Chapter 7. Simulation Methods for VaR for Options and Bonds (in R/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 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 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 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 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 R
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 = length(yield)    # number of time periods
r = 0.07   # initial yield rate
Par = 10   # par value
coupon = r * Par     # coupon payments
cc = rep(coupon, T)  # vector of cash flows
cc[T] = cc[T] + Par  # add par to cash flows
P = sum(cc/((1+yield/100)^(1:T)))     # calculate price
print(P)
		
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 R
Last updated August 2019

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=length(yield),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 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 R
Last updated November 2020

SP = rep(NA, length = S)
for (i in 1:S){      # S simulations
  SP[i] = sum(cc/((1+ysim[,i]/100)^(1:T)))
}
SP = SP-(mean(SP) - P) # correct for mean
par(mfrow=c(1,2), pty="s")
barplot(SP)
hist(SP,probability=TRUE)
x=seq(6,16,length=100)
lines(x, dnorm(x, mean = mean(SP), sd = sd(SP)))
S = 50000
r = rnorm(S, 0, sigma) # generate random numbers
ysim = matrix(nrow=length(yield),ncol=S)
for (i in 1:S) ysim[,i]=yield+r[i]
SP = rep(NA, length = S)
for (i in 1:S){      # S simulations
  SP[i] = sum(cc/((1+ysim[,i]/100)^(1:T)))
}
SP = SP-(mean(SP) - P) # correct for mean
par(mfrow=c(1,2), pty="s")
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 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 R
Last updated July 2020

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 = X, P = P0, r = r, sigma = sigma, T = 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 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 R
Last updated July 2020

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 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 R
Last updated 2011

par(mfrow=c(1,2), pty="s")
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 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 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 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 R
Last updated July 2020

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 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 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 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 R
Last updated July 2020

library (MASS)
set.seed(12)         # set seed
mu = rep(r/365, 2)   # 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

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 July 2020

K=2
P = c(100, 50)       # prices
x = rep(1, 2)        # 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 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 R with an option
Last updated July 2020

f = bs(X = P[2], P = P[2], r = r, sigma = sigma, T = TT)
fsim = bs(X = P[2], P = Psim[,2], r = r, sigma = sigma, T = 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 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)