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

Copyright 2011, 2016, 2018 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/.
The original 2011 R code will not fully work on a recent R because there have been some changes to libraries. The latest version of the Matlab code only uses functions from Matlab toolboxes.
The GARCH functionality in the econometric toolbox in Matlab is trying to be too clever, but can't deliver and could well be buggy. If you want to try that, here are the docs (estimate). Besides, it can only do univariate GARCH and so can't be used in Chapter 3. Kevin Sheppard's MFE toolbox is much better, while not as user friendly, it is much better written and is certainly more comprehensive. It can be downloaded here and the documentation here is quite detailed.

Listing 7.1/7.2: Transformation in R Last updated 2011

x=seq(-3,3,by=0.1)
plot(x,pnorm(x),type="l")

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 R Last updated 2011

set.seed(12) # set seed
S=10
runif(S)
rnorm(S)
rt(S,4)

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 R Last updated August 2016

yield=c(5.00, 5.69, 6.09, 6.38, 6.61,
6.79, 6.94, 7.07, 7.19, 7.30) # yield curve
TT=length(yield)
r=0.07                                # initial yield rate
Par=10                                # par value
coupon=r*Par                          # coupon payments
cc=1:10*0+coupon                      # vector of cash flows
cc[10]=cc[10]+Par                     # add par to cash flows
P=sum(cc/((1+yield/100)^(1:TT)))      # calculate price
print(P)

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 R Last updated August 2016

set.seed(12)                       # set seed
sigma = 1.5                        # daily yield volatiltiy
S = 8                              # number of simulations
r = rnorm(S,0,sigma)               # generate random numbers
ysim = matrix(nrow=TT,ncol=S)
for (i in 1:S) ysim[,i]=yield+r[i]
matplot(ysim,type='l')

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 R Last updated August 2016

SP = vector(length=S)
for (i in 1:S){                                  # S simulations
SP[i] = sum(cc/((1+ysim[,i]/100)^(TT)))
}
SP = SP-(mean(SP) - P)                           # correct for mean
barplot(SP)
hist(SP,probability=TRUE)
x=seq(6,16,length=100)
lines(x, dnorm(x, mean = mean(SP), sd = sd(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 R Last updated August 2016

P0 = 50                 # initial spot price
sigma = 0.2             # annual volatility
r = 0.05                # annual interest
TT = 0.5                # time to expiration
X = 40                  # strike price
f = bs(X,P0,r,sigma,TT) # analytical call price
## this calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
print(f)

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 R Last updated August 2016

set.seed(12)                                   # set seed
S = 1e6                                        # number of simulations
F = P0*exp(r*TT)                               # futures price
ysim = rnorm(S,-0.5*sigma^2*TT,sigma*sqrt(TT)) # sim returns, lognorm corrected
F=F*exp(ysim)                                  # sim futures price
SP = F-X                                       # payoff
SP[SP<0] = 0                                   # set negative outcomes to zero
fsim = SP*exp(-r*TT)                           # discount
call_sim = mean(fsim)                          # simulated price
print(call_sim)

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 = mean(fsim)                          # simulated price
print(call_sim)


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

hist(F,probability=TRUE,ylim=c(0,0.06))
x=seq(min(F),max(F),length=100)
lines(x, dnorm(x, mean = mean(F), sd = sd(SP)))
hist(fsim,nclass=100,probability=TRUE)

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 R Last updated 2011

set.seed(1)                           # 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 = rnorm(S,r/365-0.5*s2,sqrt(s2)) # sim returns
Psim = P*exp(ysim)                    # sim future prices
q = sort(Psim-P)                      # simulated P/L
VaR1 = -q[p*S]
print(VaR1)

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 R Last updated August 2016

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

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 R Last updated August 2016

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

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 R Last updated 2011

library (MASS)
set.seed(12)                                      # set seed
mu = c(r/365,r/365)                               # return mean
Sigma = matrix(c(0.01,0.0005,0.0005,0.02),ncol=2) # covariance matrix
y = mvrnorm(S,mu,Sigma)                           # 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 R Last updated 2011

K=2
P = c(100,50)                                 # prices
x = c(1,1)                                    # number of assets
Port = P %*% x                                # portfolio at t
Psim = matrix(t(matrix(P,K,S)),ncol=K)*exp(y) # simulated prices
PortSim = Psim %*% x                          # simulated portfolio value
q = sort(PortSim-Port[1,1])                   # simulated P/L
VaR4 = -q[S*p]
print(VaR4)

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 R with an option Last updated August 2016

f = bs(P[2],P[2],r,sigma,TT)
fsim = bs(P[2],Psim[,2],r,sigma,TT-(1/365))
q = sort(fsim$Call+Psim[,1]-f$Call-P[1]);
VaR5 = -q[p*S]
print(VaR5)

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)