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Libor Market Model: Specification and Calibration

Libor Market Model: Specification and Calibration. Alex Ferris May 1, 2012 ESE 499: Senior Design Project Washington University in St. Louis. Supervisor: Anatoliy Belaygorod, Ph.D. Vice President of Quantitative Risk—R.G.A. Adjunct Professor of Finance—Olin Business School

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Libor Market Model: Specification and Calibration

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  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. Outline Background Model Formulation Calibration Results Analysis

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

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

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

  7. LIBOR Background+ Model Formulation + Calibration + Results + Analysis • 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

  8. Interest-Rate Derivatives Background+ Model Formulation + Calibration + Results + Analysis • 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

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

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

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

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

  13. Lognormal Forward-LIBOR Model Background +Model Formulation+ Calibration + Results + Analysis • Forward-Rate dynamics under the LFM • Log of the Forward-Rate is Gaussian • Under the appropriate measure

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

  15. Cap Pricing Background +Model Formulation+ Calibration + Results + Analysis Cap price is the sum of Caplets Additivity is extremely convenient No reliance on correlation

  16. Model Cap pricing Background +Model Formulation+ Calibration + Results + Analysis Here BL is the Black Caplet Formula Each Caplet is independent

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

  18. Swaption Price Background +Model Formulation+ Calibration + Results + Analysis More complex than Caps Path dependent Correlations of forward-rates important

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

  20. Volatility Specification Background +Model Formulation+ Calibration + Results + Analysis Above equations are general Do not specify the nature of volatility A function form must be provided Brigo and Mercurio’s Formulation 7

  21. Correlation Specification Background +Model Formulation+ Calibration + Results + Analysis No assumption about correlation Functional form must be defined Rebonato’s Time-Homogenous Specification

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

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

  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 Seeking better fit to Caps Introduce Time-Varying Term

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

  30. Constraints Background + Model Formulation + Calibration+ Results + Analysis

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

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

  33. Parameter Values Background + Model Formulation + Calibration + Results+ Analysis

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

  35. Fit Parameter Values Background + Model Formulation + Calibration + Results+ Analysis

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

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

  38. Analysis Background + Model Formulation + Calibration + Results + Analysis • 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

  39. Analysis Background + Model Formulation + Calibration + Results + Analysis • 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

  40. References 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.

  41. Questions?

  42. Image Sources 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/

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