1 / 42

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

huela
Download Presentation

Libor Market Model: Specification and Calibration

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

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. 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/

More Related