# 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

# Review of Lattice Construction Methods

Just before year end we put out our paper, titled "Review of Lattice Construction Methods"!

This paper extends the generalized procedure for building trees for short rates by Hull & White. A generalization for any mean and standard deviation of the underlying short rate model is presented. In addition we review the methodologies for constructing lattice models and give a step-by-step explanation on how to construct trinominal trees. We apply the formalism to some explicit examples of various complexity.

Full text can be found here

# Curiosities on the Monotone Preserving Cubic Spline

In this paper we describe some new features of the monotone-preserving cubic splines and the Hyman’s monotonicity constraint, that is implemented into various spline interpolation methods to ensure monotonicity. We find that, while the Hyman constraint is in general useful to enforce monotonicity, it can be safely omitted when the monotone-preserving cubic spline is considered. We also find that, when computing sensitivities, consistency requires making some specific assumptions about how to deal with non-differentiable locations, that become relevant for special values of the parameter space.

Keywords: Yield curve, fixed-income, interpolation, Hyman, monotone preserving cubic splines.

# 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.

# On the non-differentiability of Hyman Monotonicity constraint

In this paper we describe some new features of Hyman’s monotonicity constraint, which is implemented into various cubic spline interpolation methods. We consider the problem of understanding how sensitive such methods are to small changes of the input y-values and, in particular, how relevant Hyman’s constraint is with respect to such changes. We find that many things cancel out and that eventually Hyman’s constraint can be safely omitted when the monotone-preserving cubic spline is used. We also find that consistency requires including some specific boundary conditions that become relevant for special values of the parameter space.

Keywords: Yield curve, interpolation, monotone preserving cubic splines.

# 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

# UD Fin Lib

Ugly Duckling Finance is currently working on its financial library, UDFinLib. UDFinLib will appear soon and will be advertised on this blog and website. Everybody interested is therefore invited to come back later when it will be ready to use. In this blog post I would like to anticipate some of the features of the library.

The library comes in two parts: the core and the Excel Add-in. 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.