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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. Supervisor: Anatoliy Belaygorod, Ph.D. Vice President of Quantitative Risk—R.G.A. Adjunct Professor of Finance—Olin Business School - PowerPoint PPT Presentation
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Libor Market Model:Specification and Calibration
Alex FerrisMay 1, 2012
ESE 499: Senior Design ProjectWashington University in St. Louis
Supervisor:Anatoliy Belaygorod, Ph.D.
Vice President of Quantitative Risk—R.G.A. Adjunct Professor of Finance—Olin Business School
Background
Model Formulation
Calibration
Results
Analysis
Outline
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
A Map of the World
Background + Model Formulation + Calibration + Results + Analysis
A Closer View
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
LIBOR
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
Interest-Rate Derivatives
Background + Model Formulation + Calibration + Results + Analysis
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
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
Options on swaps
Sell for a premium
Also, extremely liquid
Swaptions
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
LIBOR Market Model
Background + Model Formulation + Calibration + Results + Analysis
Forward-Rate dynamics under the LFM Log of the Forward-Rate is Gaussian
Under the appropriate measure
Lognormal Forward-LIBOR Model
Background + Model Formulation + Calibration + Results + Analysis
Full Dynamics
Background + Model Formulation + Calibration + Results + Analysis
Cap price is the sum of Caplets Additivity is extremely convenient No reliance on correlation
Cap Pricing
Background + Model Formulation + Calibration + Results + Analysis
Here BL is the Black Caplet Formula Each Caplet is independent
Model Cap pricing
Background + Model Formulation + Calibration + Results + Analysis
Model Cap Price
Background + Model Formulation + Calibration + Results + Analysis
More complex than Caps Path dependent Correlations of forward-rates important
Swaption Price
Background + Model Formulation + Calibration + Results + Analysis
Model Swaption Pricing
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
Volatility Specification
Background + Model Formulation + Calibration + Results + Analysis
No assumption about correlation Functional form must be defined
Rebonato’s Time-Homogenous Specification
Correlation Specification
Background + Model Formulation + Calibration + Results + Analysis
Volatility and Correlation Functional Forms
Find optimal parameters
Goal: Fit model to market data
Calibration
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
Preliminary Steps
Background + Model Formulation + Calibration + Results + Analysis
Cap Quotes
Background + Model Formulation + Calibration + Results + Analysis
Swaption Quotes
Background + Model Formulation + Calibration + Results + Analysis
Cap Volatility Surface
Background + Model Formulation + Calibration + Results + Analysis
Swaption Volatility Surface
Background + Model Formulation + Calibration + Results + Analysis
Seeking better fit to Caps Introduce Time-Varying Term
Additional Vol Specification
Background + Model Formulation + Calibration + Results + Analysis
Used fmincon with active-set algorithm
Linear constraints
Sought best parameter values to minimize
the SSE
Optimization
Background + Model Formulation + Calibration + Results + Analysis
Formulation 7 Rebonato 6.21a
-
-
-
Constraints
Background + Model Formulation + Calibration + Results + Analysis
Results
Background + Model Formulation + Calibration + Results + Analysis
Results
Background + Model Formulation + Calibration + Results + Analysis
Parameter Formulation 7 Rebonato 6.21a
a 12.2690 -20
b 1.7798 6.3973
c 0.8290 1.3830
d 7.7659 0.6914
0.108 0.1 (Set)
- -0.3534
- 2.1037
- 1.4645
- 3.8375
- 0.1068
Parameter Values
Background + Model Formulation + Calibration + Results + Analysis
Correlation Surface
Background + Model Formulation + Calibration + Results + Analysis
Fit Parameter Values
Background + Model Formulation + Calibration + Results + Analysis
Swaption Fit
Background + Model Formulation + Calibration + Results + Analysis
Swaption Fit (Relaxed)
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
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
Analysis
Background + Model Formulation + Calibration + Results + Analysis
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
Questions?
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