# Introduction

Standard fixed-income applications make a larger and larger use of the multi-curve framework to price products and hedge risks. For whatever reason this is the case, it is useful to know how to implement such a framework.

We have already talked about multi-curves in the past. Here we gave a list useful references and here we illustrated the mean features of risk metrics and sensitivity patterns. In this blog, we describe how to design the multi-curve framework. We do not claim that this is the only way or the best way. This is one possible way, which however turned out to work quite well within our system and happened to be easily integrated into our library.

Code snippets that will be shown below have been developed in C# using Visual Studio. Continue reading

# Intuitions From The Multi-Curve Framework

### How can we hedge within the multi-curve framework?

Let’s consider a simplified case. Our building blocks will be swaps only of various tenors and maturities with the following purposes:

• Discounting instruments: we use 1-month tenor swaps of various maturities to construct the discounting curve.
• Tenor instruments: we use 3-months, 6-months and 12-months tenor swaps to construct the forward curves. We denote them as 3M, 6M and 12M.

The exact swap data are given in the attached spreadsheets [a] and [b]. In particular:

• the Data tab contains details of the input data and convention used in the calculation;
• the Rows tab contains the matrix with the PV01s/IV01s of all the input instruments as rows;
• the Columns tab contains the actual PV01/IV01 matrices as the transpose of the matrix in the previous tab;
• our notation in these examples is that all the curves are ordered by discounting type (the first one is always the discounting curve) and increasing tenors.

# Multi-Curve - Useful References

We have recently started the project of including the multi-curve framework into the UDFinLib, our own financial library. The topic is delicate, as it consists of both research and implementation at once.

After the 2007-2008 world financial crisis it became clear that the classical single-curve framework that had been used until then was not appropriate to value products and to hedge portfolio's positions. All of a sudden credit risk was an every day's topic, collateral margins exploded and the previously small spreads between different-tenor swaps (OIS vs Libor, 3M-tenor vs 6M-tenor) could not be neglected anymore. Single-curve building, which treated instrument with different tenors in the same way, had to be upgraded to multi-curve.

In a nutshell, the multi-curve framework amounts to construct one discount curve and many tenor curves. The discount curve is typically built with OIS instruments, which are the best approximation for the risk free rate. All the other tenor curve are built with instruments with homogeneous tenors. The most used tenors are 3M, 6M, 9M, 12M. Typically, the longer the tenor the riskier the trade and hence the higher the corresponding rate.

In this blog we give a non-exhaustive list of references that helped us in both understanding the multi-curve and designing the process.

Articles

Books

# The Geometry of Interest Rate Risk

In this paper we consider the process of interest rate risk management. The yield curve construction is revisited and emphasis is given to aspects such as input instruments, bootstrap and interpolation. For various financial products we present new formulas that are crucial to define sensitivities to changes in the instruments and/or in the curve rates. Such sensitivities are exploited for hedging purposes. We construct the risk space, which eventually turns out to be a curve property, and show how to hedge any product or any portfolio of products in terms of the original curve instruments.

Keywords: Yield curve, hedging, interest rate risk management.