meanReversion = 0.1
volatilityTimes = np.array([ 2.0, 5.0, 10.0, 20.0 ])
volatilityValues = np.array([ 0.005, 0.005, 0.005, 0.005 ])
hwModel = HullWhiteModel(discCurve,meanReversion,volatilityTimes,volatilityValues)
# calibration targets
expiries = np.array([ 2, 5, 10, 20 ])
swapterms = np.array([ 2, 5, 10, 20 ])
marketVols = np.zeros([len(expiries),len(swapterms)])
swaptions = []
for i in range(expiries.shape[0]):
swaptionLine = []
for j in range(swapterms.shape[0]):
marketVols[i][j] = atmVols.loc[str(expiries[i])+'y',str(swapterms[j])+'y']
swaptionLine.append(createSwaption(str(expiries[i])+'y',str(swapterms[j])+'y',discCurve,projCurve))
swaptions.append(swaptionLine)
def objective(sigma):
hwModel = HullWhiteModel(discCurve,meanReversion,volatilityTimes,sigma)
modelVols = np.zeros(expiries.shape[0]*swapterms.shape[0])
for i in range(expiries.shape[0]):
for j in range(swapterms.shape[0]):
modelVols[i*swapterms.shape[0]+j] = swaptions[i][j].npvHullWhite(hwModel,'v')
modelVols[i*swapterms.shape[0]+j] -= marketVols[i][j]
return modelVols
#opt = least_squares(objective,volatilityValues,bounds=(0.1*volatilityValues,10*volatilityValues))
opt = least_squares(objective,volatilityValues,method='lm')
print(opt.x)
# swaption(s)
maturity = 20 # in years
swaptions = []
terms = []
inputVols = []
for k in range(1, maturity):
expTerm = str(k) + "y"
swpTerm = str(maturity - k) + "y"
sigma = 0.01 # atmVols.loc[expTerm,swpTerm]
terms.append(expTerm + "-" + swpTerm)
inputVols.append(sigma)
swaptions.append(
createSwaption(
expTerm, swpTerm, discCurve, projCurve, 0.03, ql.VanillaSwap.Receiver, sigma
)
)
bermudanSwaption = BermudanSwaption(swaptions, meanReversion)
table = pandas.DataFrame(
[
np.array(terms),
np.array(inputVols),
bermudanSwaption.model.volatilityTimes,
bermudanSwaption.model.volatilityValues,
bermudanSwaption.swaptionsNPV(),
bermudanSwaption.bondOptionsNPV(),
]
).T
def objective(sigma):
tmpModel = HullWhiteModel(discCurve, meanReversion, np.array([30.0]),
np.array([sigma]))
return (createSwaption("10y", "10y", discCurve, projCurve).npvHullWhite(
tmpModel, "v") - 0.01) # we calibrate to 100bp 10y-10y vols
4.30e-2,
4.30e-2,
4.30e-2,
]
rates2 = [r + 0.005 for r in rates]
discCurve = YieldCurve(terms, rates)
projCurve = YieldCurve(terms, rates2)
# Hull-White model mean reversion
meanReversion = 0.05
# first we have a look at the model-implied volatility smile
fig = plt.figure(figsize=(4, 6))
atmRate = createSwaption("10y", "10y", discCurve, projCurve).fairRate()
relStrikes = [-0.03 + 1e-3 * k for k in range(61)]
hwVols = [0.0050, 0.0075, 0.0100, 0.0125]
for hwVol in hwVols:
hwModel = HullWhiteModel(discCurve, meanReversion, np.array([30.0]),
np.array([hwVol]))
normalVols = [
createSwaption("10y", "10y", discCurve, projCurve,
atmRate + strike).npvHullWhite(hwModel, "v") * 1e4
for strike in relStrikes
]
plt.plot(
relStrikes,
normalVols,
label="a=" + str(meanReversion) + ", sigma_r=" + str(hwVol),
)
flatCurve = YieldCurve(['30y'],[0.03])
terms = [ '1y', '2y', '3y', '4y', '5y', '6y', '7y', '8y', '9y', '10y', '12y', '15y', '20y', '25y', '30y', '50y' ]
rates = [ 2.70e-2, 2.75e-2, 2.80e-2, 3.00e-2, 3.36e-2, 3.68e-2, 3.97e-2, 4.24e-2, 4.50e-2, 4.75e-2, 4.75e-2, 4.70e-2, 4.50e-2, 4.30e-2, 4.30e-2, 4.30e-2 ]
rates2 = [ r+0.005 for r in rates ]
discCurve = YieldCurve(terms,rates)
projCurve = YieldCurve(terms,rates2)
# Hull-White model mean reversion
meanReversion = 0.05
# first we have a look at the model-implied volatility smile
fig = plt.figure(figsize=(4, 6))
atmRate = createSwaption('10y','10y',discCurve,projCurve).fairRate()
relStrikes = [ -0.03+1e-3*k for k in range(61)]
hwVols = [ 0.0050, 0.0075, 0.0100, 0.0125 ]
for hwVol in hwVols:
hwModel = HullWhiteModel(discCurve,meanReversion,np.array([30.0]),np.array([hwVol]))
normalVols = [ createSwaption('10y','10y',discCurve,projCurve,atmRate+strike).npvHullWhite(hwModel,'v')*1e+4
for strike in relStrikes ]
plt.plot(relStrikes, normalVols, label='a='+str(meanReversion)+', sigma_r='+str(hwVol))
plt.ylim(0,250)
plt.legend()
plt.xlabel('Strike (relative to ATM)')
plt.ylabel('Model-implied normal volatility (bp)')
plt.show()
# since Hull White smile is essentially flat we now consider ATM swaptions
# we set up ATM swaptions on a grid of expiry and swap terms
4.50e-2, 4.75e-2, 4.75e-2, 4.70e-2, 4.50e-2, 4.30e-2, 4.30e-2
]
terms = ['30y'] # flat curve
rates = [5.00e-2]
rates2 = [r + 0.005 for r in rates]
discCurve = YieldCurve(terms, rates)
projCurve = YieldCurve(terms, rates)
a = 0.03 # mean reversion
vol = 0.01
h = 1.0e-12
swaptionStrike = createSwaption('10y','10y',discCurve,projCurve).fairRate() \
+ vol * np.sqrt(10.0) * np.sqrt(3.0)
swaptionStrike = 0.1048
print('swaptionStrike = ' + str(swaptionStrike))
res = []
for h in [10.0**(-k) for k in range(3, 18)]:
print('h = ' + str(h))
# swaptions
s0 = createSwaption('10y',
'10y',
discCurve,
projCurve,
strike=swaptionStrike,
normalVolatility=vol)
sp = createSwaption('10y',