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


Copyright 2011 - 2020 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 July 2020

x=-3:0.1:3;
plot(x,normcdf(x))
title("CDF of Normal Distribution")
		
Listing 7.1/7.2: Transformation in Python
Last updated July 2020

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()
		

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 July 2020

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))
		

Listing 7.5/7.6: Price bond in MATLAB
Last updated July 2020

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
		
Listing 7.5/7.6: Price bond in Python
Last updated July 2020

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)
		

Listing 7.7/7.8: Simulate yields in MATLAB
Last updated July 2020

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")
		
Listing 7.7/7.8: Simulate yields in Python
Last updated July 2020

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()
		

Listing 7.9/7.10: Simulate bond prices in MATLAB
Last updated July 2020

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")
		
Listing 7.9/7.10: Simulate bond prices in Python
Last updated November 2020

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()
		

Listing 7.11/7.12: Black-Scholes valuation in MATLAB
Last updated July 2020

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 July 2020

## This calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
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)
		

Listing 7.13/7.14: Black-Scholes simulation in MATLAB
Last updated July 2020

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
		
Listing 7.13/7.14: Black-Scholes simulation in Python
Last updated July 2020

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)
		

Listing 7.15/7.16: Option density plots in MATLAB
Last updated July 2020

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');
		
Listing 7.15/7.16: Option density plots in Python
Last updated July 2020

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()
		

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 July 2020

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)
		

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 July 2020

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)
		

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 July 2020

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)
		

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

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 July 2020

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)
		

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 July 2020

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)