Presentation () - Washington University in St. Louis

Report
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
[email protected]
Outline
 Background
 Model Formulation
 Calibration
 Results
 Analysis
Why Do Interest-Rates Matter?
 Most basic component of finance
 Allow for the exchange of capital
 Effect us every day
 Mortgages
 Car Loans
 Student Loans
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
LIBOR





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
Background + Model Formulation + Calibration + Results + Analysis
Interest-Rate Derivatives




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
Background + Model Formulation + Calibration + Results + Analysis
Caps
 Literally “caps” a floating interest-rate
 Used to limit the risk of rate increases
 Very large, liquid market
Background + Model Formulation + Calibration + Results + Analysis
Swaps
 Allow for the conversion of debt: floating to fixed
 Available in many maturities
 Have a huge market
 Cost nothing to initiate!
Background + Model Formulation + Calibration + Results + Analysis
Swaptions
 Options on swaps
 Sell for a premium
 Also, extremely liquid
Background + Model Formulation + Calibration + Results + Analysis
LIBOR Market Model




Desire to merge theoretical and practical
Fit the experience of traders
Provided rigorous framework
Two sub-types
 LFM
 LSM
Background + Model Formulation + Calibration + Results + Analysis
Lognormal Forward-LIBOR Model
  =      
 Forward-Rate dynamics under the LFM
 Log of the Forward-Rate is Gaussian
 Under the appropriate measure
 ln  
1 2
=     −   ()
2
Background + Model Formulation + Calibration + Results + Analysis
Full Dynamics
 < ,  ≤  :
 

=    
=+1
,     
 +      
1 +   
 = ,  ≤ −1 :
  =      
 > ,  ≤ −1 :  

= −   
=+1
,     
 +      
1 +   
Background + Model Formulation + Calibration + Results + Analysis
Cap Pricing

 (0,  )( −1 , −1 ,  − )+
  =
=+1
 Cap price is the sum of Caplets
 Additivity is extremely convenient
 No reliance on correlation
Background + Model Formulation + Calibration + Results + Analysis
Model Cap pricing
   0, −1 ,  ,  =    0, −1 ,  , 
=   0,    ,  0 , 
 Here BL is the Black Caplet Formula
 Each Caplet is independent
Background + Model Formulation + Calibration + Results + Analysis
Model Cap Price
 ,  0 ,  =    −1 −  +
=  0 Φ 1 ,  0 ,  − Φ 2 ,  0 , 

2
ln
+

2
1 ,  0 ,  =


2
ln
−

2
2 ,  0 ,  =

2
2
 = −1 −1−
−1
1
2−1− ∶=
 ()2 
−1 0
Background + Model Formulation + Calibration + Results + Analysis
Swaption Price
+

  =   0, 
   ,    , −1 ,  − 
=+1
 More complex than Caps
 Path dependent
 Correlations of forward-rates important
Background + Model Formulation + Calibration + Results + Analysis
Model Swaption Pricing

 2
(,
) =
,= +1
 0  0  0  0 ,
, 0 2
  =

0
    
  ,  , 

=+1  
,  , 
(, )
 , ,  =
(, )
Background + Model Formulation + Calibration + Results + Analysis
Volatility Specification
 Above equations are general
 Do not specify the nature of volatility
 A function form must be provided
  = Φ  −1 − ; , , ,  :
= Φ  −1 −  +   − −1− + 
Brigo and Mercurio’s Formulation 7
Background + Model Formulation + Calibration + Results + Analysis
Correlation Specification
 No assumption about correlation
 Functional form must be defined
, =  −  − 
Rebonato’s Time-Homogenous Specification
Background + Model Formulation + Calibration + Results + Analysis
Calibration
 Volatility and Correlation Functional Forms
 Find optimal parameters
 Goal: Fit model to market data
Background + Model Formulation + Calibration + Results + Analysis
Preliminary Steps
 Market data must first be processed
 Quoting conventions make pricing easier
 Underlying data is obscured
 Need to bootstrap additional information
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
Additional Vol Specification
 Seeking better fit to Caps
 Introduce Time-Varying Term
  = Φ  −1 − ; , , ,  () ∶
= Φ  −1 −  +   − −1 −
Background + Model Formulation + Calibration + Results + Analysis
Optimization
 Used fmincon with active-set algorithm
 Linear constraints
 Sought best parameter values to minimize the SSE
Background + Model Formulation + Calibration + Results + Analysis
Constraints
Formulation 7
Rebonato 6.21a
+ >
+ >
,  > 
,  > 
− < , , ,  < 
− < , , ,  < 
.  <  < . 
-
-
−∞ <  ,  ,  ,  < ∞
-
.  <  < ∞
Background + Model Formulation + Calibration + Results + Analysis
Results
Background + Model Formulation + Calibration + Results + Analysis
Results
Background + Model Formulation + Calibration + Results + Analysis
Parameter Values
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
Background + Model Formulation + Calibration + Results + Analysis
Correlation Surface
Background + Model Formulation + Calibration + Results + Analysis
Φ 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
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
Background + Model Formulation + Calibration + Results + Analysis
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
Background + Model Formulation + Calibration + Results + Analysis
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.
Questions?
Image Sources
 http://blog.mindbodyonline.com/wpcontent/uploads/2010/06/php2225IJPM.jpg
 http://www.forgivemystudentloans.com/wpcontent/uploads/2011/11/student-debt.gif
 http://www.advancedcarechiro.com/chiropracticresources/frequently-asked-questions/

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