Chapter 7. Simulation Methods for VaR for Options and Bonds (in R/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 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 2019

yield=c(5.00, 5.69, 6.09, 6.38, 6.61,
        6.79, 6.94, 7.07, 7.19, 7.30)       # yield curve
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:length(yield)))) # 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 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 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 2019

SP = vector(length=S)
for (i in 1:S){                                      # S simulations
  SP[i] = sum(cc/((1+ysim[,i]/100)^(length(yield))))
}
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 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 2019

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,length(yield)) # 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*length(yield))                                          # futures price
ysim = rnorm(S,-0.5*sigma^2*length(yield),sigma*sqrt(length(yield))) # 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*length(yield))                                      # 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

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 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,length(yield))               # analytical call price
fsim = bs(X,Psim,r,sigma,length(yield)-(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)]