Pricing CMS-Spread Options with QuantLib
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1 Pricing CMS-Spread Options with QuantLib Dr. André Miemiec QuantLib Workshop Düsseldorf, December 5, 2014 d-fine All rights reserved 0
2 Whatwearegoingtodo? IR-Markets& CMS Spread Options Model & QuantLib-Implementation Calibration Results A Toy Model & Conclusions d-fine All rights reserved 1
3 This are the current challenges in the Interest Rate Markets Swaptions [Hagan: (2002)] Hagan:(2014) McLoud: (2011) CMS [Hagan: (2003)] CMS-Spread [Berrahoui: (2004)] Arbitrage freeandconsistentmodellingofall threeedgesofir-marketis still a challengingopen question. d-fine All rights reserved 2
![CMS-Spread [Berrahoui: (2004)] Arbitrage freeandconsistentmodellingofall](/docs-images/40/6196154/images/page_3.jpg "threeedgesofir-marketis still a challengingopen question.")
4 CMS-Spread Options are always good for a surprise» Sample Trade (CMS 10Y-2Y), T2E 14yr, K = 0.6% Picture Removed» Two Questions: Whatis a valid rangeforthemtm? e.g. frommarket Quotes Whosepriceis therightone? Result ofa properlycalibratedmodel d-fine All rights reserved 3
5 Whereisthemarket?» Market Quotes of Single Look CMS-Spread Options (5Y): Error boundsarereasonablytight. So qualityofpricingmodelshouldbeeasilydetectable. d-fine All rights reserved 4
6 What comes next? IR-Markets& CMS Spread Options Model & QuantLib-Implementation Calibration Results A Toy Model & Conclusions d-fine All rights reserved 5
7 CMS-Spread Option Model: Forward Premium» Berrahoui sansatz: = { +, }» MarginalsF 2 (x), F 10 (x) obtainedfrom Digital CMS-Options via a modifiedconundrumpricer. = = 1 (0) ($ % )» Choise of Copulas: (asymmetric) Gaussian/Gumbel» Smile-Model: SABR d-fine All rights reserved 6
8 Marginals via CMS-Digital-Replication h = max ( ) h 11 ) & '()*+,-./ = : / 0 / & '()*+,-./ = 0 1 () () 0() + () 0() 11 3 Payer-Swaption(model free) d-fine All rights reserved 7
9 Howdo themarginalslooklike?» Example: Marginal distribution of CMS2Y in 5Y: due to SABR Herea lotof work hastobedonein ordertocureinconsistenceson thelevelofinputmarketdata. d-fine All rights reserved 8
10 Whatcomesnext? IR-Markets& CMS Spread Options Model & QuantLib-Implementation Calibration Results A Toy Model & Conclusions d-fine All rights reserved 9
11 How does the implementation perform?» Market Quotes of Single Look CMS-Spread Options: rho=0.889 QL resultsareok. Performance is critical(severalsecs per coupon). Inacceptablefora productivepricer. d-fine All rights reserved 10
12 Back on the road to improve stability and performance» Steps towards increased stability: Alter construction of marginals, i.e. retrieve it from marginals in the swap measure:? ; < = 0() 0( ) => ) McLoud has given more symmetrical versions of the original Berrahoui formula. From a numerical perspective it might be advantageous to use this symmetric version. Using a precomputedtableofmarginalsonecangetridof theperformanceissue. d-fine All rights reserved 11
13 Whatcomesnext? IR-Markets& CMS Spread Options Model & QuantLib-Implementation Calibration Results A Toy Model & Conclusions d-fine All rights reserved 12
14 Toy Model / VBA-Prototyping» Toy-Modell: lognormal marginals, no swaption smile (cf. Andersen & Piterbarg, p. 778):? ; < = 0() 0( ) => ) ATM-Density Toymodelis different fromthestandardmodel, thatquotescms-options via a normal volatility. d-fine All rights reserved 13
15 Toy Model / VBA-Prototyping» Market Quotes of Single Look CMS-Spread Options: rho= Toymodelperformsasgoodas thereal implementationandis evenfaster. Impliedparametersaresimilar. d-fine All rights reserved 14
16 Conclusions» QL-Implementation: A working implementation for CMS-Spread Options is nearly directly achievable by reusing several codepiecesalreadycontainedin QuantLib. The main task is the generation of the marginal distributions. Empirical evidence indicates that a direct construction of the marginals using the well established conundrum method is not fast enough for productivepurposes. The prerecording of the marginal distribution before entering the integration is likely to solve this deficiency but was not implemented yet. Some suggestions to improve the stability of the pricer were made. CalibrationtoMarket Quotesworks fine.» Toy-Model: In additionthestudyofa toymodelseemstoindicate, thata calibratedmodelcanbeobtainedby plugging ATM marginals from the swap measure more or less directly into the copula method. Again, calibrationtomarket Quotesworks fine. Parameters haveslightlychanged. d-fine All rights reserved 15
17 References (1) P. S. Hagan, D. Kumar, A.S. Lesniewski, andd.e. Woodward, Managing SmileRisk, Wilmott Magazine, September (2) P.S. Hagan, D. Kumar, A.S. Lesniewski, andd.e. Woodward, Arbitrage-Free SABR, Wilmott Magazine, (3) P.S. Hagan, Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors, Wilmott Magazine, September (4) S. Benaim, M. Dodgson, and D. Kainth, An arbitrage-free method for smile extrapolation, Royal Bank of Scotland, Technical report, (5) P. McLoud, The CMS Spread Triangle, RISK, 1:126, (6) M. Berrahoui, Pricing CMS Spread Options and Digital CMS Spread Options with Smile, Wilmott Magazine, (7) A.K.H. Geisselmeyer, Consistent Pricing of CMS and CMS Spread Options with SABR-like Distributions and Power-t Copulas,UniCredit& Universities, Working Paper Series, No. 29, May2012 (8) L. Andersen, V.Piterbarg, Interest Rate Modelling, Vol. 3 d-fine All rights reserved 16
18 Ihre Ansprechpartner bei d-fine Dr. André Miemiec Manager Tel Mobile d-fine GmbH Frankfurt München London Wien Zürich Zentrale Dr. Mark W. Beinker Partner Tel Mobile d-fine GmbH Opernplatz 2 D Frankfurt/Main T F: d-fine All rights reserved 17
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