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Libor Market Model: Specification and Calibration

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Presentation on theme: "Libor Market Model: Specification and Calibration"— 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 Anatoliy Belaygorod, Ph.D. Vice President of Quantitative Risk—R.G.A.
Supervisor: Anatoliy Belaygorod, Ph.D. Vice President of Quantitative Risk—R.G.A. Adjunct Professor of Finance—Olin Business School

3 Outline Background Model Formulation Calibration Results Analysis

4 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

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

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

7 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

8 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

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 Background + Model Formulation + Calibration + Results + Analysis

10 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

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

12 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

13 Lognormal Forward-LIBOR Model
𝑑 𝐹 𝑘 𝑡 = 𝜎 𝑘 𝑡 𝐹 𝑘 𝑡 𝑑 𝑍 𝑘 𝑡 Forward-Rate dynamics under the LFM Log of the Forward-Rate is Gaussian Under the appropriate measure 𝑑 ln 𝐹 𝑘 𝑡 = 𝜎 𝑘 𝑡 𝑑𝑊 𝑡 − 𝜎 2 𝑘 (𝑡)𝑑𝑡 Background + Model Formulation + Calibration + Results + Analysis

14 Full Dynamics 𝑖<𝑘, 𝑡 ≤ 𝑇 𝑖 : 𝑑 𝐹 𝑘 𝑡 = 𝜎 𝑘 𝑡 𝐹 𝑘 𝑡 𝑗=𝑖+1 𝑘 𝜌 𝑘,𝑗 𝜏 𝑗 𝜎 𝑗 𝑡 𝐹 𝑗 𝑡 1+ 𝜏 𝑗 𝐹 𝑗 𝑡 𝑑𝑡 + 𝜎 𝑘 𝑡 𝐹 𝑘 𝑡 𝑑 𝑍 𝑘 𝑡 𝑖=𝑘, 𝑡 ≤ 𝑇 𝑘−1 : 𝑑 𝐹 𝑘 𝑡 = 𝜎 𝑘 𝑡 𝐹 𝑘 𝑡 𝑑 𝑍 𝑘 𝑡 𝑖>𝑘, 𝑡 ≤ 𝑇 𝑘−1 : 𝑑 𝐹 𝑘 𝑡 = −𝜎 𝑘 𝑡 𝐹 𝑘 𝑡 𝑗=𝑘+1 𝑖 𝜌 𝑘,𝑗 𝜏 𝑗 𝜎 𝑗 𝑡 𝐹 𝑗 𝑡 1+ 𝜏 𝑗 𝐹 𝑗 𝑡 𝑑𝑡 + 𝜎 𝑘 𝑡 𝐹 𝑘 𝑡 𝑑 𝑍 𝑘 𝑡 Background + Model Formulation + Calibration + Results + Analysis

15 Cap Pricing 𝐶𝑎𝑝 𝑃𝑟𝑖𝑐𝑒= 𝑖=𝛼+1 𝛽 𝑁 𝜏 𝑖 𝐷(0, 𝑇 𝑖 ) (𝐹 𝑇 𝑖−1 , 𝑇 𝑖−1 , 𝑇 𝑖 −𝐾) + Cap price is the sum of Caplets Additivity is extremely convenient No reliance on correlation Background + Model Formulation + Calibration + Results + Analysis

16 Model Cap pricing Here BL is the Black Caplet Formula
𝐶𝑎𝑝𝑙𝑒𝑡 𝑃𝑟𝑖𝑐𝑒 𝐿𝐹𝑀 0, 𝑇 𝑖−1 , 𝑇 𝑖 ,𝐾 = 𝐶𝑎𝑝𝑙𝑒𝑡 𝑃𝑟𝑖𝑐𝑒 𝐵𝑙𝑎𝑐𝑘 0, 𝑇 𝑖−1 , 𝑇 𝑖 ,𝐾 =𝑁 𝑃 0, 𝑇 𝑖 𝜏 𝑖 𝐵𝐿 𝐾, 𝐹 𝑖 0 , 𝑣 𝑖 Here BL is the Black Caplet Formula Each Caplet is independent Background + Model Formulation + Calibration + Results + Analysis

17 Model Cap Price 𝐵𝐿 𝐾, 𝐹 𝑖 0 , 𝑣 𝑖 = 𝐸 𝑖 𝐹 𝑖 𝑇 𝑖−1 −𝐾 + = 𝐹 𝑖 0 Φ 𝑑 1 𝐾, 𝐹 𝑖 0 , 𝑣 𝑖 −𝐾Φ 𝑑 2 𝐾, 𝐹 𝑖 0 , 𝑣 𝑖 𝑑 1 𝐾, 𝐹 𝑖 0 , 𝑣 𝑖 = ln 𝐹 𝐾 + 𝑣 2 2 𝑣 𝑑 2 𝐾, 𝐹 𝑖 0 , 𝑣 𝑖 = ln 𝐹 𝐾 − 𝑣 2 2 𝑣 𝑣 𝑖 2 = 𝑇 𝑖−1 𝑣 𝑇 𝑖−1 −𝐶𝑎𝑝𝑙𝑒𝑡 2 𝑣 𝑇 𝑖−1 −𝐶𝑎𝑝𝑙𝑒𝑡 2 := 1 𝑇 𝑖−1 0 𝑇 𝑖−1 𝜎 𝑖 (𝑡) 2 𝑑𝑡 Background + Model Formulation + Calibration + Results + Analysis

18 Swaption Price More complex than Caps Path dependent
𝑆𝑤𝑎𝑝𝑡𝑖𝑜𝑛 𝑃𝑟𝑖𝑐𝑒= 𝑁 𝐷 0, 𝑇 𝛼 𝑖=𝛼+1 𝛽 𝜏 𝑖 𝑃 𝑇 𝛼 , 𝑇 𝑖 𝐹 𝑇 𝛼 , 𝑇 𝑖−1 , 𝑇 𝑖 −𝐾 + More complex than Caps Path dependent Correlations of forward-rates important Background + Model Formulation + Calibration + Results + Analysis

19 Model Swaption Pricing
(𝑣 𝛼,𝛽 𝐿𝐹𝑀 ) 2 = 𝑖,𝑗= 𝛼+1 𝛽 𝑤 𝑖 0 𝑤 𝑗 0 𝐹 𝑖 0 𝐹 𝑗 0 𝜌 𝑖,𝑗 𝑆 𝛼,𝛽 𝑇 𝛼 𝜎 𝑖 𝑡 𝜎 𝑗 𝑡 𝑑𝑡 𝑤 𝑖 𝑡 = 𝜏 𝑖 𝐹𝑃 𝑡, 𝑇 𝛼 , 𝑇 𝑖 𝑘=𝛼+1 𝛽 𝜏 𝑘 𝐹𝑃 𝑡, 𝑇 𝛼 , 𝑇 𝑘 𝐹𝑃 𝑡,𝑇,𝑆 = 𝑃(𝑡,𝑆) 𝑃(𝑡,𝑇) Background + Model Formulation + Calibration + Results + Analysis

20 Volatility Specification
Above equations are general Do not specify the nature of volatility A function form must be provided 𝜎 𝑖 𝑡 = Φ 𝑖 𝜓 𝑇 𝑖−1 −𝑡;𝑎,𝑏,𝑐,𝑑 := Φ 𝑖 𝑎 𝑇 𝑖−1 −𝑡 +𝑑 𝑒 −𝑏 𝑇 𝑖−1 −𝑡 +𝑐 Brigo and Mercurio’s Formulation 7 Background + Model Formulation + Calibration + Results + Analysis

21 Correlation Specification
No assumption about correlation Functional form must be defined 𝜌 𝑖,𝑗 = 𝑒 −𝛽 𝑇 𝑖 − 𝑇 𝑗 Rebonato’s Time-Homogenous Specification Background + Model Formulation + Calibration + Results + Analysis

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

23 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

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
Seeking better fit to Caps Introduce Time-Varying Term 𝜎 𝑖 𝑡 = Φ 𝑖 𝜓 𝑇 𝑖−1 −𝑡;𝑎,𝑏,𝑐,𝑑 𝜖(𝑡) := Φ 𝑖 𝑎 𝑇 𝑖−1 −𝑡 +𝑑 𝑒 −𝑏 𝑇 𝑖−1 −𝑡 +𝑐 𝑎 𝑇 𝑖−1 −𝑡 +𝑑 𝑒 −𝑏 𝑇 𝑖−1 −𝑡 +𝑐 𝑖=1 3 𝜖 𝑖 sin 𝑡𝜋𝑖 𝑀𝑎𝑡 + 𝜖 𝑖 𝑒 − 𝜀 7 𝑡 Background + Model Formulation + Calibration + Results + Analysis

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

30 Constraints Formulation 7 Rebonato 6.21a 𝒅+𝒄 >𝟎 𝒄,𝒅 >𝟎 −𝟐𝟎 <𝒂,𝒃,𝒄,𝒅 <𝟐𝟎 𝟎.𝟎𝟎𝟎𝟎𝟓 < 𝜷 <𝟎.𝟏𝟎𝟖 - −∞ < 𝝐 𝟏 , 𝝐 𝟐 , 𝝐 𝟑 , 𝝐 𝟒 < ∞ 𝟎.𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟏 < 𝝐 𝟕 < ∞ 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 Parameter Formulation 7 Rebonato 6.21a a -20 b 1.7798 6.3973 c 0.8290 1.3830 d 7.7659 0.6914 𝛃 0.108 0.1 (Set) 𝛜 𝟏 - 𝛜 𝟐 2.1037 𝛜 𝟑 1.4645 𝛜 𝟒 3.8375 𝛜 𝟕 0.1068 Background + Model Formulation + Calibration + Results + Analysis

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

35 Φ 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 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

39 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

40 References Bank of International Settlements: Monetary and Economic Department. OTC derivatives market activity in the first half of Basel, Switzerland: Bank of International Settlements, Belaygorod, Anatoliy. "FIN 552 Lecture Notes and Course Materials." Brigo, Damiano and Fabio Mercurio. Interest Rate Models - Theory and Practice. 2nd. Berlin: Springer Finance, 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


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