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


Copyright 2011 - 2022 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 MATLAB
Last updated July 2020

x=-3:0.1:3;
plot(x,normcdf(x))
title("CDF of Normal Distribution")
		
Listing 7.1/7.2: Transformation in Julia
Last updated July 2020

x = -3:0.1:3
using Distributions, Plots;
return plot(x, cdf.(Normal(0,1), x), title = "CDF of Standard Normal", legend = false)
		

Listing 7.3/7.4: Various RNs in MATLAB
Last updated August 2016

rng default; % set seed
S=10;
rand(S,1)
randn(S,1)
trnd(4,S,1)
		
Listing 7.3/7.4: Various RNs in Julia
Last updated July 2020

using Random;
Random.seed!(12);    # set seed
S = 10;
println(S, " draws from Uniform(0,1): \n", rand(Uniform(0,1), S), "\n") # alternatively, rand(S)
println(S, " draws from Normal(0,1): \n", rand(Normal(0,1), S), "\n")   # alternatively, randn(S)
println(S, " draws from Student-t(4): \n", rand(TDist(4), S), "\n")
		

Listing 7.5/7.6: Price bond in MATLAB
Last updated July 2020

yield = [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);
r=0.07;    % initial yield rate
Par=10;    % par value
coupon=r*Par;        % coupon payments
cc=zeros(1,T)+coupon;          % vector of cash flows
cc(T)=cc(T)+Par;     % add par to cash flows
P=sum(cc./((1+yield./100).^(1:T)))  % calculate price
		
Listing 7.5/7.6: Price bond in Julia
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_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[T] += Par         # add par to cash flows
P = sum(cc./((1 .+ yield_c/100).^(1:T))) # calc price
println("Price of the bond: ", round(P, digits = 3), " USD")
		

Listing 7.7/7.8: Simulate yields in MATLAB
Last updated July 2020

randn('state',123);   % set the seed
sigma = 1.5;         % daily yield volatility
S = 8;     % number of simulations
r = randn(1,S)*sigma; % generate random numbers
ysim=nan(T,S);
for i=1:S
    ysim(:,i)=yield+r(i);
end
ysim=repmat(yield',1,S)+repmat(r,T,1);
plot(ysim)
title("Simulated yield curves")
		
Listing 7.7/7.8: Simulate yields in Julia
Last updated July 2020

Random.seed!(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(NaN, (T,S))
for i in 1:S
    ysim[:,i] = yield_c .+ r[i]
end
plot(ysim, title = "Simulated yield curves")
		

Listing 7.9/7.10: Simulate bond prices in MATLAB
Last updated July 2020

SP = nan(S,1);
for s = 1:S          % S simulations
    SP(s) = sum(cc./(1+ysim(:,s)'./100).^((1:T)));
end
SP = SP-(mean(SP) - P); % correct for mean
bar(SP)
S = 50000;
rng("default")
r = randn(S,1) * sigma;
ysim = nan(T,S);
for i = 1:S
    ysim(:,i) = yield' + r(i);
end
SP = nan(S,1);
for i = 1:S
    SP(i) = sum(cc./(1+ysim(:,i)'./100).^((1:T)));
end
SP = SP  - (mean(SP)-P);
histfit(SP)
title("Histogram of simulated bond prices with fitted normal")
		
Listing 7.9/7.10: Simulate bond prices in Julia
Last updated July 2020

using Statistics, StatsPlots;
SP = fill(NaN, S)
for i in 1:S         # S simulations
    SP[i] = sum(cc./(1 .+ ysim[:,i]./100).^(1:T))
end
SP .-= (mean(SP) - P) # correct for mean
bar(SP)
S = 50000
r = randn(S) * sigma
ysim = fill(NaN, (T,S))
for i in 1:S
    ysim[:,i] = yield_c .+ r[i]
end
SP = fill(NaN, S)
for i in 1:S
    SP[i] = sum(cc./(1 .+ ysim[:,i]./100).^(1:T))
end
SP .-= (mean(SP) - P)
histogram(SP,nbins=100,normed=true,xlims=(7,13), legend = false, title = "Histogram of simulated bond prices with fitted normal")
res = fit_mle(Normal, SP)
plot!(Normal(res.μ, res.σ), linewidth = 4, legend = false)
		

Listing 7.11/7.12: Black-Scholes valuation in MATLAB
Last updated July 2020

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 7.11/7.12: Simulate bond prices in Julia
Last updated June 2022

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

randn('state',0);    % set seed
S = 1e6;   % number of simulations
ysim = randn(S,1)*sigma*sqrt(T)-0.5*T*sigma^2; % sim returns, lognorm corrected
F = P0*exp(r*T)*exp(ysim);     % sim future prices
SP = F-X;  % payoff
SP(find(SP < 0)) = 0;          % set negative outcomes to zero
fsim = SP * exp(-r*T) ;        % discount
mean(fsim)           % simulated price
		
Listing 7.13/7.14: Black-Scholes simulation in Julia
Last updated July 2020

Random.seed!(12)     # set seed
S = 10^6   # number of simulations
ysim = randn(S) * sigma * sqrt(T) .- 0.5 * sigma^2 * T # sim returns, lognorm corrected
F = P0 * exp(r * T) * exp.(ysim)         # simulated future prices
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
println("Simulated call price: ", round(call_sim, digits = 3))
		

Listing 7.15/7.16: Option density plots in MATLAB
Last updated July 2020

subplot(1,2,1)
histfit(F);
title("Simulated prices");
xline(X, 'LineWidth', 1, 'label', 'Strike');
subplot(1,2,2)
hist(fsim,100);
title("Option price density");
xline(mean(fsim), 'LineWidth', 1, 'label', 'Call');
		
Listing 7.15/7.16: Option density plots in Julia
Last updated July 2020

histogram(F, bins = 100, normed = true, xlims = (20,80), title = "Simulated prices density", label = false)
res = fit_mle(Normal, F)
plot!(Normal(res.μ, res.σ), linewidth = 4, label = "Fitted normal")
vline!([X], linewidth = 4, color = "black", label = "Strike price")
histogram(fsim, bins = 110, normed = true, xlims = (0,35), title = "Option density", label = false)
vline!([f["Call"]], linewidth = 4, color = "black", label = "Call price")
		

Listing 7.17/7.18: Simulate VaR in MATLAB
Last updated 2011

randn('state',0);    % 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 = randn(S,1)*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(S*p)
		
Listing 7.17/7.18: Simulate VaR in Julia
Last updated July 2020

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 = 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[ceil(Int, p*S)]
println("Simulated VaR", ceil(Int, p*100), "%: ", round(VaR1, digits = 3), " USD")
		

Listing 7.19/7.20: Simulate option VaR in MATLAB
Last updated 2011

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(p*S)
		
Listing 7.19/7.20: Simulate option VaR in Julia
Last updated June 2022

T = 0.25   # time to expiration
X = 100    # strike price
sigma = sqrt(s2 * 250)         # annual volatility
f = bs(X = X,P = P,r = r,sigma = sigma,T = T)      # analytical call price
fsim = bs(X = X,P = Psim, r = r, sigma = sigma, T = T-(1/365)) # sim option prices
q = sort(fsim["Call"] .- f["Call"])      # simulated P/L
VaR2 = -q[ceil(Int, p*S)]
println("Simulated Option VaR", ceil(Int, p*100), "%: ", round(VaR2, digits = 3), " USD")
		

Listing 7.21/7.22: Example 7.3 in MATLAB
Last updated 2011

X1 = 100;
X2 = 110;
f1 = bs(X1,P,r,sigma,T);
f2 = bs(X2,P,r,sigma,T);
f1sim=bs(X1,Psim,r,sigma,T-(1/365));
f2sim=bs(X2,Psim,r,sigma,T-(1/365));
q = sort(f1sim.Call+f2sim.Put+Psim-f1.Call-f2.Put-P);
VaR3 = -q(p*S)
		
Listing 7.21/7.22: Example 7.3 in Julia
Last updated June 2022

X1 = 100
X2 = 110
f1 = bs(X = X1,P = P,r = r,sigma = sigma,T = T)
f2 = bs(X = X2,P = P,r = r,sigma = sigma,T = T)
f2sim = bs(X = X2,P = Psim,r = r,sigma = sigma,T = T-(1/365))
f1sim = bs(X = X1,P = Psim,r = r,sigma = sigma,T = T-(1/365))
q = sort(f1sim["Call"] .+ f2sim["Put"] .+ Psim .- f1["Call"] .- f2["Put"] .- P)
VaR3 = -q[ceil(Int, p*S)]
println("Portfolio Option VaR", ceil(Int, p*100), "%: ", round(VaR3, digits = 3), " USD")
		

Listing 7.23/7.24: Simulated two-asset returns in MATLAB
Last updated 2011

randn('state',12)    % set seed
mu = [r/365 r/365]';           % return mean
Sigma=[0.01 0.0005; 0.0005 0.02]; % covariance matrix
y = mvnrnd(mu,Sigma,S);        % simulated returns
		
Listing 7.23/7.24: Simulated two-asset returns in Julia
Last updated July 2020

Random.seed!(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 MATLAB
Last updated 2011

K = 2;
P = [100 50]';       % prices
x = [1 1]';          % number of assets
Port = P'*x;         % portfolio at t
Psim = repmat(P,1,S)' .*exp(y); % simulated prices
PortSim=Psim * x;    % simulated portfolio value
q = sort(PortSim-Port);        % simulated P/L
VaR4 = -q(S*p)
		
Listing 7.25/7.26: Two-asset VaR in Julia
Last updated July 2020

K = 2
P = [100 50]         # prices
x = [1 1]  # number of assets
Port = reshape(P * x', 1)[1]   # portfolio at t
Psim = repeat(P, outer = [S,1]).*exp.(y)' # simulated prices
PortSim = reshape(Psim * x', S)          # simulated portfolio value
q = sort(PortSim .- Port)      # simulated P/L
VaR4 = -q[ceil(Int, p * S)]
println("Two-asset VaR", ceil(Int, p*100), "%: ", round(VaR4, digits = 3), " USD")
		

Listing 7.27/7.28: A two-asset case in MATLAB with an option
Last updated 2011

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(p*S)
		
Listing 7.27/7.28: A two-asset case in Julia with an option
Last updated June 2022

f = bs(X = P[2], P = P[2], r = r, sigma = sigma, T = T)
fsim = bs(X = P[2], P = Psim[:,2], r = r, sigma = sigma, T = T-(1/365))
q = sort(fsim["Call"] .+ Psim[:,1] .- f["Call"] .- P[1])
VaR5 = -q[ceil(Int, p * S)]
println("Two-asset with option VaR", ceil(Int, p*100), "%: ", round(VaR5, digits = 3), " USD")