Convexity Adjustment for Constant maturity Swaps in a Multi-Curve Framework

In this paper we propose a double curving setup with distinct forward and discount curves to price constant maturity swaps (CMS). Using separate curves for discounting and forwarding, we develop a new convexity adjustment, by departing from the restrictive assumption of a flat term structure, and expand our setting to incorporate the more realistic and even challenging case of term structure tilts. We calibrate CMS spreads to market data and numerically compare our adjustments against the Black and SABR (stochastic alpha beta rho) CMS adjustments widely used in the market. Our analysis suggests that the proposed convexity adjustment is significantly larger compared to the Black and SABR adjustments and offers a consistent and robust valuation of CMS spreads across different market conditions

Attributing returns and optimising United States swaps portfolios using an intertemporally-consistent and arbitrage-free model of the yield curve

This paper uses the volatility-adjusted orthonormalised Laguerre polynomial model of the yield curve (the VAO model) from Krippner (2005), an intertemporally-consistent and arbitrage-free version of the popular Nelson and Siegel (1987) model, to develop a multi-dimensional yield-curve-based risk framework for fixed interest portfolios. The VAO model is also used to identify relative value (i.e. potential excess returns) from the universe of securities that define the yield curve. In combination, these risk and return elements provide an intuitive framework for attributing portfolio returns ex-post, and for optimising portfolios ex-ante. The empirical applications are to six years of daily United States interest rate swap data. The first application shows that the main sources of fixed interest portfolio risk (i.e. unanticipated variability in ex-post returns) are first-order (‘duration’) effects from stochastic shifts in the level and shape of the yield curve; second-order (‘convexity’) effects and other contributions are immaterial. The second application shows that fixed interest portfolios optimised ex-ante using the VAO model risk/relative framework significantly outperform a naive evenly-weighted benchmark over time

Convexity Adjustments Made Easy: An Overview of Convexity Adjustment Methodologies in Interest Rate Markets

Interest rate instruments are typically priced by creating a non-arbitrage replicating portfolio in a risk-neutral framework. Bespoke instruments with timing, quanto1 and other adjustments often present arbitrage opportunities, particularly in complete markets where the difference can be monetized. To eliminate arbitrage opportunities we are required to adjust bespoke instrument prices appropriately, such adjustments are typically non-linear and described as convexity adjustments. We review convexity adjustments firstly using a linear rate model and then consider a more advanced static replication approach. We outline and derive the analytical formulae for Libor and Swap Rate adjustments in a single and multicurve environment, providing examples and case studies for Libor In-Arrears, CMS Caplet, Floorlet and Swaplet adjustments in particular. In this paper we aim to review convexity adjustments with extensive reference to popular market literature to make what is traditionally an opaque subject more transparent and heuristic

Valuation of Convexity Related Derivatives

We will investigate valuation of derivatives with payoff defined as a nonlinear though close to linear function of tradable underlying assets. Derivatives involving Libor or swap rates in arrears, i.e. rates paid in a wrong time, are a typical example. It is generally tempting to replace the future unknown interest rates with the forward rates. We will show rigorously that indeed this is not possible in the case of Libor or swap rates in arrears. We will introduce formally the notion of plain vanilla derivatives as those that can be replicated by a finite set of elementary operations and show that derivatives involving the rates in arrears are not plain vanilla. We will also study the issue of valuation of such derivatives. Beside the popular convexity adjustment formula, we will develop an improved two or more variable adjustment formula applicable in particular on swap rates in arrears. Finally, we will get a precise fully analytical formula based on the usual assumption of log-normality of the relevant tradable underlying assets applicable to a wide class of convexity related derivatives. We will illustrate the techniques and different results on a case study of a real life controversial exotic swap.interest rate derivatives, Libor in arrears, constant maturity swap, valuation models, convexity adjustment

FX Modelling in Collateralized Markets: foreign measures, basis curves, and pricing formulae

We present a general derivation of the arbitrage-free pricing framework for multiple-currency collateralized products. We include the impact on option pricing of the policy adopted to fund in foreign currency, so that we are able to price contracts with cash flows and/or collateral accounts expressed in foreign currencies inclusive of funding costs originating from dislocations in the FX market. Then, we apply these results to price cross-currency swaps under different market situations, to understand how to implement a feasible curve bootstrap procedure. We present the main practical problems arising from the way the market is quoting liquid instruments: uncertainties about collateral currencies and renotioning features. We discuss the theoretical requirements to implement curve bootstrapping and the approximations usually taken to practically implement the procedure. We also provide numerical examples based on real market data

CALLABLE SWAPS, SNOWBALLS AND VIDEOGAMES

Although economically more meaningful than the alternatives, short rate models have been dismissed for financial engineering applications in favor of market models as the latter are more flexible and best suited to cluster computing implementations. In this paper, we argue that the paradigm shift toward GPU architectures currently taking place in the high performance computing world can potentially change the situation and tilt the balance back in favor of a new generation of short rate models. We find that operator methods provide a natural mathematical framework for the implementation of realistic short rate models that match features of the historical process such as stochastic monetary policy, calibrate well to liquid derivatives and provide new insights on complex structures. In this paper, we show that callable swaps, callable range accruals, target redemption notes (TARNs) and various flavors of snowballs and snowblades can be priced with methods numerically as precise, fast and stable as the ones based on analytic closed form solutions by means of BLAS level-3 methods on massively parallel GPU architectures.Interest Rate Derivatives; stochastic monetary policy; callable swaps; snowballs; GPU programming; operator methods