Copyright (c) 2015 Matthias Groncki
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this list of conditions and the following disclaimer.
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This disclaimer is taken from the QuantLib license
# import the used libraries
import numpy as np
import matplotlib.pyplot as plt
import QuantLib as ql
%matplotlib inline
# Setting evaluation date
today = ql.Date(7,4,2015)
ql.Settings.instance().setEvaluationDate(today)
# Setup Marketdata
rate = ql.SimpleQuote(0.03)
rate_handle = ql.QuoteHandle(rate)
dc = ql.Actual365Fixed()
yts = ql.FlatForward(today, rate_handle, dc)
yts.enableExtrapolation()
hyts = ql.RelinkableYieldTermStructureHandle(yts)
t0_curve = ql.YieldTermStructureHandle(yts)
euribor6m = ql.Euribor6M(hyts)
# Setup a dummy portfolio with two Swaps
def makeSwap(start, maturity, nominal, fixedRate, index, typ=ql.VanillaSwap.Payer):
"""
creates a plain vanilla swap with fixedLegTenor 1Y
parameter:
start (ql.Date) : Start Date
maturity (ql.Period) : SwapTenor
nominal (float) : Nominal
fixedRate (float) : rate paid on fixed leg
index (ql.IborIndex) : Index
return: tuple(ql.Swap, list<Dates>) Swap and all fixing dates
"""
end = ql.TARGET().advance(start, maturity)
fixedLegTenor = ql.Period("1y")
fixedLegBDC = ql.ModifiedFollowing
fixedLegDC = ql.Thirty360(ql.Thirty360.BondBasis)
spread = 0.0
fixedSchedule = ql.Schedule(start,
end,
fixedLegTenor,
index.fixingCalendar(),
fixedLegBDC,
fixedLegBDC,
ql.DateGeneration.Backward,
False)
floatSchedule = ql.Schedule(start,
end,
index.tenor(),
index.fixingCalendar(),
index.businessDayConvention(),
index.businessDayConvention(),
ql.DateGeneration.Backward,
False)
swap = ql.VanillaSwap(typ,
nominal,
fixedSchedule,
fixedRate,
fixedLegDC,
floatSchedule,
index,
spread,
index.dayCounter())
return swap, [index.fixingDate(x) for x in floatSchedule][:-1]
portfolio = [makeSwap(today + ql.Period("2d"),
ql.Period("5Y"),
1e6,
0.03,
euribor6m),
makeSwap(today + ql.Period("2d"),
ql.Period("4Y"),
5e5,
0.03,
euribor6m,
ql.VanillaSwap.Receiver),
]
#%%timeit
# Setup pricing engine and calculate the npv
engine = ql.DiscountingSwapEngine(hyts)
for deal, fixingDates in portfolio:
deal.setPricingEngine(engine)
deal.NPV()
#print(deal.NPV())
# Stochastic Process
# Assume the model is already calibrated either historical or market implied
volas = [ql.QuoteHandle(ql.SimpleQuote(0.0075)),
ql.QuoteHandle(ql.SimpleQuote(0.0075))]
meanRev = [ql.QuoteHandle(ql.SimpleQuote(0.02))]
model = ql.Gsr(t0_curve, [today+100], volas, meanRev, 16.)
process = model.stateProcess()
# Define evaluation grid
date_grid = [today + ql.Period(i,ql.Months) for i in range(0,12*6)]
for deal in portfolio:
date_grid += deal[1]
date_grid = np.unique(np.sort(date_grid))
time_grid = np.vectorize(lambda x: ql.ActualActual().yearFraction(today, x))(date_grid)
dt = time_grid[1:] - time_grid[:-1]
print(len(time_grid)*1500*2*29e-6)
6.786
# Random number generator
seed = 1
urng = ql.MersenneTwisterUniformRng(seed)
usrg = ql.MersenneTwisterUniformRsg(len(time_grid)-1,urng)
generator = ql.InvCumulativeMersenneTwisterGaussianRsg(usrg)
#%%timeit
# Generate N paths
N = 1500
x = np.zeros((N, len(time_grid)))
y = np.zeros((N, len(time_grid)))
pillars = np.array([0.0, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
zero_bonds = np.zeros((N, len(time_grid), 12))
for j in range(12):
zero_bonds[:, 0, j] = model.zerobond(pillars[j],
0,
0)
for n in range(0,N):
dWs = generator.nextSequence().value()
for i in range(1, len(time_grid)):
t0 = time_grid[i-1]
t1 = time_grid[i]
x[n,i] = process.expectation(t0,
x[n,i-1],
dt[i-1]) + dWs[i-1] * process.stdDeviation(t0,
x[n,i-1],
dt[i-1])
y[n,i] = (x[n,i] - process.expectation(0,0,t1)) / process.stdDeviation(0,0,t1)
for j in range(12):
zero_bonds[n, i, j] = model.zerobond(t1+pillars[j],
t1,
y[n, i])
# plot the paths
for i in range(0,N):
plt.plot(time_grid, x[i,:])
# generate the discount factors
discount_factors = np.vectorize(t0_curve.discount)(time_grid)
#%%timeit
#Swap pricing under each scenario
npv_cube = np.zeros((N,len(date_grid), len(portfolio)))
for p in range(0,N):
for t in range(0, len(date_grid)):
date = date_grid[t]
ql.Settings.instance().setEvaluationDate(date)
ycDates = [date,
date + ql.Period(6, ql.Months)]
ycDates += [date + ql.Period(i,ql.Years) for i in range(1,11)]
yc = ql.DiscountCurve(ycDates,
zero_bonds[p, t, :],
ql.Actual365Fixed())
yc.enableExtrapolation()
hyts.linkTo(yc)
if euribor6m.isValidFixingDate(date):
fixing = euribor6m.fixing(date)
euribor6m.addFixing(date, fixing)
for i in range(len(portfolio)):
npv_cube[p, t, i] = portfolio[i][0].NPV()
ql.IndexManager.instance().clearHistories()
ql.Settings.instance().setEvaluationDate(today)
hyts.linkTo(yts)
# Calculate the discounted npvs
discounted_cube = np.zeros(npv_cube.shape)
for i in range(npv_cube.shape[2]):
discounted_cube[:,:,i] = npv_cube[:,:,i] * discount_factors
# Calculate the portfolio npv by netting all NPV
portfolio_npv = np.sum(npv_cube,axis=2)
discounted_npv = np.sum(discounted_cube, axis=2)
# Plot the first 30 NPV paths
n_0 = 0
n = 30
f, (ax1, ax2) = plt.subplots(2, 1, figsize=(12,10), sharey=True)
for i in range(n_0,n):
ax1.plot(time_grid, portfolio_npv[i,:])
for i in range(n_0,n):
ax2.plot(time_grid, discounted_npv[i,:])
ax1.set_xlabel("Time in years")
ax1.set_ylabel("NPV in time t Euros")
ax1.set_title("Simulated npv paths")
ax2.set_xlabel("Time in years")
ax2.set_ylabel("NPV in time 0 Euros")
ax2.set_title("Simulated discounted npv paths")
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# Calculate the exposure and discounted exposure
E = portfolio_npv.copy()
dE = discounted_npv.copy()
E[E<0] = 0
dE[dE<0] = 0
# Plot the first 30 exposure paths
n = 30
f, (ax1, ax2) = plt.subplots(2, 1, figsize=(12,10))
for i in range(0,n):
ax1.plot(time_grid, E[i,:])
for i in range(0,n):
ax2.plot(time_grid, dE[i,:])
ax1.set_xlabel("Time in years")
ax1.set_ylabel("Exposure")
ax1.set_ylim([-10000,70000])
ax1.set_title("Simulated exposure paths")
ax2.set_xlabel("Time in years")
ax2.set_ylabel("Discounted Exposure")
ax2.set_ylim([-10000,70000])
ax2.set_title("Simulated discounted exposure paths")
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# Calculate the expected exposure
E = portfolio_npv.copy()
E[E<0]=0
EE = np.sum(E, axis=0)/N
# Calculate the discounted expected exposure
dE = discounted_npv.copy()
dE[dE<0] = 0
dEE = np.sum(dE, axis=0)/N
# plot the expected exposure path
n = 30
f, (ax1, ax2) = plt.subplots(2, 1, figsize=(8,10))
ax1.plot(time_grid, EE)
ax2.plot(time_grid, dEE)
ax1.set_xlabel("Time in years")
ax1.set_ylabel("Exposure")
ax1.set_title("Expected exposure")
ax2.set_xlabel("Time in years")
ax2.set_ylabel("Discounted Exposure")
ax2.set_title("Discounted expected exposure")
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# plot the expected exposure path
plt.figure(figsize=(7,5), dpi=300)
plt.plot(time_grid, dEE)
plt.xlabel("Time in years")
plt.ylabel("Discounting Expected Exposure")
plt.ylim([-2000,10000])
plt.title("Expected Exposure (netting set)")
<matplotlib.text.Text at 0x103fca890>
# Calculate the PFE curve (95% quantile)
PFE_curve = np.apply_along_axis(lambda x: np.sort(x)[0.95*N],0, E)
plt.figure(figsize=(7,5), dpi=300)
plt.plot(time_grid,PFE_curve)
plt.xlabel("Time in years")
plt.ylabel("PFE")
plt.ylim([-2000,35000])
plt.title("PFE (netting set)")
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# calculate the maximum pfe
MPFE = np.max(PFE_curve)
MPFE
32034.275198581003
# alternative pfe 95% quantile of the maxima of each exposure paths
PFE = np.sort(np.max(E,axis=1))[0.95*N]
PFE
38683.65078838585
# Setup Default Curve
pd_dates = [today + ql.Period(i, ql.Years) for i in range(11)]
hzrates = [0.02 * i for i in range(11)]
pd_curve = ql.HazardRateCurve(pd_dates,hzrates,ql.Actual365Fixed())
pd_curve.enableExtrapolation()
# Plot curve
# Calculate default probs on grid *times*
times = np.linspace(0,30,100)
dp = np.vectorize(pd_curve.defaultProbability)(times)
sp = np.vectorize(pd_curve.survivalProbability)(times)
dd = np.vectorize(pd_curve.defaultDensity)(times)
hr = np.vectorize(pd_curve.hazardRate)(times)
f, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(10,10))
ax1.plot(times, dp)
ax2.plot(times, sp)
ax3.plot(times, dd)
ax4.plot(times, hr)
ax1.set_xlabel("Time in years")
ax2.set_xlabel("Time in years")
ax3.set_xlabel("Time in years")
ax4.set_xlabel("Time in years")
ax1.set_ylabel("Probability")
ax2.set_ylabel("Probability")
ax3.set_ylabel("Density")
ax4.set_ylabel("HazardRate")
ax1.set_title("Default Probability")
ax2.set_title("Survival Probability")
ax3.set_title("Default density")
ax4.set_title("Harzard rate")
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# Calculation of the default probs
defaultProb_vec = np.vectorize(pd_curve.defaultProbability)
dPD = defaultProb_vec(time_grid[:-1], time_grid[1:])
# Calculation of the CVA
recovery = 0.4
CVA = (1-recovery) * np.sum(dEE[1:] * dPD)
CVA
586.17763177944278