• QuantLib 金融計算——案例之普通利率互換分析（1）
  • 概述
  • 合約條款
  • 實踐
    • 設定期限結構
    • 添加歷史浮動利率
    • 設定合約
    • 估值
  • 估值差異可能的來源
  • 下一步
  • 擴展閱讀

QuantLib 金融計算——案例之普通利率互換分析（1）

概述

QuantLib 中涉及利率互換的功能大致分為兩大類：

• 對存續的利率互換合約估值；
• 根據利率互換合約的成交報價推算隱含的期限結構，

這兩類功能是緊密聯系的，根據最新報價推算出的期限結構通常可以用來對存續合約進行估值，

本文接下來介紹如何具體實作對合約的估值，并以 Real world tidy interest rate swap pricing 中 Bloomberg 的結果作為比較基準，

Bloomberg 的結果：

合約條款

對存續的利率互換合約進行估值，通常是根據當前的期限結構計算出浮動端（floating leg）和固定端（fixed leg）的“預期貼現現金流”，兩者之差即合約的估值，需要注意的是，利率互換的估值對合約條款比較敏感，

示例中的合約是一個 Euribor 6M 的利率互換，條款細則如下：

• 浮動利率：Euribor 6M
• 固定利率：0.059820%
• 利差：0.0%
• 生效期：2007-01-19
• 期限：25 Y
• 型別：支付浮動利率，收取固定利率
• 浮動端支付頻率：半年一次
• 浮動端天數計算規則：ACT/360
• 固定端支付頻率：一年一次
• 固定端天數計算規則：30U/360
• 日歷：TARGET（匹配 Trans-European Automated Real-time Gross Settlement Express Transfer System 的日歷）
• 估值日期：2019-04-15

實踐

import QuantLib as ql
import prettytable as pt

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

設定期限結構

估值的核心是當前的期限結構，根據 Real world tidy interest rate swap pricing 中的貼現因子資料設定估值用的期限結構，

# 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(),  # 與浮動端一致
    calendar)

discountCurveHandle = ql.YieldTermStructureHandle(discountCurve)

添加歷史浮動利率

估值利率互換需要用到一個重要的類——IborIndex，它負責根據期限結構以及合約的條款推算出隱含的遠期利率，進而得到浮動端的預期現金流，

由于是對存續合約估值，需要為期限結構添加“歷史浮動利率”——歷史上 fixing date 上的 Euribor 6M 資料，盡管只有最近一次 fixing 的 Euribor 6M 利率會參與估值，但用戶還是要添加更早期 fixing date 的利率，否則會報錯，幸運的是它們不參與估值，可以用 0 來填充，

euriborIndex = ql.Euribor6M(discountCurveHandle)

# add fixing dates and rates for floating leg

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

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)

注：Euribor6M 是 IborIndex 的派生類，

設定合約

一些基本設定：

# swap contract

nominal = 10000000.0
spread = 0.0
swapType = ql.VanillaSwap.Receiver
lengthInYears = 25
effectiveDate = ql.Date(19, ql.January, 2007)
terminationDate = effectiveDate + ql.Period(lengthInYears, ql.Years)

設定固定端與浮動端的支付時間表（schedule），計算出現金流的發生日期：

# fixed leg

fixedLegFrequency = ql.Period(ql.Annual)
fixedLegConvention = ql.ModifiedFollowing
fixedLegDayCounter = ql.Thirty360(ql.Thirty360.USA)
fixedDateGeneration = ql.DateGeneration.Forward
fixedRate = 0.059820 / 100.0

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)

VanillaSwap 類實作了普通利率互換，VanillaSwap 類將接受一個定價引擎——DiscountingSwapEngine，并根據前面配置好的現金流日期計算浮動端和固定端的預期貼現現金流，

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

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

估值

Bloomberg 對浮動端和固定端的估值考慮了本金，而 QuantLib 默認不考慮本金，所以浮動端和固定端的 NPV 要自己計算，

fixedNpv = 0.0
floatingNpv = 0.0

fixedTable = pt.PrettyTable(['date', 'amount'])

for cf in spot25YearSwap.fixedLeg():
    if cf.date() > evaluationDate:
        fixedTable.add_row([str(cf.date()), cf.amount()])
        fixedNpv = fixedNpv + discountCurveHandle.discount(cf.date()) * cf.amount()

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

floatingTable = pt.PrettyTable(['date', 'amount'])

for cf in spot25YearSwap.floatingLeg():
    if cf.date() > evaluationDate:
        floatingTable.add_row([str(cf.date()), cf.amount()])
        floatingNpv = floatingNpv + discountCurveHandle.discount(cf.date()) * cf.amount()

floatingNpv = floatingNpv + discountCurveHandle.discount(
    spot25YearSwap.floatingLeg()[-1].date()) * nominal

npvTable = pt.PrettyTable(['NPVs', 'amount'])
npvTable.add_row(['total', spot25YearSwap.NPV()])
npvTable.add_row(['fixed leg NPV', fixedNpv])
npvTable.add_row(['floating leg NPV', floatingNpv])

npvTable.align = 'r'
npvTable.float_format = '.2'
print('NPVs:')
print(npvTable)
print()

fixedTable.align = 'r'
fixedTable.float_format = '.4'
print('Fixed Leg Cash Flows (no nominal):')
print(fixedTable)
print()

floatingTable.align = 'r'
floatingTable.float_format = '.4'
print('Floating Leg Cash Flows (no nominal):')
print(floatingTable)

結果：

NPVs:
+------------------+------------+
|             NPVs |     amount |
+------------------+------------+
|            total | -877065.26 |
|    fixed leg NPV | 9119162.21 |
| floating leg NPV | 9996227.47 |
+------------------+------------+

Fixed Leg Cash Flows (no nominal):
+--------------------+-----------+
|               date |    amount |
+--------------------+-----------+
| January 20th, 2020 | 5965.3833 |
| January 19th, 2021 | 5965.3833 |
| January 19th, 2022 | 5982.0000 |
| January 19th, 2023 | 5982.0000 |
| January 19th, 2024 | 5982.0000 |
| January 20th, 2025 | 5998.6167 |
| January 19th, 2026 | 5965.3833 |
| January 19th, 2027 | 5982.0000 |
| January 19th, 2028 | 5982.0000 |
| January 19th, 2029 | 5982.0000 |
| January 21st, 2030 | 6015.2333 |
| January 20th, 2031 | 5965.3833 |
| January 19th, 2032 | 5965.3833 |
+--------------------+-----------+

Floating Leg Cash Flows (no nominal):
+--------------------+-------------+
|               date |      amount |
+--------------------+-------------+
|    July 19th, 2019 | -11734.4444 |
| January 20th, 2020 |  -9883.8108 |
|    July 20th, 2020 | -10526.4706 |
| January 19th, 2021 |  -8236.3411 |
|    July 19th, 2021 |  -3118.9504 |
| January 19th, 2022 |    290.3520 |
|    July 19th, 2022 |   6002.4970 |
| January 19th, 2023 |  11853.1381 |
|    July 19th, 2023 |  16803.6213 |
| January 19th, 2024 |  22032.9795 |
|    July 19th, 2024 |  27371.4264 |
| January 20th, 2025 |  33134.5740 |
|    July 21st, 2025 |  38526.4090 |
| January 19th, 2026 |  43841.7013 |
|    July 20th, 2026 |  49022.3996 |
| January 19th, 2027 |  54065.4048 |
|    July 19th, 2027 |  58756.3184 |
| January 19th, 2028 |  64707.0763 |
|    July 19th, 2028 |  67946.7395 |
| January 19th, 2029 |  72599.6699 |
|    July 19th, 2029 |  74090.1906 |
| January 21st, 2030 |  78693.2841 |
|    July 19th, 2030 |  77892.3324 |
| January 20th, 2031 |  82544.3792 |
|    July 21st, 2031 |  83194.2799 |
| January 19th, 2032 |  84980.7972 |
+--------------------+-------------+

估值差異可能的來源

與 Bloomberg 的結果相比盡管非常接近，但還是存在差異，估值差異的來源可能如下：

• 在期限結構上插值的技術細節不一致，DiscountCurve 對貼現因子進行對數線性插值，Bloomberg 的技術細節不得而知，
• 浮動端和固定端的天數計算規則不一致，而期限結構的天數計算規則與浮動端保持一致，天數計算規則的不一致使得同一“日期”對浮動端和固定端來說意味著不同的“時間”，Bloomberg 如何處理這種不一致也不得而知，

下一步

• 分析國內市場上的利率互換，
• 從利率互換的成交報價中推算期限結構，

擴展閱讀

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