Constant Maturity Swap Research Articles

Valuation of callable range accrual linked to CMS Spread under generalized swap market model

In this paper, we price a widely-used financial instrument, the callable range accrual linked to constant maturity swap (CMS) spread, with the least square Monte Carlo method (LSMC) under the generalized swap market model (GSMM). This method, based on the swap rate, does not only provide an intuitive pricing solution, but also captures the characteristics of the swap market, which helps market participants better face the challenge of the London interbank offered rate (LIBOR) phase-out. By choosing the prices valued by the two-factor Hull–White model in Bloomberg as a benchmark, our method provides an accurate result, following the economic intuitions. Last but not least, we examine the impacts from the shifting of the yield curve and volatility term structure by sensitivity analysis. We find that the price positively correlates with the volatility term structure among 1% shifts, while the relationship can be uncertain in larger shifts. The changes in the estimated parameters, such as the yield curve and volatility term structure, might reflect the occurrences of macroeconomic events, such as the COVID-19 pandemic.

On the Deterministic-Shift Extended CIR Model in a Negative Interest Rate Framework

In this paper, we propose a new exogenous model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model with a perfect fit to the observed term-structure. We use the difference between two independent CIR processes and apply the deterministic-shift extension technique. To allow for a fast calibration to the market swaption surface, we apply the Gram–Charlier expansion to calculate the swaption prices in our model. We run several numerical tests to demonstrate the strengths of this model by using Monte-Carlo techniques. In particular, the model produces close Bermudan swaption prices compared to Bloomberg’s Hull–White one-factor model. Moreover, it finds constant maturity swap (CMS) rates very close to Bloomberg’s CMS rates.

Interest rate structured products: can they improve the risk–return profile?

In this paper, we investigate the contribution of interest rate structured bonds to portfolios of risk-averse retail investors. We conduct our analysis by simulating the term structure according to a multifactor no-arbitrage interest rate model and comparing the performance of a portfolio consisting of basic products (zero-coupon bonds, coupon bonds and floating rate notes) with a portfolio containing more sophisticated exotic products (like constant maturity swaps, collars, spread and volatility notes). Our analysis, performed under different market environments, as well as volatility and correlation levels, takes into account the combined effects of risk premiums required by investors and fees that they have to pay. Our results show that capital protected interest rate structured products allow investors to improve risk–return trade-off if no fees are considered. With fees, our simulations show that structured products add value to the basic portfolio in a very limited number of cases. We believe our paper contributes to understanding the role of structured products in investors portfolios also in light of the current regulatory debate on the use of complex financial products by retail investors.

A UNIFIED MARKET MODEL FOR SWAPTIONS AND CONSTANT MATURITY SWAPS

Internal-rate-of-return (IRR) settled swaptions are the main interest rate volatility instruments in the European interest rate markets. Industry practice is to use an approximation formula to price IRR swaptions based on Black model, which is not arbitrage-free. We formulate a unified market model to incorporate both swaptions and constant maturity swaps (CMS) pricing under a single, self-consistent framework. We demonstrate that the model is able to calibrate to market quotes well, and is also able to efficiently price both IRR-settled and swap-settled swaptions, along with CMS products. We use the model to illustrate the difference in implied volatilities for IRR-settled payer and receiver swaptions, the pricing of zero-wide collars and in-the-money (ITM) swaptions, the implication on put-call parity, and the issue of negative vega. These findings offer important insights to the ongoing reform in the European swaption market.

Closed-form Arrow-Debreu pricing for the Hull-White short rate model

We consider the Hull-White short rate model and provide a systematic derivation of an Arrow-Debreu pricing formula for European-style options in closed form, applying it to cap/floor pricing and CMS (constant maturity swap) index calculation. We present in the process a useful exponential expansion formula which facilitates the analytic solution of pricing equations. We propose that the methodology here described is of interest insofar as it is applicable to a wider range of short rate modelling problems potentially involving lognormal rates/credit volatility, local-stochastic volatility and/or multiple stochastic factors or underlyings.

Model-free stochastic collocation for an arbitrage-free implied volatility: Part I

This paper explains how to calibrate a stochastic collocation polynomial against market option prices directly. The method is first applied to the interpolation of short-maturity equity option prices in a fully arbitrage-free manner and then to the joint calibration of the constant maturity swap convexity adjustments with the interest rate swaptions smile. To conclude, we explore some limitations of the stochastic collocation technique.

A Unified Market Model for Swaptions and Constant Maturity Swaps

Internal-rate-of-return (IRR) settled swaptions are the main interest rate volatility instruments in the European interest rate markets. Industry practice is to use an approximation formula to price IRR swaptions based on Black model, which is not arbitrage-free. We formulate a unified market model to incorporate both swaptions and constant maturity swaps (CMS) pricing under a single, self-consistent framework. We demonstrate that the model is able to calibrate to market quotes well, and is also able to efficiently price both IRR-settled and swap-settled swaptions, along with CMS products. We use the model to illustrate the difference in implied volatilities for IRR-settled payer and receiver swaptions, the pricing of zero-wide collars and in-the-money (ITM) swaptions, the implication on put-call parity, and the issue of negative vega. These findings offer important insights to the ongoing reform in the European swaption market.

Swap Curve Steepener

The case is about a decision problem facing James on whether or not to invest in a structured product called the “CMS Steepener” issued by a large US investment bank. The payoff from the product is linked to two constant maturity swap (CMS) rates, and the investor profits if the difference between the two CMS rates increases, or alternatively if the CMS curve steepens. The case describes the risks that investing in such a product poses, and presents relevant data on the CMS rates, term structure and recent financial history of the issuer to help resolve James's decision problem.

Valuation of certain CMS spreads

In this paper, we derive an approximate lognormal process for the swap rate under the multifactor LIBOR market model using a Levy approach. Using the approximate dynamics for the swap rate, the constant maturity swap spread digital range notes with different strike rates are valued in analytic and semi-analytic form. The CMS spread digital range notes are widely traded in the marketplace, or embedded in structure notes.

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.

A Review of CMS Swap Pricing Approaches

Evaluating Constant Maturity Swap (CMS) derivatives is a lot more complex than plain vanilla interest rate swaps, because of the unnatural schedule of their payments. Their pricing requires either a convexity adjustment or the use of a model. Hence multiple approaches have been proposed. We selected the most common ones in the marketplace, to analyse and compare them by examining their characteristics. We found that the most accurate prices are generated by Monte Carlo simulations with BGM model for forward rates, while the continuous swaption replication method offers prices consistent with the other instruments of the trading book.

Valuation of CMS range notes in a multifactor LIBOR market model

In the framework of a multifactor LIBOR market model (LMM), this paper presents a new approach for finding an approximate distribution of constant maturity swap (CMS) rates under the terminal martingale measure. With this approach, we derive an analytical pricing formula for CMS range notes, which is both intuitive and tractable. Many exotic CMS rate derivatives are widely traded in the marketplace or embedded in structure notes.

Too big to ignore? Hedge fund flows and bond yields

Abstract This paper investigates the information content of aggregate hedge fund flow and its predictive power with respect to bond yields. Using a sample of 9725 hedge funds from 1994 to 2012, we find that fund flow is negatively related to the changes in 10-year Treasury and Moody’s Baa bond yields one month ahead. This relation is still pronounced after controlling for other determinants of yield changes, including the amount of arbitrage capital available in the economy, suggesting a non-trivial effect of flow-induced hedge fund trading on bond yields. Flow impact on corporate bonds is further amplified during periods of decreasing market liquidity, consistent with a fire-sale hypothesis. Hedge fund flow also predicts convergence between constant maturity swap rate and constant maturity Treasury rate, as well as between the TIPS and Treasury bond yields, suggesting that hedge funds exploit arbitrage opportunities in these fixed-income markets.

Identifying, valuing and hedging of embedded options in non-maturity deposits

Non-maturity deposits like savings accounts or demand deposits contain significant option risks caused by the bank’s discretionary pricing and the customers’ withdrawal right. Option risks follow from inherent non-linear factor exposures. I propose an ordinal response model for deposit rate jumps to identify non-linear factor exposures and a discrete-time term structure model to value the resulting option risks and to derive hedge measures “outside the model”. My delta profile resembles a constant maturity swap, but vega and gamma are more pronounced, which demonstrates that the widespread practice of static hedging with zero bonds is inadequate.

Management Framework for Non-Maturity Accounts: From the Marketing to the Hedging Strategy

Non-maturity deposits like savings accounts or demand deposits contain significant option risks caused by the bank's discretionary pricing and the customers' withdrawal right. Option risks follow from inherent non-linear factor exposures. I propose an ordinal response model for deposit rate jumps to identify non-linear factor exposures and a discrete-time term structure model to value the resulting option risks and to derive hedge measures "outside the model''. My delta profile resembles a constant maturity swap, but vega and gamma are more pronounced, which demonstrates that the widespread practice of static hedging with zero bonds is inadequate.

Pricing a Callable Leveraged Constant Maturity Swap Spread Note

A callable leveraged constant maturity swap (CMS) spread note allows the holder to benefit from future changes in the spread between two swap interest rates. The issues retains the right to call the note at pre-specified times in the future. The note is priced via Monte Carlo simulation using the current term structure of interest rates and at-the-money implied swaption volatilities.

CMS, CMS SPREADS AND SIMILAR OPTIONS IN THE MULTI-FACTOR HJM FRAMEWORK

Constant maturity swaps (CMS) and CMS spread options are analysed in the multi-factor HJM framework. For Gaussian models, which include a version of the Libor Market Models and the G2++ model, explicit approximated pricing formulae are provided. Two approximating approaches are proposed: an exact solution to an approximated equation and an approximated solution to the exact equation. The first approach borrows from previous literature on other models; the second approach is new. For the latter, the price approximation errors are smaller than in the previous literature and negligible in practice. These approaches are being used here to price standard CMS and CMS spreads and can be used for other European exotic products.

On Valuing Constant Maturity Swap Spread Derivatives

Motivated by statistical tests on historical data that confirm the normal distribution assumption on the spreads between major constant maturity swap (CMS) indexes, we propose an easy-to-implement two-factor model for valuing CMS spread link instruments, in which each forward CMS spread rate is modeled as a Gaussian process under its relevant measure, and is related to the lognormal martingale process of a corresponding maturity forward LIBOR rate through a Brownian motion. An illustrating example is provided. Closed-form solutions for CMS spread options are derived.

Affine Recursion Problem and a General Framework for Adjoint Methods for Calculating Sensitivities for Financial Instruments

Options with embedded early exercise features are of fundamental importance in finance. A simple example is the hedge of a multi-callable bond. This instrument is hedged using a Bermudan swaption.Bermudan swaptions also play a key role when pricing callable constant maturity swaps or flexible swaps and are thus relevant for risk management of Pension funds and retail banks. It is market practice to consider multi-factor models such as the Libor Market model to price and hedge such instruments.For hedging the key figures are the sensitivities to market and model parameters. For instance the Delta is the sensitivity of changing the underlying rate. To this end it is necessary to calculate such sensitivities in a fast and robust fashion. In this paper we propose a general framework for dealing with adjoints. We call it the Affine Recursion Problem (ARP). There are two solutions to the ARP. One is the forward method and the other one is the adjoint method. After a short review of approximating diffusion processes by numerical schemes we approximate the sensitivities Delta, Gamma and Vega using the general procedure. We treat the application of the general framework to Bermudan Swaptions and Trigger Swaps in Libor Market Model. For all steps of the approach we provide source code in Matlab such that not only the general approach is outlined and can be adapted to computer programs but also the reader can use this approach to set up adjoint methods.

Efficient Pricing of constant maturity swap spread options in a stochastic volatility LIBOR market model

Efficient Pricing of constant maturity swap spread options in a stochastic volatility LIBOR market model