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

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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 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 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 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 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 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 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 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 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 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 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: 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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)]