# 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

# Interpolation methods and the Hagan-West paper

Interpolation is a very useful technique for extracting data when the available information does not come in a continuous form.

From a non-technical point of view, any inference or decision process (sometimes subconsciously) is based on a kind of interpolation or best fitting or regression of the available informations. We as people are normally quite good at generalising (often too fast) from the little amount of information that we have about other people, situations, or even numerical data. This is possible because our brain can recognise patterns and see trends in any kind of data. However, technically speaking, interpolation is more that just finding a trend.

Technically, we are often given a discrete set of data corresponding to a certain function which is known at specific points, or nodes (for example, we have made an experiment for specific input values and measured the outputs  corresponding to that input), and is otherwise unknown.  In principle this is a multi-dimensional problem, and the interpolating hyper-surface will give an idea of the missing information. In fact, even if it is true that such a hyper-surface can always be numerically constructed,  however the uniqueness issue remains. Given the same input data, many different constructions can be engineered, all satisfying to various -more or less realistic- criteria, and all passing through the same input points. Continue reading

# Note Why Multi Curve is Essential

The need for Multi Curve is becoming more clear in the marketplace. As a result many banks are moving from Single Curve valuation to Multi Curve, or have already done so. For a client we wrote a short paper list some of the pro’s and con’s of implementing Multi Curve. This paper was meant to trigger the discussion in the team and beyond.

# Validation at Kempen & Co

In collaboration with Valu8, we acted for Kempen & Co on a model validation.

In collaboration with Valu8 we performed a model validation for Kempen & relating to the methodology used to valuate and hedge the Inflation Breaker, a portfolio of ground lease contracts. The contracts in the portfolio generate monthly cash flows based on the rent which increase annually with the Dutch pricing inflation index. At maturity of the contract, the tenant has the option, but not the obligation, to buy back the land.

As part of the validation we checked the cash flow model, the interest rate curve construction methodology and the inflation model. Finally, the implementation of the models in Excel/VBA was checked, and the code used to implement the models was validated.