libor market model: specification and calibration
DESCRIPTION
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 PresentationTRANSCRIPT
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Libor Market Model:Specification and Calibration
Alex FerrisMay 1, 2012
ESE 499: Senior Design ProjectWashington University in St. Louis
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Supervisor:Anatoliy Belaygorod, Ph.D.
Vice President of Quantitative Risk—R.G.A. Adjunct Professor of Finance—Olin Business School
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Background
Model Formulation
Calibration
Results
Analysis
Outline
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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
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A Map of the World
Background + Model Formulation + Calibration + Results + Analysis
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A Closer View
Background + Model Formulation + Calibration + Results + Analysis
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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
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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
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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
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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
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Options on swaps
Sell for a premium
Also, extremely liquid
Swaptions
Background + Model Formulation + Calibration + Results + Analysis
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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
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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
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Full Dynamics
Background + Model Formulation + Calibration + Results + Analysis
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Cap price is the sum of Caplets Additivity is extremely convenient No reliance on correlation
Cap Pricing
Background + Model Formulation + Calibration + Results + Analysis
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Here BL is the Black Caplet Formula Each Caplet is independent
Model Cap pricing
Background + Model Formulation + Calibration + Results + Analysis
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Model Cap Price
Background + Model Formulation + Calibration + Results + Analysis
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More complex than Caps Path dependent Correlations of forward-rates important
Swaption Price
Background + Model Formulation + Calibration + Results + Analysis
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Model Swaption Pricing
Background + Model Formulation + Calibration + Results + Analysis
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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
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No assumption about correlation Functional form must be defined
Rebonato’s Time-Homogenous Specification
Correlation Specification
Background + Model Formulation + Calibration + Results + Analysis
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Volatility and Correlation Functional Forms
Find optimal parameters
Goal: Fit model to market data
Calibration
Background + Model Formulation + Calibration + Results + Analysis
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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
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Cap Quotes
Background + Model Formulation + Calibration + Results + Analysis
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Swaption Quotes
Background + Model Formulation + Calibration + Results + Analysis
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Cap Volatility Surface
Background + Model Formulation + Calibration + Results + Analysis
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Swaption Volatility Surface
Background + Model Formulation + Calibration + Results + Analysis
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Seeking better fit to Caps Introduce Time-Varying Term
Additional Vol Specification
Background + Model Formulation + Calibration + Results + Analysis
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Used fmincon with active-set algorithm
Linear constraints
Sought best parameter values to minimize
the SSE
Optimization
Background + Model Formulation + Calibration + Results + Analysis
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Formulation 7 Rebonato 6.21a
-
-
-
Constraints
Background + Model Formulation + Calibration + Results + Analysis
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Results
Background + Model Formulation + Calibration + Results + Analysis
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Results
Background + Model Formulation + Calibration + Results + Analysis
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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
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Correlation Surface
Background + Model Formulation + Calibration + Results + Analysis
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Fit Parameter Values
Background + Model Formulation + Calibration + Results + Analysis
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Swaption Fit
Background + Model Formulation + Calibration + Results + Analysis
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Swaption Fit (Relaxed)
Background + Model Formulation + Calibration + Results + Analysis
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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
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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
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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
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Questions?
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