def test_Evolve(self):
# we set up a test case via QuantLib
zeroRate = 0.03
d = 3
times = np.array([ 10.0 ])
sigma = np.array([ [ 0.0060 ],
[ 0.0050 ],
[ 0.0040 ] ])
slope = np.array([ [ 0.10 ],
[ 0.20 ],
[ 0.30 ] ])
curve = np.array([ [ 0.05 ],
[ 0.10 ],
[ 0.20 ] ])
delta = np.array([ 1.0, 5.0, 20.0 ])
chi = np.array([ 0.01, 0.05, 0.15 ])
Gamma = np.array([ [1.0, 0.8, 0.6],
[0.8, 1.0, 0.8],
[0.6, 0.8, 1.0] ])
# try QuantLib
try:
import QuantLib as ql
ytsH = ql.YieldTermStructureHandle(ql.FlatForward(0,ql.NullCalendar(),zeroRate,ql.Actual365Fixed()))
qg = ql.QuasiGaussianModel(ytsH,d,times,sigma,slope,curve,[ 0.0 ],delta,chi,Gamma,0.1)
mc = ql.RealMCSimulation(qg,[0.0, 5.0, 10.0],[0.0, 5.0, 10.0],1,1234,False,False,True)
mc.simulate()
dW = np.array(mc.brownian(0))
dW = dW[:,:3]
#print(dW)
X = np.array(mc.observedPath(0))
Xref = np.delete(X,12,1)
#print(Xref)
qlzcb = ql.RealMCZeroBond(5.0,10.0,'')
zcbRef = ql.RealMCPayoffPricer_at(qlzcb,mc)[0]
#
atol = 1.0e-14
equalPlaces = 15
except:
# we save QL results in case we don't have QL available
ytsH = YieldCurve(zeroRate)
dW = np.array([
[-0.8723107, -0.00585635, 0.31102388],
[-0.15673275, 0.28482762, 0.79041947]])
Xref = np.array([
[ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ],
[-0.02182255, 0.0397456 , -0.01717905, 0.0006412 , -0.00116167, 0.00059747, -0.00116167, 0.00259077, -0.00141925, 0.00059747, -0.00141925, 0.00088815, 0.00185998],
[-0.02476015, 0.04705765, -0.0083194 , 0.00127306, -0.00232704, 0.00102049, -0.00232704, 0.00518216, -0.00243878, 0.00102049, -0.00243878, 0.00133699, 0.03866521]])
#
zcbRef = 0.851676232405887
#
atol = 1.0e-8
equalPlaces = 9
#
model = QuasiGaussianModel(ytsH,d,times,sigma,slope,curve,delta,chi,Gamma)
X = np.zeros([3,13])
model.evolve(0.0,X[0],5.0,dW[0],X[1])
model.evolve(5.0,X[1],5.0,dW[1],X[2])
self.assertTrue(np.allclose(X,Xref,rtol=0.0,atol=atol))
zcb = model.zeroBond(5.0,10.0,X[1],None)
self.assertAlmostEqual(zcb,zcbRef,equalPlaces)
def test_ZeroBond(self):
yts = YieldCurve(0.03)
d = 3
times = np.array([ 10.0 ])
sigma = np.array([ [ 0.0060 ],
[ 0.0050 ],
[ 0.0040 ] ])
slope = np.array([ [ 0.10 ],
[ 0.20 ],
[ 0.30 ] ])
curve = np.array([ [ 0.05 ],
[ 0.10 ],
[ 0.20 ] ])
delta = np.array([ 1.0, 5.0, 20.0 ])
chi = np.array([ 0.01, 0.05, 0.15 ])
Gamma = np.array([ [1.0, 0.8, 0.6],
[0.8, 1.0, 0.8],
[0.6, 0.8, 1.0] ])
model = QuasiGaussianModel(yts,d,times,sigma,slope,curve,delta,chi,Gamma)
#
x = np.random.uniform(0.0,0.05,d)
y = np.random.uniform(0.0,0.05,[d,d])
y = y @ y.T
t = 5.0
T = 10.0
#
X = np.append(x,y.reshape((d*d,)))
X = np.append(X,[0.0])
zcb = model.zeroBond(t,T,X,None)
#
G = (1.0 - np.exp(-chi*(T-t)))/chi
zcbRef = yts.discount(T) / yts.discount(t) * np.exp(-G.dot(x) - 0.5 * G.dot(y).dot(G) )
self.assertAlmostEqual(zcb,zcbRef,14)
def test_AuxilliaryStateVsQuasiGaussian(self):
curve = YieldCurve(0.03)
d = 3
times = np.array([1.0, 2.0, 5.0, 10.0])
sigma = np.array([[0.0060, 0.0070, 0.0080, 0.0100],
[0.0030, 0.0035, 0.0040, 0.0050],
[0.0025, 0.0025, 0.0025, 0.0025]])
slope = np.zeros([3, 4])
curve = np.zeros([3, 4])
delta = np.array([1.0, 5.0, 20.0])
chi = np.array([0.01, 0.05, 0.15])
Gamma = np.array([[1.0, 0.8, 0.6], [0.8, 1.0, 0.8], [0.6, 0.8, 1.0]])
qGmodel = QuasiGaussianModel(curve, d, times, sigma, slope, curve,
delta, chi, Gamma)
X0 = qGmodel.initialValues()
#T = np.linspace(-1.0, 11.0, 121)
#vols = np.array([ qGmodel.sigma_xT(t,X0) for t in T ])
#plt.plot(T,vols[:,0,0],label='sigma[0,0]')
vols = np.array([qGmodel.sigma_xT(t, X0) for t in times])
#plt.plot(times,vols[:,0,0],'*',label='sigma[0,0]')
#plt.show()
#
mVmodel = MarkovFutureModel(None, d, times, vols, chi)
# we need to evolve the Quasi Gaussian model to obtain y(t)
times = np.linspace(0.0, 11.0, 111)
sim = McSimulation(qGmodel, times, 1, 1, showProgress=False)
Y0 = sim.X[0, :, d:d + d * d]
Y0.shape = [111, 3, 3]
Y1 = np.array([mVmodel.y(t) for t in times])
# print(np.max(np.abs(Y1[1:]/Y0[1:] - 1.0)))
self.assertLess(np.max(np.abs(Y1[1:] / Y0[1:] - 1.0)), 6.2e-14)
def test_CreditHybridModel(self):
# Rates-FX
eurRates = HullWhiteModel(YieldCurve(0.015),0.03,np.array([10.0]),np.array([0.0050]))
usdRates = HullWhiteModel(YieldCurve(0.015),0.03,np.array([10.0]),np.array([0.0050]))
usdFx = AssetModel(1.25,0.15)
hybModel = HybridModel('EUR',eurRates,['USD'],[usdFx],[usdRates],np.eye(3))
# Credit
spreadLevel = 0.05
spreadCurve = YieldCurve(spreadLevel) # use 5% instantanous default probablility
mean = 0.0001 # 1bp
sigma = 0.0100 # 100bp
skew = 0.5*sigma/spreadLevel
#
spreadModel = QuasiGaussianModel(spreadCurve,1,np.array([10.0]),np.array([[sigma]]),
np.array([[skew]]),np.array([[-skew]]),np.array([0.01]),np.array([mean]),np.identity(1))
corr = np.eye(4)
corr[1,3] = -0.85 # credit-FX
corr[3,1] = -0.85
creditModel = CreditModel(hybModel,['CP'],[spreadModel],corr)
# print(creditModel.factorAliases())
#
obsTimes = np.linspace(0.0,10.0,3)
nPaths = 2
seed = 314159265359
mcSim = McSimulation(creditModel,obsTimes,nPaths,seed,True,False)
def test_Sigma_xT(self):
yts = YieldCurve(0.03)
d = 3
times = np.array([ 10.0 ])
sigma = np.array([ [ 0.0060 ],
[ 0.0050 ],
[ 0.0040 ] ])
slope = np.array([ [ 0.10 ],
[ 0.20 ],
[ 0.30 ] ])
curve = np.array([ [ 0.05 ],
[ 0.10 ],
[ 0.20 ] ])
delta = np.array([ 1.0, 5.0, 20.0 ])
chi = np.array([ 0.01, 0.05, 0.15 ])
Gamma = np.array([ [1.0, 0.8, 0.6],
[0.8, 1.0, 0.8],
[0.6, 0.8, 1.0] ])
model = QuasiGaussianModel(yts,d,times,sigma,slope,curve,delta,chi,Gamma)
#
x = np.array([0.00206853, 0.01093424, 0.00696041]) # np.random.uniform(0.0,0.05,d)
y = np.array([[0.0298081, 0.00656372, 0.0041344 ],
[0.04282753, 0.02406725, 0.0001211 ],
[0.02906687, 0.01078535, 0.02328022]]) # np.random.uniform(0.0,0.05,[d,d])
y = y @ y.T
t = 5.0
#
X = np.append(x,y.reshape((d*d,)))
X = np.append(X,[0.0])
sigma_xT = model.sigma_xT(t,X)
# reference result from QuantLib
sigma_xT_Ref = np.array([
[ 0.031658475910, 0.017375011335, 0.084573118461 ],
[-0.007058576846, 0.021654204395, -0.134352049767 ],
[-0.016417240779, -0.043917617245, 0.051199735933 ] ])
self.assertTrue(np.allclose(sigma_xT,sigma_xT_Ref,rtol=0.0,atol=1.0e-12))
def setUp(self):
today = ql.Date(5,ql.October,2020)
ql.Settings.instance().evaluationDate = today
#
discYtsH = ql.YieldTermStructureHandle(
ql.FlatForward(today,0.015,ql.Actual365Fixed()))
projYtsH = ql.YieldTermStructureHandle(
ql.FlatForward(today,0.020,ql.Actual365Fixed()))
index = ql.Euribor6M(projYtsH)
startDate = ql.Date(12,ql.October,2020)
endDate = ql.Date(12,ql.October,2030)
calendar = ql.TARGET()
fixedTenor = ql.Period('1y')
floatTenor = ql.Period('6m')
fixedSchedule = ql.MakeSchedule(startDate,endDate,tenor=fixedTenor,calendar=calendar)
floatSchedule = ql.MakeSchedule(startDate,endDate,tenor=floatTenor,calendar=calendar)
couponDayCount = ql.Thirty360()
notional = 1.0
fixedRate = 0.02
fixedLeg = ql.FixedRateLeg(fixedSchedule,couponDayCount,[notional],[fixedRate])
floatingLeg = ql.IborLeg([notional],floatSchedule,index)
#
swap = Swap([fixedLeg,floatingLeg],[1.0,-1.0],discYtsH)
#
observationTimes = np.linspace(0.0,10.0,11)
self.timeline = swap.timeLine(observationTimes)
#
yc = YieldCurve(0.02)
d = 2
times = np.array( [ 1.0, 5.0, 10.0 ])
sigma = np.array([ [ 0.0050, 0.0060, 0.0070 ],
[ 0.0050, 0.0060, 0.0070 ] ])
slope = np.array([ [ 0.0100, 0.0100, 0.0100 ],
[ 0.0200, 0.0200, 0.0200 ] ])
curve = np.array([ [ 0.0000, 0.0000, 0.0000 ],
[ 0.0000, 0.0000, 0.0000 ] ])
delta = np.array( [ 1.0, 20.0 ])
chi = np.array( [ 0.01, 0.15 ])
Gamma = np.identity(2)
model = QuasiGaussianModel(yc,d,times,sigma,slope,curve,delta,chi,Gamma)
#
nPaths = 1
simTimes = observationTimes
seed = 314159265359
timeInterpolation = True
self.sim = McSimulation(model,simTimes,nPaths,seed,timeInterpolation,False)
def test_ModelSetup(self):
curve = YieldCurve(0.03)
d = 3
times = np.array([ 1.0, 2.0, 5.0, 10.0 ])
sigma = np.array([ [ 0.0060, 0.0070, 0.0080, 0.0100 ],
[ 0.0030, 0.0035, 0.0040, 0.0050 ],
[ 0.0025, 0.0025, 0.0025, 0.0025 ] ])
slope = np.array([ [ 0.10, 0.15, 0.20 , 0.25 ],
[ 0.20, 0.35, 0.40 , 0.55 ],
[ 0.10, 0.10, 0.10 , 0.10 ] ])
curve = np.array([ [ 0.05, 0.05, 0.05 , 0.05 ],
[ 0.10, 0.10, 0.10 , 0.10 ],
[ 0.20, 0.15, 0.10 , 0.00 ] ])
delta = np.array([ 1.0, 5.0, 20.0 ])
chi = np.array([ 0.01, 0.05, 0.15 ])
Gamma = np.array([ [1.0, 0.8, 0.6],
[0.8, 1.0, 0.8],
[0.6, 0.8, 1.0] ])
# test matricies
model = QuasiGaussianModel(curve,d,times,sigma,slope,curve,delta,chi,Gamma)
self.assertTrue(np.allclose(model._DfT @ model._DfT.T,Gamma))
Hf_H_inv = np.exp(np.array([ [ -chi_ * delta_ for chi_ in chi ] for delta_ in delta ]))
self.assertTrue(np.allclose(Hf_H_inv @ model._HHfInv,np.eye(d)))
aliases = model.stateAliases()
aliasesRef = ['x_0', 'x_1', 'x_2', 'y_0_0', 'y_0_1', 'y_0_2', 'y_1_0', 'y_1_1', 'y_1_2', 'y_2_0', 'y_2_1', 'y_2_2', 's']
self.assertListEqual(aliases,aliasesRef)
aliases = model.factorAliases()
aliasesRef = ['x_0', 'x_1', 'x_2']
self.assertListEqual(aliases,aliasesRef)
# test parameter functions
times = np.linspace(0.0, 11.0, 23)
#
sigma = np.array([ [t] + [ model.sigma(i,t) for i in range(d) ]
for t in times ])
sigmaRef = np.array([
[0.00e+00, 6.00e-03, 3.00e-03, 2.50e-03],
[5.00e-01, 6.00e-03, 3.00e-03, 2.50e-03],
[1.00e+00, 6.00e-03, 3.00e-03, 2.50e-03],
[1.50e+00, 7.00e-03, 3.50e-03, 2.50e-03],
[2.00e+00, 7.00e-03, 3.50e-03, 2.50e-03],
[2.50e+00, 8.00e-03, 4.00e-03, 2.50e-03],
[3.00e+00, 8.00e-03, 4.00e-03, 2.50e-03],
[3.50e+00, 8.00e-03, 4.00e-03, 2.50e-03],
[4.00e+00, 8.00e-03, 4.00e-03, 2.50e-03],
[4.50e+00, 8.00e-03, 4.00e-03, 2.50e-03],
[5.00e+00, 8.00e-03, 4.00e-03, 2.50e-03],
[5.50e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[6.00e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[6.50e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[7.00e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[7.50e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[8.00e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[8.50e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[9.00e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[9.50e+00, 1.00e-02, 5.00e-03, 2.50e-03],
[1.00e+01, 1.00e-02, 5.00e-03, 2.50e-03],
[1.05e+01, 1.00e-02, 5.00e-03, 2.50e-03],
[1.10e+01, 1.00e-02, 5.00e-03, 2.50e-03] ])
self.assertTrue(np.allclose(sigma,sigmaRef))
#
slope = np.array([ [t] + [ model.slope(i,t) for i in range(d) ]
for t in times ])
slopeRef = np.array([
[ 0. , 0.1 , 0.2 , 0.1 ],
[ 0.5, 0.1 , 0.2 , 0.1 ],
[ 1. , 0.1 , 0.2 , 0.1 ],
[ 1.5, 0.15, 0.35, 0.1 ],
[ 2. , 0.15, 0.35, 0.1 ],
[ 2.5, 0.2 , 0.4 , 0.1 ],
[ 3. , 0.2 , 0.4 , 0.1 ],
[ 3.5, 0.2 , 0.4 , 0.1 ],
[ 4. , 0.2 , 0.4 , 0.1 ],
[ 4.5, 0.2 , 0.4 , 0.1 ],
[ 5. , 0.2 , 0.4 , 0.1 ],
[ 5.5, 0.25, 0.55, 0.1 ],
[ 6. , 0.25, 0.55, 0.1 ],
[ 6.5, 0.25, 0.55, 0.1 ],
[ 7. , 0.25, 0.55, 0.1 ],
[ 7.5, 0.25, 0.55, 0.1 ],
[ 8. , 0.25, 0.55, 0.1 ],
[ 8.5, 0.25, 0.55, 0.1 ],
[ 9. , 0.25, 0.55, 0.1 ],
[ 9.5, 0.25, 0.55, 0.1 ],
[10. , 0.25, 0.55, 0.1 ],
[10.5, 0.25, 0.55, 0.1 ],
[11. , 0.25, 0.55, 0.1 ] ])
self.assertTrue(np.allclose(slope,slopeRef))
#
curve = np.array([ [t] + [ model.curve(i,t) for i in range(d) ]
for t in times ])
curveRef = np.array([
[ 0. , 0.05, 0.1, 0.2 ],
[ 0.5, 0.05, 0.1, 0.2 ],
[ 1. , 0.05, 0.1, 0.2 ],
[ 1.5, 0.05, 0.1, 0.15],
[ 2. , 0.05, 0.1, 0.15],
[ 2.5, 0.05, 0.1, 0.1 ],
[ 3. , 0.05, 0.1, 0.1 ],
[ 3.5, 0.05, 0.1, 0.1 ],
[ 4. , 0.05, 0.1, 0.1 ],
[ 4.5, 0.05, 0.1, 0.1 ],
[ 5. , 0.05, 0.1, 0.1 ],
[ 5.5, 0.05, 0.1, 0. ],
[ 6. , 0.05, 0.1, 0. ],
[ 6.5, 0.05, 0.1, 0. ],
[ 7. , 0.05, 0.1, 0. ],
[ 7.5, 0.05, 0.1, 0. ],
[ 8. , 0.05, 0.1, 0. ],
[ 8.5, 0.05, 0.1, 0. ],
[ 9. , 0.05, 0.1, 0. ],
[ 9.5, 0.05, 0.1, 0. ],
[10. , 0.05, 0.1, 0. ],
[10.5, 0.05, 0.1, 0. ],
[11. , 0.05, 0.1, 0. ] ])
self.assertTrue(np.allclose(curve,curveRef))
def setUp(self):
### full smile/skew model
# domestic rates
domAlias = 'EUR'
eurCurve = YieldCurve(0.03)
d = 2
times = np.array([10.0])
sigma = np.array([[0.0060], [0.0040]])
slope = np.array([[0.10], [0.15]])
curve = np.array([[0.05], [0.10]])
delta = np.array([1.0, 10.0])
chi = np.array([0.01, 0.15])
Gamma = np.array([[1.0, 0.6], [0.6, 1.0]])
eurRatesModel = QuasiGaussianModel(eurCurve, d, times, sigma, slope,
curve, delta, chi, Gamma)
# assets
forAliases = ['USD', 'GBP']
spotS0 = [1.0, 2.0]
spotVol = [0.3, 0.2]
forAssetModels = [
AssetModel(S0, vol) for S0, vol in zip(spotS0, spotVol)
]
# USD rates
usdCurve = YieldCurve(0.02)
d = 3
times = np.array([10.0])
sigma = np.array([[0.0060], [0.0050], [0.0040]])
slope = np.array([[0.10], [0.20], [0.30]])
curve = np.array([[0.05], [0.10], [0.20]])
delta = np.array([1.0, 5.0, 20.0])
chi = np.array([0.01, 0.05, 0.15])
Gamma = np.array([[1.0, 0.8, 0.6], [0.8, 1.0, 0.8], [0.6, 0.8, 1.0]])
usdRatesModel = QuasiGaussianModel(usdCurve, d, times, sigma, slope,
curve, delta, chi, Gamma)
#
gbpRatesModel = HWModel()
#
# 'EUR_x_0', 'EUR_x_1', 'USD_logS', 'USD_x_0', 'USD_x_1', 'USD_x_2', 'GBP_logS', 'GBP_x'
corr = np.array([
[1.0, 0.0, 0.5, -0.5, 0.0, 0.0, -0.5, -0.5], # EUR_x_0
[0.0, 1.0, 0.0, 0.0, -0.5, 0.0, -0.5, 0.0], # EUR_x_1
[0.5, 0.0, 1.0, -0.5, -0.5, -0.5, 0.0, 0.0], # USD_logS
[-0.5, 0.0, -0.5, 1.0, 0.0, 0.0, 0.0, 0.0], # USD_x_0
[0.0, -0.5, -0.5, 0.0, 1.0, 0.0, 0.0, 0.0], # USD_x_1
[0.0, 0.0, -0.5, 0.0, 0.0, 1.0, 0.0, 0.0], # USD_x_2
[-0.5, -0.5, 0.0, 0.0, 0.0, 0.0, 1.0, 0.5], # GBP_logS
[-0.5, 0.0, 0.0, 0.0, 0.0, 0.0, 0.5, 1.0], # GBP_x
])
#
# corr = np.identity(2 + 1 + 3 + 1 + 1 ) # overwrite
#
self.model = HybridModel(domAlias, eurRatesModel, forAliases,
forAssetModels,
[usdRatesModel, gbpRatesModel], corr)
### Gaussian model
# domestic rates
domAlias = 'EUR'
eurCurve = YieldCurve(0.03)
d = 2
times = np.array([10.0])
sigma = np.array([[0.0060], [0.0040]])
slope = np.array([[0.00], [0.00]])
curve = np.array([[0.00], [0.00]])
delta = np.array([1.0, 10.0])
chi = np.array([0.01, 0.15])
Gamma = np.array([[1.0, 0.6], [0.6, 1.0]])
eurRatesModel = QuasiGaussianModel(eurCurve, d, times, sigma, slope,
curve, delta, chi, Gamma)
# assets
forAliases = ['USD', 'GBP']
spotS0 = [1.0, 2.0]
spotVol = [0.3, 0.2]
forAssetModels = [
AssetModel(S0, vol) for S0, vol in zip(spotS0, spotVol)
]
# USD rates
usdCurve = YieldCurve(0.02)
d = 3
times = np.array([10.0])
sigma = np.array([[0.0060], [0.0050], [0.0040]])
slope = np.array([[0.10], [0.20], [0.30]])
curve = np.array([[0.05], [0.10], [0.20]])
delta = np.array([1.0, 5.0, 20.0])
chi = np.array([0.01, 0.05, 0.15])
Gamma = np.array([[1.0, 0.8, 0.6], [0.8, 1.0, 0.8], [0.6, 0.8, 1.0]])
self.gaussianModel = HybridModel(domAlias, eurRatesModel, forAliases,
forAssetModels,
[usdRatesModel, gbpRatesModel], corr)