Re: [Quantlib-users] CMS Spread Cap Pricing

[Peter C. <pca...@gm...>]

Hi Aditya

2 the error shouldn't show up unless you pass a volatilityType to the
pricer. Might have to do with how SWIG calls the constructor, not sure at
the moment.

3 what I meant was: the cms price depends on the volatilities outside the
quoted part ATM-200 / ATM+200 as well, so maybe BBG uses different
extrapolated vols than you do. And as I said, the model itself (and the
mean reversion value) have a big impact, so you might first want check what
BBG is doing

Thanks
Peter



On Fri, 12 Aug 2022 at 10:43, Aditya Gupta <adi...@co...>
wrote:

> Hi Peter,
>
>
>
> For point 2) the LinearTsrPricer can take the input as
> ql.ConstantSwaptionVolatilityStructure, but the Lognormal pricer complains
> by saying: " if only an atm surface is given, the volatility type must be
> inherited" So I guess it is not possible. To create a ‘constant volatility
> cube’, I instantiated an atmMatrix with a constant value, but all the OTM
> and ITM swaptions had a volatility of 0. This way, for every triplet
> (strike, expiry, tenor), QL would return a constant volatility. It is a
> workaround, but it sufficed for my purpose.
>
>
>
> For 3) The code and data for the extrapolation is below. Some notes:
>
>
>
> volHandle at the end is passed into the pricer. The data is not complete
> but should be enough for debugging purposes. The BBG/QL volatility cubes
> seem to match to a decent degree (i.e. I don’t believe it is the source of
> the error). The volatilities are within 10 bps when testing random data
> points. However, the atm strike does not change (even if the volatility is
> constant) and does not agree with Bloomberg.
>
>
>
> normalVolCube =
>
> Expiry
>
> 1Yr
>
> 2Yr
>
> 3Yr
>
> 4Yr
>
> 5Yr
>
> 7Yr
>
> 10Yr
>
> 12Yr
>
> 15Yr
>
> 20Yr
>
> 25Yr
>
> 30Yr
>
> 1Mo
>
>      153.65
>
> 160.51
>
> 151.81
>
> 143.8
>
> 141.03
>
> 129.88
>
> 123.16
>
> 118.88
>
> 110.61
>
> 108.9
>
> 104.21
>
> 101.91
>
> 3Mo
>
>      150.06
>
> 152.12
>
> 149.09
>
> 140.08
>
> 134.79
>
> 125.74
>
> 118.65
>
> 115.21
>
> 112.26
>
> 107.13
>
> 103.41
>
> 99.43
>
> 6Mo
>
>      154.30
>
> 150.74
>
> 143.9
>
> 137.09
>
> 132.84
>
> 123.31
>
> 115.03
>
> 111.31
>
> 107.88
>
> 103.41
>
> 100.5
>
> 96.31
>
> 9Mo
>
>      155.29
>
> 149.86
>
> 142.87
>
> 135.17
>
> 130.97
>
> 121.33
>
> 112.06
>
> 108.57
>
> 105.04
>
> 99.95
>
> 97.12
>
> 93.19
>
> 1Yr
>
>      155.98
>
> 149.11
>
> 140.78
>
> 133.39
>
> 127.82
>
> 118.13
>
> 109.42
>
> 106.32
>
> 102.06
>
> 97.15
>
> 94.05
>
> 90.8
>
> 2Yr
>
>      142.61
>
> 135.71
>
> 129.64
>
> 123.34
>
> 118.42
>
> 110
>
> 102.05
>
> 99.13
>
> 95.18
>
> 89.95
>
> 86.93
>
> 84.38
>
> 3Yr
>
>      130.05
>
> 124.03
>
> 117.71
>
> 113.72
>
> 109.87
>
> 102.87
>
> 96.37
>
> 93.46
>
> 89.5
>
> 84.21
>
> 81.15
>
> 79.07
>
> 4Yr
>
>      118.85
>
> 113.22
>
> 108.13
>
> 104.87
>
> 102.41
>
> 97.49
>
> 91.33
>
> 88.78
>
> 85.31
>
> 79.42
>
> 77.09
>
> 74.93
>
> 5Yr
>
>      109.73
>
> 104.94
>
> 102.2
>
> 99.09
>
> 96.85
>
> 92.53
>
> 87.38
>
> 84.84
>
> 81.37
>
> 76.05
>
> 73.36
>
> 72.24
>
> 6Yr
>
>      102.35
>
> 99.42
>
> 97.38
>
> 94.42
>
> 92.18
>
> 88.36
>
> 83.64
>
> 81.19
>
> 77.88
>
> 72.88
>
> 70.54
>
> 69.33
>
> 7Yr
>
>        96.52
>
> 93.55
>
> 91.44
>
> 89.36
>
> 87.41
>
> 84.15
>
> 79.99
>
> 77.68
>
> 74.58
>
> 70.13
>
> 68.25
>
> 66.85
>
> 8Yr
>
>        91.55
>
> 89.03
>
> 87.38
>
> 85.58
>
> 83.78
>
> 80.83
>
> 77.08
>
> 74.82
>
> 71.81
>
> 67.42
>
> 65.22
>
> 64.37
>
> 9Yr
>
>        86.91
>
> 85.11
>
> 83.57
>
> 81.89
>
> 80.19
>
> 77.48
>
> 74.07
>
> 71.89
>
> 68.99
>
> 64.76
>
> 62.82
>
> 62.05
>
> 10Yr
>
>        83.39
>
> 81.29
>
> 79.55
>
> 77.84
>
> 76.4
>
> 74.03
>
> 71.19
>
> 69.2
>
> 66.54
>
> 62.56
>
> 60.74
>
> 60.1
>
> 12Yr
>
>        77.24
>
> 75.44
>
> 74
>
> 72.38
>
> 70.97
>
> 69.1
>
> 66.47
>
> 64.68
>
> 62.27
>
> 58.88
>
> 57.3
>
> 56.85
>
> 15Yr
>
>        68.34
>
> 67.24
>
> 65.61
>
> 64.58
>
> 64.34
>
> 62.11
>
> 60.25
>
> 58.73
>
> 56.7
>
> 53.89
>
> 52.72
>
> 52.28
>
> 20Yr
>
>        60.48
>
> 60.35
>
> 59.15
>
> 58.46
>
> 57.75
>
> 56.64
>
> 55.07
>
> 53.74
>
> 51.97
>
> 49.87
>
> 48.96
>
> 48.63
>
> 25Yr
>
>        58.04
>
> 56.62
>
> 55.74
>
> 55.16
>
> 54.62
>
> 53.59
>
> 52.25
>
> 51.07
>
> 49.5
>
> 47.49
>
> 46.72
>
> 46.08
>
> 30Yr
>
>        54.83
>
> 53.31
>
> 52.73
>
> 52.34
>
> 51.96
>
> 51.06
>
> 49.7
>
> 48.76
>
> 47.52
>
> 45.21
>
> 44.11
>
> 44.1
>
>
>
>
>
> strikeSpreadData Snippet:
>
> Term x Tenor
>
> -200bps
>
> -100bps
>
> -50bps
>
> -25bps
>
> ATM
>
> 25bps
>
> 50bps
>
> 100bps
>
> 200bps
>
> 3Mo X 1Yr
>
> 163.06
>
> 147.61
>
> 145.65
>
> 147.02
>
> 150.06
>
> 154.64
>
> 160.47
>
> 174.74
>
> 208.13
>
> 3Mo X 2Yr
>
> 168.32
>
> 152.63
>
> 149.59
>
> 150.1
>
> 152.12
>
> 155.6
>
> 160.36
>
> 172.72
>
> 203.32
>
> 3Mo X 3Yr
>
> 165.94
>
> 150.14
>
> 146.87
>
> 147.22
>
> 149.09
>
> 152.42
>
> 157.05
>
> 169.18
>
> 199.38
>
> 3Mo X 5Yr
>
> 154.44
>
> 137.68
>
> 133.49
>
> 133.36
>
> 134.79
>
> 137.76
>
> 142.09
>
> 153.73
>
> 182.89
>
> 3Mo X 7Yr
>
> 145.98
>
> 128.92
>
> 124.5
>
> 124.31
>
> 125.74
>
> 128.76
>
> 133.16
>
> 144.93
>
> 173.96
>
> 3Mo X 10Yr
>
> 141.64
>
> 123.6
>
> 118.33
>
> 117.66
>
> 118.65
>
> 121.3
>
> 125.42
>
> 136.78
>
> 165.14
>
> 3Mo X 12Yr
>
> 124.87
>
> 115.74
>
> 113.96
>
> 114.17
>
> 115.21
>
> 117.06
>
> 119.67
>
> 126.74
>
> 145.33
>
> 3Mo X 15Yr
>
> 137.53
>
> 118.56
>
> 112.58
>
> 111.56
>
> 112.26
>
> 114.72
>
> 118.73
>
> 130
>
> 158.17
>
> 3Mo X 20Yr
>
> 133.96
>
> 114.4
>
> 107.91
>
> 106.63
>
> 107.13
>
> 109.46
>
> 113.42
>
> 124.65
>
> 152.72
>
> 3Mo X 25Yr
>
> 130.47
>
> 110.79
>
> 104.19
>
> 102.88
>
> 103.41
>
> 105.83
>
> 109.9
>
> 121.34
>
> 149.56
>
> 3Mo X 30Yr
>
> 126.66
>
> 106.9
>
> 100.19
>
> 98.87
>
> 99.43
>
> 101.94
>
> 106.13
>
> 117.77
>
> 146.12
>
> 6Mo X 1Yr
>
> 153.08
>
> 147.26
>
> 148.88
>
> 151.13
>
> 154.3
>
> 158.3
>
> 163.02
>
> 174.1
>
> 199.93
>
> 6Mo X 2Yr
>
> 155.61
>
> 147.49
>
> 147.3
>
> 148.56
>
> 150.74
>
> 153.82
>
> 157.69
>
> 167.34
>
> 191.18
>
> 6Mo X 3Yr
>
> 151.41
>
> 141.74
>
> 140.85
>
> 141.85
>
> 143.9
>
> 146.93
>
> 150.84
>
> 160.73
>
> 185.27
>
> 6Mo X 5Yr
>
> 144.96
>
> 132.89
>
> 130.7
>
> 131.19
>
> 132.84
>
> 135.64
>
> 139.46
>
> 149.43
>
> 174.44
>
> 6Mo X 7Yr
>
> 137.51
>
> 124.48
>
> 121.67
>
> 121.87
>
> 123.31
>
> 125.95
>
> 129.67
>
> 139.51
>
> 164.24
>
> 6Mo X 10Yr
>
> 131.51
>
> 117.49
>
> 113.98
>
> 113.86
>
> 115.03
>
> 117.47
>
> 121.05
>
> 130.73
>
> 155.14
>
> 6Mo X 12Yr
>
> 119.63
>
> 111.38
>
> 109.93
>
> 110.24
>
> 111.31
>
> 113.15
>
> 115.68
>
> 122.45
>
> 140.14
>
> 6Mo X 15Yr
>
> 126.39
>
> 111.49
>
> 107.36
>
> 106.95
>
> 107.88
>
> 110.16
>
> 113.64
>
> 123.22
>
> 147.39
>
> 6Mo X 20Yr
>
> 123.32
>
> 107.85
>
> 103.29
>
> 102.66
>
> 103.41
>
> 105.58
>
> 109
>
> 118.53
>
> 142.61
>
> 6Mo X 25Yr
>
> 120.86
>
> 105.2
>
> 100.48
>
> 99.78
>
> 100.5
>
> 102.67
>
> 106.1
>
> 115.7
>
> 139.82
>
>
>
>
>
> # ATM Volatility matrix
>
> volType = ql.Normal
>
> flatExtrapolation = False
>
>
>
> atmSwapTenors = [ql.Period(tenor[:-1]) for tenor in
> normalVolCube.columns[1:]]
>
> atmOptionTenors = [ql.Period(tenor[:-1]) for tenor in
> normalVolCube['Expiry']]
>
>
>
> normalVols = normalVolCube[normalVolCube.columns[1:]].apply(
>
>     lambda x: x * 1e-4).values.tolist()
>
>
>
> swaptionVolMatrix = ql.SwaptionVolatilityMatrix(calendar, convention,
>
>                                                 atmOptionTenors,
> atmSwapTenors,
>
>                                                 ql.Matrix(normalVols),
>
>                                                 dayCounter,
> flatExtrapolation,
>
>                                                 volType)
>
>
>
> swaptionVolMatrixHandle = ql.SwaptionVolatilityStructureHandle(
>
>     swaptionVolMatrix)
>
>
>
> strikeSpreads = [-200, -100, -50, -25, 0, 25, 50, 100, 200]
>
> strikeSpreads = [x * 1e-4 for x in strikeSpreads]
>
>
>
> optionTenors = [
>
>     ql.Period(3, ql.Months),
>
>     ql.Period(6, ql.Months),
>
>     ql.Period(9, ql.Months),
>
>     ql.Period(1, ql.Years),
>
>     ql.Period(3, ql.Years),
>
>     ql.Period(5, ql.Years),
>
>     ql.Period(7, ql.Years),
>
>     ql.Period(10, ql.Years),
>
>     ql.Period(15, ql.Years),
>
>     ql.Period(20, ql.Years),
>
>     ql.Period(30, ql.Years)
>
> ]
>
>
>
> swapTenors = [
>
>     ql.Period(1, ql.Years),
>
>     ql.Period(2, ql.Years),
>
>     ql.Period(3, ql.Years),
>
>     ql.Period(5, ql.Years),
>
>     ql.Period(7, ql.Years),
>
>     ql.Period(10, ql.Years),
>
>     ql.Period(12, ql.Years),
>
>     ql.Period(15, ql.Years),
>
>     ql.Period(20, ql.Years),
>
>     ql.Period(25, ql.Years),
>
>     ql.Period(30, ql.Years)
>
> ]
>
>
>
> nRows = len(optionTenors) * len(swapTenors)
>
> nCols = len(strikeSpreads)
>
>
>
> volSpreadsMatrix = strikeSpreadData[strikeSpreadData.columns[1:]].apply(
>
>     lambda x: x).values.tolist()
>
>
>
> volSpreads = []
>
>
>
> for i in range(nRows):
>
>     volSpreadsRow = []
>
>     for j in range(nCols):
>
>         volSpreadsRow.append(
>
>             ql.QuoteHandle(ql.SimpleQuote(volSpreadsMatrix[i][j])))
>
>
>
>     volSpreads.append(volSpreadsRow)
>
>
>
> vegaWeightedSmileFit = False
>
>
>
> volCube = ql.SwaptionVolCube2(swaptionVolMatrixHandle, optionTenors,
>
>                               swapTenors, strikeSpreads, volSpreads,
>
>                               swapIndex1, swapIndex2, vegaWeightedSmileFit)
>
>
>
> volHandle = ql.SwaptionVolatilityStructureHandle(volCube)
>
> volHandle.enableExtrapolation()
>
>
>
> Thank you!
>
>
>
> -----Original Message-----
> From: Peter Caspers <pca...@gm...>
> Sent: Friday, August 12, 2022 4:20 PM
> To: Aditya Gupta <adi...@co...>
> Cc: qua...@li...
> Subject: Re: [Quantlib-users] CMS Spread Cap Pricing
>
>
>
> [You don't often get email from pca...@gm.... Learn why this is
> important at https://aka.ms/LearnAboutSenderIdentification ]
>
>
>
> CAUTION: EXTERNAL Sender
>
>
>
>
>
> Hi Aditya
>
>
>
> 1. correct
>
> 2. the pricer takes any SwaptionVolatilityStructure, i.e. a
> ConstantSwaptionVolatilityStructure should work. Is that what you mean?
>
> 3. this might be related to smile extrapolation, which strike range do you
> cover in your swaption cube and how do you set up the extrapolation? The
> pricer uses an integration over [-200%, +200%] by default.
>
> 4. I agree 500% is much to high, let's discuss further once 3 is resolved,
> i.e. QL / BBG matches
>
>
>
> In addition, the linear TSR model might not work well when the time to
> expiry is large. Do you have more info on which model BBG uses?
>
>
>
> Thank you
>
> Peter
>
>
>
> On Fri, 12 Aug 2022 at 09:36, Aditya Gupta <adi...@co...>
> wrote:
>
> >
>
> > Hi,
>
> >
>
> >
>
> >
>
> > I am trying to price a CMS Spread Cap using Quantlib. I am using data
> from Bloomberg, and I am using the following classes:
>
> >
>
> >
>
> >
>
> > yts is a bootstrapped Sofr curve
>
> >
>
> > Instrument = ql.CappedFlooredCmsSpreadCoupon()
>
> >
>
> > volCube = ql.SwaptionVolCube2() with the correct data and volType =
>
> > ql.Normal
>
> >
>
> > cmsPricer = ql.LinearTSRPricer(volHandle, meanReversion, yts)
>
> >
>
> > pricer = ql.LognormalCmsSpreadPricer(cmsPricer, correlation, yts, 16)
>
> >
>
> >
>
> >
>
> > I have a few questions.
>
> >
>
> >
>
> >
>
> > It seems to me that to price a ql. CappedFlooredCmsSpreadCoupon, the
> only option is to use a ql.LognormalCmsSpreadPricer, and hence if we use
> Normal volatility, the input cmsPricer must be the LinearTSRPricer (Hagan
> cannot be used). Is my assessment correct?
>
> > In order to check the pricing, I used a ‘constant’ volatility cube (set
> the atm matrix to a constant, rest to 0) and tested the pricing. The
> instrument was priced correctly (within the error from the discount curve).
> Can the pricer accept a constant volatility otherwise?
>
> > The volCube seems to be returning the correct volatilities (within
> reason) when compared to Bloomberg, and is being passed to the pricer
> without error. But for some reason, the price is completely different
> (about half of what it should be). What could be the cause for the error,
> or how could I go about debugging this?
>
> > If I use scipy.optimize to ‘artificially’ optimize the price as a
> function of the mean reversion, for a value of around 5 (which is way too
> high for the mean correlation), I was able to bring the error down to less
> than 1 dollar. Clearly this approach is not sensible. So I used the
> HullWhite model on the swaption matrix, and calibrated the mean reversion.
> But the error for this value of the mean reversion is much too high. How
> can I determine a good value for the mean reversion?
>
> >
>
> >
>
> >
>
> > Right now I am content with there being some error compared to
> Bloomberg, but I am unable to understand the blackbox behind the pricing
> engine and the mean reversion calibration, and if there is any way to solve
> these issues. Any and all help would be greatly appreciated. Thank you.
>
> >
>
> >
>
> >
>
> > Some of the code and data is as follows:
>
> >
>
> >
>
> >
>
> > today = ql.Date(25, 7, 2022)
>
> >
>
> > ql.Settings.instance().evaluationDate = today
>
> >
>
> >
>
> >
>
> > #Global variables
>
> >
>
> > calendar = ql.UnitedStates(ql.UnitedStates.FederalReserve)
>
> >
>
> > convention = ql.ModifiedFollowing
>
> >
>
> > endOfMonth = False
>
> >
>
> > dayCounter = ql.Actual360()
>
> >
>
> > fixingDays = 0
>
> >
>
> > compounding = ql.Compounded
>
> >
>
> > compoundingFrequency = ql.Annual
>
> >
>
> >
>
> >
>
> > yts = ql.RelinkableYieldTermStructureHandle()
>
> >
>
> > index = ql.Sofr(yts)
>
> >
>
> >
>
> >
>
> > helper = []
>
> >
>
> >
>
> >
>
> > # sofrData are just the rates from a dataframe
>
> >
>
> >
>
> >
>
> > helper += [
>
> >
>
> >     ql.DepositRateHelper(
>
> >
>
> >         ql.QuoteHandle(ql.SimpleQuote(sofrData['Shifted Rate'][0] /
>
> > 100.0)),
>
> >
>
> >         index)
>
> >
>
> > ]
>
> >
>
> >
>
> >
>
> > helper += [
>
> >
>
> >     ql.OISRateHelper(0,
>
> >
>
> >                      ql.Period(expiration),
>
> >
>
> >                      ql.QuoteHandle(ql.SimpleQuote(rate / 100.0)),
>
> >
>
> >                      index,
>
> >
>
> >                      paymentLag=0,
>
> >
>
> >                      paymentConvention=convention,
>
> >
>
> >                      paymentFrequency=ql.Annual,
>
> >
>
> >                      paymentCalendar=calendar,
>
> >
>
> >                      averagingMethod=ql.RateAveraging.Compound) for
>
> > rate,
>
> >
>
> >     expiration in zip(sofrData['Shifted Rate'][1:],
>
> > sofrData["Term"][1:])
>
> >
>
> > ]
>
> >
>
> >
>
> >
>
> > curve = ql.PiecewiseSplineCubicDiscount(today, helper, ql.Actual360())
>
> >
>
> > curve.enableExtrapolation()
>
> >
>
> > yts.linkTo(curve)
>
> >
>
> >
>
> >
>
> > meanReversion = ql.QuoteHandle(ql.SimpleQuote(5.037702498779181))
>
> > #this is the ‘optimised’ value  but I need further help
>
> >
>
> > correlation = ql.QuoteHandle(ql.SimpleQuote(36.6 * 1e-4))
>
> >
>
> >
>
> >
>
> > nominal = 1e9
>
> >
>
> > gearing = 1.0
>
> >
>
> > spread = 0.0
>
> >
>
> > cap = -8 * 1e-4
>
> >
>
> > floor = ql.nullDouble()  #infinite floor
>
> >
>
> >
>
> >
>
> > startDate = ql.Date(26, 9, 2022)
>
> >
>
> > endDate = ql.Date(26, 9, 2023)
>
> >
>
> > paymentDate = ql.Date(26, 9, 2023)
>
> >
>
> >
>
> >
>
> > isInArrears = False
>
> >
>
> > exCouponDate = ql.Date()
>
> >
>
> >
>
> >
>
> > fixingDays = 0
>
> >
>
> >
>
> >
>
> > swapIndex1 = ql.UsdLiborSwapIsdaFixAm(ql.Period('30y'), yts, yts)
>
> >
>
> > ql.IndexManager.instance().clearHistory(swapIndex1.name())
>
> >
>
> >
>
> >
>
> > swapIndex2 = ql.UsdLiborSwapIsdaFixAm(ql.Period('10y'), yts, yts)
>
> >
>
> > ql.IndexManager.instance().clearHistory(swapIndex2.name())
>
> >
>
> >
>
> >
>
> > spreadIndex = ql.SwapSpreadIndex("Joint Index", swapIndex1,
>
> > swapIndex2)
>
> >
>
> >
>
> >
>
> > instrument = ql.CappedFlooredCmsSpreadCoupon(paymentDate, nominal,
>
> > startDate,
>
> >
>
> >                                              endDate, fixingDays,
>
> > spreadIndex,
>
> >
>
> >                                              gearing, spread, cap,
>
> > floor,
>
> >
>
> >                                              ql.Date(), ql.Date(),
>
> > dayCounter,
>
> >
>
> >                                              isInArrears,
>
> > exCouponDate)
>
> >
>
> >
>
> >
>
> > cmsPricer = ql.LinearTsrPricer(volHandle, meanReversion, yts)
>
> >
>
> >
>
> >
>
> > # cmsPricer = ql.NumericHaganPricer(volHandle,
>
> > ql.GFunctionFactory.ParallelShifts , meanReversion)#, Rate
>
> > lowerLimit=0.0, Rate upperLimit=1.0, Real precision=1.0e-6, Real
>
> > hardUpperLimit=QL_MAX_REAL)
>
> >
>
> >
>
> >
>
> > integrationPoints = 16
>
> >
>
> >
>
> >
>
> > pricer = ql.LognormalCmsSpreadPricer(cmsPricer, correlation, yts,
>
> >
>
> >                                      integrationPoints)
>
> >
>
> >
>
> >
>
> > instrument.setPricer(pricer)
>
> >
>
> >
>
> >
>
> > Regards,
>
> >
>
> > Aditya Gupta.
>
> >
>
> >
>
> >
>
> > ________________________________
>
> > Complus Asset Management Limited ("CAML") is licensed by the Securities
> and Futures Commission of Hong Kong (“SFC”) for the Regulated Activities of
> Advising on Securities and Asset Management with CE number AWX256.  CAML is
> subject to certain SFC codes and regulations, details of which can be found
> on the SFC's website at http://www.sfc.hk.  CAML is a company
> incorporated in Hong Kong with its registered office and principal place of
> business as indicated above.
>
> >
>
> > Unless specifically indicated, this message should not be construed as
> an offer to sell or solicitation to purchase any securities, financial
> products or services or the giving of investment advice within or outside
> Hong Kong.  This message is intended solely for the person(s) to whom it is
> addressed.  If you receive this message in error, please immediately delete
> it and all copies of it from your computer network or hard copies, and
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>
> > _______________________________________________
>
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>
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