Analytical Finance II Swaps In Multi Curve Framework December 2012 Jingjing Guo Elsa Han Neda Kazemie

Introduction “The crisis that affected financial markets in the last years leaded market practitioners to revise well known basic concepts like the ones of discount factors and forward rates. A single yield curve is not sufficient any longer to describe the market of interest rate products.”[1] 2

EONIA swap rate with maturity one year (white line) and EURIBOR swap-rate (orange line) on the same maturity. Reference: Bloomberg platform. 3

basis-swap spread for six-months vs. three-months EURIBOR rates on a swap with maturity one year. Reference: Bloomberg platform. 4

[1],[2] Interest-Rate Modeling with Multiple Yield Curves,Andrea Pallaviciniy and Marco Tarenghiz,First Version: October 13, This version: June 25, 2010 Financial crisis implies that using a unique term structure is not sufficient any more and we need different term-structure for building discounting curve and forwarding curves. In the Euro area the discounting curve can be obtained starting from the EONIA-based Overnight Indexed Swaps, that typically cover a time horizon up to thirty years. The use of a discounting curve obtained from overnight rates is a typical choice in the multi-curve setting [2] 5

Steps of the project: First: To build the yield curve from market quotes of deposits, Forward Rate Agreements (FRA), short futures and standard Interest Rate Swaps (IRS) from quotes in EUR for different tenors, OIS,1M,3M and 6M. Second : Calculate 1M forward rate from the 1M curve and 3M forward rate from 3M curve. Third: Comparing the value of two given swaps, if we have a collateral agreement we will use the OIS for discounting otherwise we will use the 6M curve for discounting. Instead of bootstrapping we use the extended Nelson-Siegel method (also known as Nelson-Siegel-Svensson). We optimize the parameters of the model that the rates given by the Nelson-Siegel-Svensson model best fit the market values, to do so we use solver in Excel. 6

Nelson-Siegel-Svensson Model Formulation 7

Nelson-Siegel-Svensson Parameters β ₀ = 1.55 β ₁ = β ₂ = β ₃ = 9.01 τ ₁ = 3.89 τ ₂ =

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Nelson-Siegel-Svensson Parameters β ₀ =1.58 β ₁ =-1.25 β ₂ = β ₃ = τ ₁ =4.18 τ ₂ =

Discount & Spot Rate Curves Given by Nelson-Siegel-Svensson Model 12

Case 1 Maturity 20 years, Swap Rate 1.4 % annually, Frequency 1M Calculated 1M forward rate from 1M curve 1. Collateral agreement Discount the cash-flows using OIS curve Swap Rate = Not a collateral agreement Discount cash-flows using EUR_6M Swap Rate =

Case 2 Maturity 20 years, Swap Rate 1.6 % annually, Frequency 3M Calculated 3M forward rate from 3M curve 1. Collateral agreement Discount the cash-flows using OIS curve Swap Rate = Not a collateral agreement Discount cash-flows using EUR_6M Swap Rate =

We would like to express our special thanks of gratitude to our teacher Jan Röman who gave us the golden opportunity to do this project and providing the necessary date and valuable guidance.