             % par value
coupon=r*Par;                       % coupon payments
cc=zeros(1,T)+coupon;               % vector of cash flows 
cc(T)=cc(T)+Par;                    % add par to cash flows
P=sum(cc./((1+yield./100).^(1:T)))  % calculate price

yield_c = [5.00, 5.69, 6.09, 6.38, 6.61, 
           6.79, 6.94, 7.07, 7.19, 7.30]    # yield curve
T = len(yield_c)
r = 0.07                                    # initial yield rate
Par = 10                                    # par value
coupon = r * Par                            # coupon payments 
cc = [coupon] * 10                          # vector of cash flows
cc[T-1] += Par                              # add par to cash flows
P=np.sum(cc/(np.power((1+np.divide(yield_c,100)),
                      list(range(1,T+1))))) # calc price
print(P)

randn('state',123);         % set the seed
sigma = 1.5;                % daily yield volatility
S = 8;                      % number of simulations
r = randn(1,S)*sigma;       % generate random numbers
ysim=nan(T,S);
for i=1:S
    ysim(:,i)=yield+r(i);
end
ysim=repmat(yield',1,S)+repmat(r,T,1);
plot(ysim)
title("Simulated yield curves")

np.random.seed(12)                   # set seed
sigma = 1.5                          # daily yield volatility
S = 8                                # number of simulations
r = np.random.normal(0,sigma,size=S) # generate random numbers
ysim = np.zeros([T,S])
for i in range(S):
    ysim[:,i] = yield_c + r[i]
plt.plot(ysim)
plt.title("Simulated yield curves")
plt.show()
plt.close()

SP = nan(S,1);
for s = 1:S                                        % S simulations
    SP(s) = sum(cc./(1+ysim(:,s)'./100).^((1:T)));
end
SP = SP-(mean(SP) - P);                            % correct for mean
bar(SP)
S = 50000;
rng("default")
r = randn(S,1) * sigma;
ysim = nan(T,S);
for i = 1:S
    ysim(:,i) = yield' + r(i);
end
SP = nan(S,1);
for i = 1:S
    SP(i) = sum(cc./(1+ysim(:,i)'./100).^((1:T)));
end
SP = SP  - (mean(SP)-P);
histfit(SP)
title("Histogram of simulated bond prices with fitted normal")

S = 8                   # simulation size
SP = np.zeros([S])      # initializing vector with zeros
for i in range(S):      # S simulations
    SP[i] = np.sum(cc/(np.power((1+ysim[:,i]/100), list(range(1,T+1)))))
SP -= (np.mean(SP) - P) # correct for mean
plt.bar(range(1,S+1), SP)
plt.title("Simulated bond prices, S = 8")
plt.show()
plt.close()
S = 50000
np.random.seed(12)
r = np.random.normal(0, sigma, size = S)
ysim = np.zeros([T,S])
for i in range(S):
    ysim[:,i] = yield_c + r[i]
SP = np.zeros([S])
for i in range(S):     
    SP[i] = np.sum(cc/(np.power((1+ysim[:,i]/100), list(range(1,T+1)))))
SP -= (np.mean(SP) - P)
plt.hist(SP, bins = 30, range = (7, 13), density = True)
fitted_norm=stats.norm.pdf(np.linspace(7,13,30),
                           np.mean(SP),np.std(SP,ddof=1))
plt.plot(np.linspace(7,13,30), fitted_norm)
plt.title("Histogram for simulated bond prices, S = 50000")
plt.show()
plt.close()

P0 = 50;                      % initial spot price
sigma = 0.2;                  % annual volatility
r = 0.05;                     % annual interest
T = 0.5;                      % time to expiration
X = 40;                       % strike price
f = bs(X,P0,r,sigma,T)        % analytical call price

P0 = 50                     # initial spot price
sigma = 0.2                 # annual volatility
r = 0.05                    # annual interest
T = 0.5                     # time to expiration
X = 40                      # strike price
f = bs(X, P0, r, sigma, T)  # analytical call price
print(f)

randn('state',0);            % set seed
S = 1e6;                     % number of simulations
ysim = randn(S,1)*sigma*sqrt(T)-0.5*T*sigma^2; % sim returns, lognorm corrected
F = P0*exp(r*T)*exp(ysim);   % sim future prices
SP = F-X;                    % payoff
SP(find(SP < 0)) = 0;        % set negative outcomes to zero
fsim = SP * exp(-r*T) ;      % discount
mean(fsim)                   % simulated price  

np.random.seed(12)          # set seed
S = 10**6                   # number of simulations
ysim=np.random.normal(-0.5*sigma**2*T,
                      sigma*np.sqrt(T),size=S) # sim returns, lognorm corrected
F = P0 * np.exp(r * T) * np.exp(ysim)          # sim future prices
SP = F - X                  # payoff
SP[SP < 0] = 0              # set negative outcomes to zero
fsim = SP * np.exp(-r * T)  # discount
call_sim = np.mean(fsim)    # simulated price
print(call_sim)

subplot(1,2,1)
histfit(F);
title("Simulated prices");
xline(X, 'LineWidth', 1, 'label', 'Strike');
subplot(1,2,2)
hist(fsim,100);
title("Option price density");
xline(mean(fsim), 'LineWidth', 1, 'label', 'Call');

plt.hist(F, bins = 60, range = (20,80), density = True)
fitted_norm=stats.norm.pdf(np.linspace(20,80,60),np.mean(F),np.std(F,ddof=1))
plt.plot(np.linspace(20,80,60), fitted_norm)
plt.axvline(x=X, color='k')    
plt.title("Density of future prices")
plt.show()
plt.close()
plt.hist(fsim, bins = 60, range = (0, 35), density = True)
plt.axvline(x=f['Call'], color='k')
plt.title("Density of Call option prices")
plt.show()
plt.close()

randn('state',0);   % set seed
S = 1e7;            % number of simulations
s2 = 0.01^2;        % daily variance
p = 0.01;           % probability
r = 0.05;           % annual riskfree rate
P = 100;            % price today
ysim = randn(S,1)*sqrt(s2)+r/365-0.5*s2; % sim returns
Psim = P*exp(ysim);  % sim future prices 
q = sort(Psim-P);  % simulated P/L
VaR1 = -q(S*p)

from math import ceil
np.random.seed(1)       # set seed
S = 10**7               # number of simulations
s2 = 0.01**2            # daily variance
p = 0.01                # probability
r = 0.05                # annual riskfree rate
P = 100                 # price today
ysim=np.random.normal(r/365-0.5*s2,np.sqrt(s2),size=S) # sim returns
Psim = P * np.exp(ysim) # sim future prices
q = np.sort(Psim - P)   # simulated P/L
VaR1 = -q[ceil(p*S) - 1]
print(VaR1)

T = 0.25;                           % time to expiration
X = 100;                            % strike price
sigma = sqrt(s2*250);               % annual volatility
f = bs(X,P,r,sigma,T);              % analytical call price
fsim=bs(X,Psim,r,sigma,T-(1/365));  % sim option prices
q = sort(fsim.Call-f.Call);         % simulated P/L
VaR2 = -q(p*S)

T = 0.25       # time to expiration
X = 100                                 # strike price
sigma = np.sqrt(s2 * 250)               # annual volatility
f = bs(X, P, r, sigma, T)               # analytical call price
fsim = bs(X, Psim, r, sigma, T-(1/365)) # sim option prices
q = np.sort(fsim['Call']-f['Call'])     # simulated P/L
VaR2 = -q[ceil(p*S) - 1] 
print(VaR2)

X1 = 100;
X2 = 110;
f1 = bs(X1,P,r,sigma,T);
f2 = bs(X2,P,r,sigma,T);  
f1sim=bs(X1,Psim,r,sigma,T-(1/365));
f2sim=bs(X2,Psim,r,sigma,T-(1/365));
q = sort(f1sim.Call+f2sim.Put+Psim-f1.Call-f2.Put-P); 
VaR3 = -q(p*S)

X1 = 100
X2 = 110
f1 = bs(X1, P, r, sigma, T)
f2 = bs(X2, P, r, sigma, T)
f2sim = bs(X2, Psim, r, sigma, T-(1/365))
f1sim = bs(X1, Psim, r, sigma, T-(1/365))
q = np.sort(f1sim['Call'] + f2sim['Put'] + Psim - f1['Call'] - f2['Put'] - P)
VaR3 = -q[ceil(p*S) - 1]
print(VaR3)

randn('state',12)                 % set seed
mu = [r/365 r/365]';              % return mean
Sigma=[0.01 0.0005; 0.0005 0.02]; % covariance matrix
y = mvnrnd(mu,Sigma,S);           % simulated returns

np.random.seed(12)                                      # set seed
mu = np.transpose([r/365, r/365])                       # return mean
Sigma = np.matrix([[0.01, 0.0005],[0.0005, 0.02]])      # covariance matrix
y = np.random.multivariate_normal(mu, Sigma, size = S)  # simulated returns

K = 2; 
P = [100 50]';                        % prices
x = [1 1]';                           % number of assets
Port = P'*x;                          % portfolio at t
Psim = repmat(P,1,S)' .*exp(y);       % simulated prices
PortSim=Psim * x;                     % simulated portfolio value
q = sort(PortSim-Port);               % simulated P/L
VaR4 = -q(S*p)

import numpy.matlib
P = np.asarray([100, 50])              # prices
x = np.asarray([1, 1])                 # number of assets
Port = np.matmul(P, x)                 # portfolio at t
Psim = np.matlib.repmat(P,S,1)*np.exp(y) # simulated prices
PortSim = np.matmul(Psim, x)           # simulated portfolio value
q = np.sort(PortSim - Port)            # simulated P/L
VaR4 = -q[ceil(p*S) - 1]
print(VaR4)

f = bs(P(2),P(2),r,sigma,T);
fsim=bs(P(2),Psim(:,2),r,sigma,T-(1/365));
q = sort(fsim.Call+Psim(:,1)-f.Call-P(1)); 
VaR5 = -q(p*S)

f = bs(X = P[1], P = P[1], r = r, sigma = sigma, T = T)
fsim = bs(X = P[1], P = Psim[:,1], r = r, sigma = sigma, T = T-(1/365))
q = np.sort(fsim['Call'] + Psim[:,0] - f['Call'] - P[0])
VaR5 = -q[ceil(p*S) - 1]
print(VaR5)