Python and Julia Chapter 7. Simulation Methods for VaR for Options and Bonds

# Chapter 7. Simulation Methods for VaR for Options and Bonds

### Python and Julia

Copyright 2011 - 2023 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: www.gnu.org/licenses.

##### Transformation in Python
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()

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


##### Various RNs in Python
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))

##### Various RNs in Julia
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")


##### Price bond in Python
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)

##### Price bond in Julia
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")


##### Simulate yields in Python
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()

##### Simulate yields in Julia
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")


##### Simulate bond prices in Python
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()

##### Simulate bond prices in Julia
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)


##### Black-Scholes valuation in Python
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)

##### Simulate bond prices in Julia
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


##### Black-Scholes simulation in Python
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)

##### Black-Scholes simulation in Julia
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))


##### Option density plots in Python
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()

##### Option density plots in Julia
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")


##### Simulate VaR in Python
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)

##### Simulate VaR in Julia
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")


##### Simulate option VaR in Python
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)

##### Simulate option VaR in Julia
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")


##### Example 7.3 in Python
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)

##### Example 7.3 in Julia
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")


##### Simulated two-asset returns in Python
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

##### Simulated two-asset returns in Julia
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


##### Two-asset VaR in Python
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)

##### Two-asset VaR in Julia
K = 2
P = [100 50]                       # prices
x = [1 1]                          # number of assets
Port = reshape(P * x', 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")


##### A two-asset case in Python with an option
f = bs(X = P, P = P, r = r, sigma = sigma, T = T)
fsim = bs(X = P, P = Psim[:,1], r = r, sigma = sigma, T = T-(1/365))
q = np.sort(fsim['Call'] + Psim[:,0] - f['Call'] - P)
VaR5 = -q[ceil(p*S) - 1]
print(VaR5)

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


##### Financial Risk Forecasting
Market risk forecasting with R, Julia, Python and Matlab. Code, lecture slides, implementation notes, seminar assignments and questions.