Chapter 7. Simulation Methods for VaR for Options and Bonds (in MATLAB/Python)


Copyright 2011 - 2019 Jon Danielsson. This code is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This code is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The GNU General Public License is available at: https://www.gnu.org/licenses/.


Listing 7.1/7.2: Transformation in MATLAB
Last updated 2011

x=-3:0.1:3;
plot(x,normcdf(x))
		
Listing 7.1/7.2: Transformation in Python
Last updated June 2018

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.show()
plt.close()
		

Listing 7.3/7.4: Various RNs in MATLAB
Last updated August 2016

rng default; % set seed
S=10;
rand(S,1)
randn(S,1)
trnd(4,S,1)
		
Listing 7.3/7.4: Various RNs in Python
Last updated June 2018

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

Listing 7.5/7.6: Price bond in MATLAB
Last updated 2011

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(10)=cc(10)+Par;                   % add par to cash flows
P=sum(cc./((1+yield./100).^(1:T)))   % calculate price
		
Listing 7.5/7.6: Price bond in Python
Last updated June 2018

import numpy as np
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[9] += 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)
		

Listing 7.7/7.8: Simulate yields in MATLAB
Last updated 2011

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)
		
Listing 7.7/7.8: Simulate yields in Python
Last updated June 2018

import numpy as np
import matplotlib.pyplot as plt
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.show()
plt.close()
		

Listing 7.9/7.10: Simulate bond prices in MATLAB
Last updated 2011

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)
		
Listing 7.9/7.10: Simulate bond prices in Python
Last updated June 2018

import numpy as np
import matplotlib.pyplot as plt
S = 8
SP = np.zeros([S])
for i in range(S):                                        # S simulations
    SP[i] = np.sum(cc/((1+ysim[:,i]/100)**T))
SP -= (np.mean(SP) - P)                                   # correct for mean
plt.bar(range(1,S+1), SP)
plt.show()
plt.close()
S = 50000
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/((1+ysim[:,i]/100)**T))
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.show()
plt.close()
		

Listing 7.11/7.12: Black-Scholes valuation in MATLAB
Last updated 2011

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
%% this calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
		
Listing 7.11/7.12: Black-Scholes valuation in Python
Last updated June 2018

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
## this calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
print (f)
		

Listing 7.13/7.14: Black-Scholes simulation in MATLAB
Last updated 2011

randn('state',0);                            % set seed
S = 1e6;                                     % number of simulations
F = P0*exp(r*T);                             % futures price
ysim=randn(S,1)*sigma*sqrt(T)-0.5*T*sigma^2; % sim returns, lognorm corrected
F=F*exp(ysim);                               % sim futures price
SP = F-X;                                    % payoff
SP(find(SP < 0)) = 0;                        % set negative outcomes to zero
fsim =SP*exp(-r*T) ;                         % discount
mean(fsim)                                   % simulated price
		
Listing 7.13/7.14: Black-Scholes simulation in Python
Last updated June 2018

import numpy as np
np.random.seed(12)                             # set seed
S = 10**6                                      # number of simulations
F = P0 * np.exp(r * T)                         # futures price
ysim=np.random.normal(-0.5*sigma**2*T,
                      sigma*np.sqrt(T),size=S) # sim returns, lognorm corrected
F = F * np.exp(ysim)                           # sim futures price
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)
		

Listing 7.15/7.16: Option density plots in MATLAB
Last updated 2011

subplot(1,2,1)
histfit(F);
subplot(1,2,2)
hist(fsim,100);
		
Listing 7.15/7.16: Option density plots in Python
Last updated June 2018

import numpy as np
import matplotlib.pyplot as plt
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.show()
plt.close()
plt.hist(fsim, bins = 60, range = (0, 35), density = True)
plt.axvline(x=f['Call'], color='k')
plt.show()
plt.close()
		

Listing 7.17/7.18: Simulate VaR in MATLAB
Last updated 2011

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)
		
Listing 7.17/7.18: Simulate VaR in Python
Last updated June 2018

import numpy as np
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[int(p*S) - 1]
print(VaR1)
		

Listing 7.19/7.20: Simulate option VaR in MATLAB
Last updated 2011

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)
		
Listing 7.19/7.20: Simulate option VaR in Python
Last updated June 2018

import numpy as np
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[int(p*S) - 1]
print(VaR2)
		

Listing 7.21/7.22: Example 7.3 in MATLAB
Last updated 2011

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)
		
Listing 7.21/7.22: Example 7.3 in Python
Last updated June 2018

import numpy as np
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[int(p*S) - 1]
print(VaR3)
		

Listing 7.23/7.24: Simulated two-asset returns in MATLAB
Last updated 2011

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
		
Listing 7.23/7.24: Simulated two-asset returns in Python
Last updated June 2018

import numpy as np
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
		

Listing 7.25/7.26: Two-asset VaR in MATLAB
Last updated 2011

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)
		
Listing 7.25/7.26: Two-asset VaR in Python
Last updated June 2018

import numpy as np
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[int(p*S) - 1]
print(VaR4)
		

Listing 7.27/7.28: A two-asset case in MATLAB with an option
Last updated 2011

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)
		
Listing 7.27/7.28: A two-asset case in Python with an option
Last updated June 2018

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