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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

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**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

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Outline Background Model Formulation Calibration Results Analysis

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**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

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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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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

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**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

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**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

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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

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

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**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

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**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

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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

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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

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**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

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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

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**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

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**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

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**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

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**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

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**Calibration Volatility and Correlation Functional Forms**

Find optimal parameters Goal: Fit model to market data Background + Model Formulation + Calibration + Results + Analysis

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**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

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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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**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

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**Optimization Used fmincon with active-set algorithm Linear constraints**

Sought best parameter values to minimize the SSE Background + Model Formulation + Calibration + Results + Analysis

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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

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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 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

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

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Φ 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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**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

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**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

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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.

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Questions?

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