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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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Swaps Swaps involve exchange of one set of financial obligations with another e.g. fixed rate of interests with floating rate of interest, one currency.

Swaps Swaps involve exchange of one set of financial obligations with another e.g. fixed rate of interests with floating rate of interest, one currency.

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