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


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 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 7.1/7.2: Transformation in Julia
Last updated June 2018

x = collect(linspace(-3, 3, 61))
using Plots;
using Distributions;
return plot(x, cdf.(Normal(0,1), x))
		

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 7.3/7.4: Various RNs in Julia
Last updated June 2018

srand(12);                     # set seed
S = 10;
println(rand(Uniform(0,1), S)) # alternatively, rand(S)
println(rand(Normal(0,1), S))  # alternatively, randn(S)
println(rand(TDist(4), S))
		

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 7.5/7.6: Price bond in Julia
Last updated June 2018

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_c)
r = 0.07                                 # initial yield rate
Par = 10                                 # par value
coupon = r * Par                         # coupon payments
cc = repeat([coupon], outer = 10)        # vector of cash flows
cc[10] += Par                            # add par to cash flows
P = sum(cc./((1+yield_c/100).^(1:T)))    # calc price
		

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 7.7/7.8: Simulate yields in Julia
Last updated June 2018

srand(12)                              # set seed
sigma = 1.5                            # daily yield volatility
S = 8                                  # number of simulations
r = rand(Normal(0,1), S)               # generate random numbers
ysim = fill!(Array{Float64}(T,S), NaN)
for i in range(1, S)
    ysim[:,i] = yield_c + r[i]
end
using Plots;
plot(ysim)
		

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 7.9/7.10: Simulate bond prices in Julia
Last updated June 2018

SP = fill!(Array{Float64}(S,1), NaN)
for i in range(1,S)                              # S simulations
    SP[i] = sum(cc./(1+ysim[:,i]./100).^T)
end
SP -= (mean(SP) - P)                             # correct for mean
using Plots;
bar(SP)
S = 50000
r = randn(S) * sigma
ysim = fill!(Array{Float64}(T,S), NaN)
for i in range(1, S)
    ysim[:,i] = yield_c + r[i]
end
SP = fill!(Array{Float64}(S,1), NaN)
for i in range(1,S)
    SP[i] = sum(cc./(1+ysim[:,i]./100).^T)
end
SP -= (mean(SP) - P)
using Plots;
histogram(SP,nbins=100,normed=true,xlims=(7,13))
res = fit_mle(Normal, SP)
plot!(Normal(res.μ, res.σ), linewidth = 4)
		

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 7.11/7.12: Simulate bond prices in Julia
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)
		

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 7.13/7.14: Black-Scholes simulation in Julia
Last updated June 2018

srand(12)                                   # set seed
S = 10^6                                    # number of simulations
F = P0 * exp(r * T)                         # futures price
ysim = randn(S)*sigma*sqrt(T)-0.5*sigma^2*T # 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 * T)                     # discount
call_sim = mean(fsim)                       # simulated price
		

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 7.15/7.16: Option density plots in Julia
Last updated June 2018

using Plots;
histogram(F, bins = 100, normed = true, xlims = (20,80))
res = fit_mle(Normal, F)
plot!(Normal(res.μ, res.σ), linewidth = 4)
vline!([X], linewidth = 4, color = "black")
histogram(fsim, bins = 110, normed = true, xlims = (0,35))
vline!([f["Call"]], linewidth = 4, color = "black")
		

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 7.17/7.18: Simulate VaR in Julia
Last updated June 2018

srand(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 = randn(S) * 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[convert(Int, 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 7.19/7.20: Simulate option VaR in Julia
Last updated June 2018

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[convert(Int, 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 7.21/7.22: Example 7.3 in Julia
Last updated June 2018

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 = sort(f1sim["Call"] + f2sim["Put"] + Psim - f1["Call"] - f2["Put"] - P)
VaR2 = -q[convert(Int, p*S)]
		

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

using Distributions;
srand(12);                          # set seed
mu = Vector([r/365, r/365]);        # return mean
Sigma = [0.01 0.0005; 0.0005 0.02]; # covariance matrix
y = rand(MvNormal(mu,Sigma), S);    # simulated returns
		

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 7.25/7.26: Two-asset VaR in Julia
Last updated June 2018

K = 2
P = [100 50]                     # prices
x = [1 1]                        # number of assets
Port = reshape(P * x', 1)[1]     # portfolio at t
Psim = repmat(P, S, 1).*exp.(y)' # simulated prices
PortSim = reshape(Psim * x', S)  # simulated portfolio value
q = sort(PortSim - Port)         # simulated P/L
VaR4 = -q[convert(Int, 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)
		
Listing 7.27/7.28: A two-asset case in Julia with an option
Last updated June 2018

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[convert(Int, p * S)]