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Beside the custom elements within the AdditionalFields node, the envelope contains the same elements for all Trade types. The meanings and allowable values of the various elements in the Envelope node follow below.

Id: The Id element in the envelope is used to identify trades within a portfolio. Counterparty: Specifies the name of the counterparty of the trade. It is used to show exposure analytics by counterparty. Underscores and blank spaces may be used as well. The specified NettingSetId must be a unique identifier for the netting set.

If left blank or omitted the trade will not belong to any netting set, and thus not be eligible for netting. Note that ORE does not assume a hierarchical organisation of such portfolios. Allowable values for each PortfolioId: Any string. For example, el- ements such as Sector, Desk or Folder can be used. The elements within the AdditionalFields node are used for informational purposes only, and do not affect any analytics in ORE.

The main problem with previous contributions is that the yield curve is assumed to be flat and only parallel shifts are allowed. However, in a swap where one pays Libor plus spread and receives a year CMS rate, the structure is mainly sensitive to the slope of the interest rate yield curve and is almost immunised against any parallel shift. In this paper, following Hagan , we apply the commonly used convexity adjustment in a new framework of double curving. We then develop a new convexity adjustment, by departing from the restricting assumption that the term structure is flat, and we allow for a tilt.

Using market data for Euro money market instruments Eonia, Euribor , CMS spreads and swaption volatilities, we find out that the new convexity adjustment is significantly larger than the one commonly made in the literature. The remaining of this paper is organised as follows. Section 3 shows the main result of our work, that is a new convexity adjustment that takes into account the tilt in the term structure, under a double curving framework.

Section 4 briefly depicts the smile-consistent convexity adjustment using SABR model. Sections 5 and 6 present the market data that have been used and describe numerical calculations. Finally, Sect. The valuation framework This section introduces the definitions of basic instruments under the muti-curve environment. It mostly follows the works of Brigo and Mercurio and Mercurio Forward rates can be defined for both curves.

We assume a given single discount curve for use in the calculation of all net present values NPVs , i. Following Mercurio , we adopt the standard definition for the FRA rate. In the multi-curve framework, however, Eq. Therefore, the present value of a future Libor rate is no longer obtained by discounting the corresponding forward rate, but by discounting the corresponding FRA rate. According to Mercurio , the FRA rate is the natural generalization of a forward rate to the multi-curve case.

This has a straightforward implication, when it comes to the valuation of Interest Rate Swaps. Interest rate swap We show how to evaluate an IRS under the multi-curve framework. For simplicity, we assume that IRS tenors for fixed and floating legs are the same.

Constant maturity swap A constant maturity swap contract, is a swap where one of the legs pays receives periodically a swap rate with a fixed time to maturity, c, while the other leg receives pays either fixed or floating. More commonly, one term is set to a short term floating index such as the 3-month Libor rate, while the other leg is set to a long term fixed rate such as the year swap rate.

The convexity adjustment arises since the expected payoff is calculated in a world which is forward risk neutral with respect to a zero coupon bond. In that world, the expected underlying swap rate upon which the payoff is based , does not equal the forward swap rate.

The convexity is just the difference between the expected swap rate and the forward swap rate. When we consider pricing CMS-type derivatives, it is convenient to compute the expectation of the future CMS rates under the forward measure, that is associated with the payment dates. However, the natural martingale measure of the CMS rate is the underlying forward swap measure.

Convexity correction arises when one computes the expected value of the CMS rate under the forward measure that differs from the natural swap measure with the underlying forward swap measure as numeraire. Convexity adjustment Following Pelsser , we define the convexity adjustment as the difference in expectation of some quantity i. Flat term structure with parallel shifts Following Hagan and Brigo and Mercurio , we initially derive an expression for the convexity adjustment when the term structure is flat and can only evolve with parallel shifts.

More specifically, following Liu et al. A term structure with tilts In this section, we depart from the restrictive and unrealistic assumption of a flat term structure and we extend our analysis by allowing for a tilt. Using Eqs. Finally, for function f we use the following parametric functional form, based on the well-known Nelson and Siegel model. In the presence of a market smile, when the term structure is not flat, but may tilt, the adjustment is necessarily more involved, if we aim to incorporate consistently the information coming from the quoted implied volatilities.

The procedure to derive a smile consistent convexity adjustment is described in Mercurio and Pallavicini and Pallavicini and Tarenghi , and is the one we will use here. For the consistent derivation of CMS convexity adjustment, volatility modelling is required.

We use the SABR model a popular market choice for swaption smile analysis for the swap rate in order to infer from it the volatility smile surface. See Mercurio and Pallavicini and Pallavicini and Tarenghi for a detailed description of the calibration procedure. Market data We use three data sets for this study, one containing Euro money market instruments for the construction of the yield curves, a second one containing CMS swap spreads with a maturity of 5-years, where the associated underlying swaps have a year maturity i.

All market data was collected from Bloomberg. Data sets are presented in detail below: For the discounting curve, we use Eonia Fixing and OIS rates from 3-months to years. For the 3-month curve, we use Euribor 6-months fixing, FRA rates up to 15 months, and swaps from 2 to 30 years, paying an annual fix rate in exchange for the Euribor 3-month rate. For the 6-month curve, we use Euribor 6-months fixing, FRA rates up to 2 years, and swaps from 2 to 30 years, paying an annual fix rate in exchange for the Euribor 6-month rate.

However, it quotes the spread only for a few CMS swap maturities and tenors usually 5, 10, 15, 20 and 30 years. Thus, CMS spreads depend on three different curves in our framework; first, the funding curve used to discount the cash flows of the CMS swap, which we consider to be the risk-free curve i. OIS curve ; second, the 3-month forwarding curve for the Euribor rates paid in the second leg of the CMS; and third, the 6-month or 1-year forwarding curve for the Euribor rates paid by the indexation IRS.

Empirical results In this section, we compare numerically the accuracy of the approximations for the CMS convexity adjustments against the Black and SABR models convexity adjustments presented in Sect. An empirical illustration Our first numerical example is based on Euro data as of 3 February We test a CMS with maturity of 5 years i.

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