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Libor Market Model: Specification and Calibration Alex Ferris May 1, 2012 ESE 499: Senior Design Project Washington University in St. Louis.

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Presentation on theme: "Libor Market Model: Specification and Calibration Alex Ferris May 1, 2012 ESE 499: Senior Design Project Washington University in St. Louis."— Presentation transcript:

1 Libor Market Model: Specification and Calibration Alex Ferris May 1, 2012 ESE 499: Senior Design Project Washington University in St. Louis

2 Supervisor: Anatoliy Belaygorod, Ph.D. Vice President of Quantitative Risk—R.G.A. Adjunct Professor of Finance—Olin Business School belaygorod@wustl.edu

3  Background  Model Formulation  Calibration  Results  Analysis Outline

4  Most basic component of finance  Allow for the exchange of capital  Effect us every day  Mortgages  Car Loans  Student Loans Why Do Interest-Rates Matter? Background + Model Formulation + Calibration + Results + Analysis

5 A Map of the World Background + Model Formulation + Calibration + Results + Analysis

6 A Closer View Background + Model Formulation + Calibration + Results + Analysis

7  The London Interbank Offered Rate  Set by independent reporting of banks  By far the most important interest-rate  Changes daily  Has various maturities  3 month is most important for this discussion LIBOR Background + Model Formulation + Calibration + Results + Analysis

8  Allow for the hedging of interest-rate risk  Also used for speculation  Used by companies and investors world-wide  Come in many flavors  Plain Vanilla  Exotic Interest-Rate Derivatives Background + Model Formulation + Calibration + Results + Analysis

9  Literally “caps” a floating interest-rate  Used to limit the risk of rate increases  Very large, liquid market Caps Background + Model Formulation + Calibration + Results + Analysis

10  Allow for the conversion of debt: floating to fixed  Available in many maturities  Have a huge market  Cost nothing to initiate! Swaps Background + Model Formulation + Calibration + Results + Analysis

11  Options on swaps  Sell for a premium  Also, extremely liquid Swaptions Background + Model Formulation + Calibration + Results + Analysis

12  Desire to merge theoretical and practical  Fit the experience of traders  Provided rigorous framework  Two sub-types  LFM  LSM LIBOR Market Model Background + Model Formulation + Calibration + Results + Analysis

13 Lognormal Forward-LIBOR Model Background + Model Formulation + Calibration + Results + Analysis

14 Full Dynamics Background + Model Formulation + Calibration + Results + Analysis

15 Cap Pricing Background + Model Formulation + Calibration + Results + Analysis

16 Model Cap pricing Background + Model Formulation + Calibration + Results + Analysis

17 Model Cap Price Background + Model Formulation + Calibration + Results + Analysis

18 Swaption Price Background + Model Formulation + Calibration + Results + Analysis

19 Model Swaption Pricing Background + Model Formulation + Calibration + Results + Analysis

20 Volatility Specification Background + Model Formulation + Calibration + Results + Analysis

21 Correlation Specification Background + Model Formulation + Calibration + Results + Analysis

22  Volatility and Correlation Functional Forms  Find optimal parameters  Goal: Fit model to market data Calibration Background + Model Formulation + Calibration + Results + Analysis

23  Market data must first be processed  Quoting conventions make pricing easier  Underlying data is obscured  Need to bootstrap additional information Preliminary Steps Background + Model Formulation + Calibration + Results + Analysis

24 Cap Quotes Background + Model Formulation + Calibration + Results + Analysis

25 Swaption Quotes Background + Model Formulation + Calibration + Results + Analysis

26 Cap Volatility Surface Background + Model Formulation + Calibration + Results + Analysis

27 Swaption Volatility Surface Background + Model Formulation + Calibration + Results + Analysis

28 Additional Vol Specification Background + Model Formulation + Calibration + Results + Analysis

29  Used fmincon with active-set algorithm  Linear constraints  Sought best parameter values to minimize the SSE Optimization Background + Model Formulation + Calibration + Results + Analysis

30 Formulation 7Rebonato 6.21a - - - Constraints Background + Model Formulation + Calibration + Results + Analysis

31 Results Background + Model Formulation + Calibration + Results + Analysis

32 Results Background + Model Formulation + Calibration + Results + Analysis

33 ParameterFormulation 7Rebonato 6.21a a12.2690-20 b1.77986.3973 c0.82901.3830 d7.76590.6914 0.1080.1 (Set) --0.3534 -2.1037 -1.4645 -3.8375 -0.1068 Parameter Values Background + Model Formulation + Calibration + Results + Analysis

34 Correlation Surface Background + Model Formulation + Calibration + Results + Analysis

35

36 Swaption Fit Background + Model Formulation + Calibration + Results + Analysis

37 Swaption Fit (Relaxed) Background + Model Formulation + Calibration + Results + Analysis

38  Art versus Science of calibration  Models are largely used to price exotics  Many decisions impact results  What data to use  What data to prioritize  Seed values  Constraints Analysis Background + Model Formulation + Calibration + Results + Analysis

39  Model performed very well for Caps  Fit to Swaptions was less accurate  Relaxing constraints improved results  Limitations  Approximation of swap-rate volatility  Limited parameters  Need to include new market developments Analysis Background + Model Formulation + Calibration + Results + Analysis

40 Bank of International Settlements: Monetary and Economic Department. OTC derivatives market activity in the first half of 2011. Basel, Switzerland: Bank of International Settlements, 2011. Belaygorod, Anatoliy. "FIN 552 Lecture Notes and Course Materials." 2011. Brigo, Damiano and Fabio Mercurio. Interest Rate Models - Theory and Practice. 2nd. Berlin: Springer Finance, 2006. Levin, Kirill. "Bloomberg Volatility Cube." n.d. Rebonato, Riccardo. Modern Pricing of Interest-Rate Derivatives. Princeton, New Jersey: Princeton University Press, 2002. References

41 Questions?

42  http://blog.mindbodyonline.com/wp- content/uploads/2010/06/php2225IJPM.jpg  http://www.forgivemystudentloans.com/wp- content/uploads/2011/11/student-debt.gif  http://www.advancedcarechiro.com/chiropractic- resources/frequently-asked-questions/ Image Sources


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