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I am trying to price plain vanilla interest rate swap (IRS) using QuantLib. What I am trying to do is to generate a path of simulated IRS price by simulating the interest rates using HW 1 Factor model. Here is what I have accomplished so far:

import prettytable as pt
import matplotlib.pyplot as plt
import numpy as np

calendar = ql.TARGET()
evaluationDate = ql.Date(15, ql.April, 2019)
ql.Settings.instance().evaluationDate = evaluationDate
# discount curve

curveDates = [
    ql.Date(15, ql.April, 2019), ql.Date(23, ql.April, 2019), ql.Date(16, ql.May, 2019), ql.Date(16, ql.July, 2019),
    ql.Date(16, ql.October, 2019), ql.Date(16, ql.April, 2020), ql.Date(16, ql.October, 2020), ql.Date(16, ql.April, 2021),
    ql.Date(19, ql.April, 2022), ql.Date(17, ql.April, 2023), ql.Date(16, ql.April, 2024), ql.Date(16, ql.April, 2025),
    ql.Date(16, ql.April, 2026), ql.Date(16, ql.April, 2027), ql.Date(18, ql.April, 2028), ql.Date(16, ql.April, 2029),
    ql.Date(16, ql.April, 2030), ql.Date(16, ql.April, 2031), ql.Date(17, ql.April, 2034), ql.Date(18, ql.April, 2039),
    ql.Date(19, ql.April, 2044), ql.Date(20, ql.April, 2049), ql.Date(16, ql.April, 2054), ql.Date(16, ql.April, 2059),
    ql.Date(16, ql.April, 2064), ql.Date(16, ql.April, 2069)]

discountFactors = [
    1.0, 1.0000735, 1.0003059, 1.0007842, 1.0011807, 1.0023373, 1.0033115,
    1.0039976, 1.0039393, 1.0015958, 0.9972325, 0.9907452, 0.9820912, 0.9715859,
    0.9591332, 0.9455427, 0.9311096, 0.9161298, 0.8705738, 0.8017461, 0.7464983,
    0.7010373, 0.6626670, 0.6289098, 0.5974307, 0.5684840]

discountCurve = ql.DiscountCurve(
    curveDates,
    discountFactors,
    ql.Actual360(),  # 
    calendar)

discountCurveHandle = ql.YieldTermStructureHandle(discountCurve)
euriborIndex = ql.Euribor6M(discountCurveHandle)

# add fixing dates and rates for floating leg

unusedRate = 0.0  # not used in pricing
#euribor-6M at 2019-01-17
rate20190117 = -0.00236  

euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2007), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.July, 2007), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2008), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.July, 2008), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(15, ql.January, 2009), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(16, ql.July, 2009), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(15, ql.January, 2010), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(15, ql.July, 2010), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2011), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(15, ql.July, 2011), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2012), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.July, 2012), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2013), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.July, 2013), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(16, ql.January, 2014), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.July, 2014), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(15, ql.January, 2015), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(16, ql.July, 2015), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(15, ql.January, 2016), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(15, ql.July, 2016), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2017), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.July, 2017), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2018), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.July, 2018), fixing=unusedRate)
euriborIndex.addFixing(fixingDate=ql.Date(17, ql.January, 2019), fixing=rate20190117)

# swap contract
nominal = 36200000.0
spread = 0.009
swapType = ql.VanillaSwap.Receiver
lengthInYears = 25
effectiveDate = ql.Date(19, ql.January, 2007)
terminationDate = effectiveDate + ql.Period(lengthInYears, ql.Years)

# fixed leg

fixedLegFrequency = ql.Period(ql.Semiannual)
fixedLegConvention = ql.ModifiedFollowing
fixedLegDayCounter = ql.Thirty360(ql.Thirty360.USA)
fixedDateGeneration = ql.DateGeneration.Forward
fixedRate = 0.045

fixedSchedule = ql.Schedule(
    effectiveDate,
    terminationDate,
    fixedLegFrequency,
    calendar,
    fixedLegConvention,
    fixedLegConvention,
    fixedDateGeneration,
    False)

# floating leg

floatingLegFrequency = ql.Period(ql.Semiannual)
floatingLegConvention = ql.ModifiedFollowing
floatingLegDayCounter = ql.Actual360()
floatingDateGeneration = ql.DateGeneration.Forward

floatSchedule = ql.Schedule(
    effectiveDate,
    terminationDate,
    floatingLegFrequency,
    calendar,
    floatingLegConvention,
    floatingLegConvention,
    floatingDateGeneration,
    False)

spot25YearSwap = ql.VanillaSwap(
    swapType,
    nominal,
    fixedSchedule,
    fixedRate,
    fixedLegDayCounter,
    floatSchedule,
    euriborIndex,
    spread,
    floatingLegDayCounter)

swapEngine = ql.DiscountingSwapEngine(discountCurveHandle)
spot25YearSwap.setPricingEngine(swapEngine)

fixedNpv = 0.0
floatingNpv = 0.0


for cf in spot25YearSwap.fixedLeg():
    if cf.date() > evaluationDate:
        fixedNpv = fixedNpv + discountCurveHandle.discount(cf.date()) * cf.amount()
        print('Fixed Leg Cash Flows (no nominal):',fixedNpv)

fixedNpv = fixedNpv + discountCurveHandle.discount(spot25YearSwap.fixedLeg()[-1].date()) * nominal

for cf in spot25YearSwap.floatingLeg():
    if cf.date() > evaluationDate:
        floatingNpv = floatingNpv + discountCurveHandle.discount(cf.date()) * cf.amount()
        print('Floating Leg Cash Flows (no nominal):', floatingNpv)

floatingNpv = floatingNpv + discountCurveHandle.discount(spot25YearSwap.floatingLeg()[-1].date()) * nominal
print('Sum of Fixed Leg Cash Flows (no nominal):',fixedNpv)
print('Sum of Floating Leg Cash Flows (no nominal):', floatingNpv)
print('Price of the Swap is ',fixedNpv - floatingNpv)


###################################################Simulating Short Rates##################################################
sigma = 0.1
a = 0.03
timestep = 360
length = 15 # in years
forward_rate = 0.05
day_count = ql.Thirty360()
todays_date = ql.Date(15, 4, 2019)
ql.Settings.instance().evaluationDate = todays_date

#spot_curve = ql.FlatForward(todays_date, ql.QuoteHandle(ql.SimpleQuote(forward_rate)), day_count)
#spot_curve_handle = ql.YieldTermStructureHandle(spot_curve)

hw_process = ql.HullWhiteProcess(discountCurveHandle, a, sigma)
rng = ql.GaussianRandomSequenceGenerator(ql.UniformRandomSequenceGenerator(timestep, ql.UniformRandomGenerator()))
seq = ql.GaussianPathGenerator(hw_process, length, timestep, rng, False)

def generate_paths(num_paths, timestep):
    arr = np.zeros((num_paths, timestep+1))
    for i in range(num_paths):
        sample_path = seq.next()
        path = sample_path.value()
        time = [path.time(j) for j in range(len(path))]
        value = [path[j] for j in range(len(path))]
        arr[i, :] = np.array(value)
    return np.array(time), arr

num_paths = 1
time, paths = generate_paths(num_paths, timestep)
for i in range(num_paths):
    plt.plot(time, paths[i, :], lw=0.8, alpha=0.6)
plt.title("Hull-White Short Rate Simulation")
plt.show()

num_paths = 1000
time, paths = generate_paths(num_paths, timestep)

vol = [np.var(paths[:, i]) for i in range(timestep+1)]
plt.plot(time, vol, "r-.", lw=3, alpha=0.6)
plt.plot(time,sigma*sigma/(2*a)*(1.0-np.exp(-2.0*a*np.array(time))), "b-", lw=2, alpha=0.5)
plt.title("Variance of Short Rates")

def alpha(forward, sigma, a, t):
    return forward + 0.5* np.power(sigma/a*(1.0 - np.exp(-a*t)), 2)

avg = [np.mean(paths[:, i]) for i in range(timestep+1)]
plt.plot(time, avg, "r-.", lw=3, alpha=0.6)
plt.plot(time,alpha(forward_rate, sigma, a, time), "b-", lw=2, alpha=0.6)
plt.title("Mean of Short Rates")
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  • $\begingroup$ It seems to be that the code both finds the price of an IRS and simulates paths from the Hull-White 1-factor model, so I am not quite sure what the question is? $\endgroup$
    – mmencke
    Commented Sep 1, 2021 at 21:25
  • $\begingroup$ Can we simulate the price of IRS using Hull White 1-Factor model? $\endgroup$
    – Desi_Quant
    Commented Sep 3, 2021 at 6:24
  • $\begingroup$ Sorry, I misread the question. I have tried to answer it to the best of my ability. $\endgroup$
    – mmencke
    Commented Sep 4, 2021 at 20:03

1 Answer 1

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For a plain vanilla swap we can define the annuity $$A(t)\equiv\sum_{i=0}^{N-1} \tau_i P(t,T_{i+1})$$ and write the swap rate as $$S(t)\equiv\frac{P(t,T_0)-P(t,T_N)}{A(t)}$$ such that the swap price can be written as $$V_{swap}(t)=A(t)(S(t)-c)$$ where $c$ is the fixed rate coupon. It would be great if we were able to simulate the swap rate (and annuity) directly, but I am not quite sure whether this is possible in the Hull-White model. On pages 422 to 423 in Andersen & Piterbarg (2010) there is an approximation for the distribution of the swap rate for a general Gaussian one-factor short rate model and how this can be used for pricing swaptions. Maybe the results can be used for simulating the swap rate directly, but I am not sure. Be aware that the specification is a bit more general with Hull-White being a special case.

Another possibility would be to construct the implied discount curve on each step of the path, update the interest rate index and re-price the swap. This is however very computationally expensive and not very elegant. I have made some C++ code here that does this to price Credit Value Adjustment, which you can use for inspiration if you choose to pursue this option.

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  • $\begingroup$ Thanks mmencke! Let me try running the C++ code. $\endgroup$
    – Desi_Quant
    Commented Sep 5, 2021 at 13:54

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