Constant Maturity Swaps, Forward Measure and LIBOR Market Model

Valuation of Swaps and Options on Constant Maturity CDS Spreads

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.

A Review of CMS Swap Pricing Approaches

A Libor Market Model with Default Risk

On the Information in the Interest Rate Term Structure and Option Prices

We introduce a general class of analytically tractable diffusions for modelling forward LIBOR rates under their canonical measure. The class, which is based on assuming a smooth functional dependence at expiry between a forward rate and an associated Brownian motion, is highly tractable. It implies explicit dynamics, known marginal and transition densities and explicit caplet prices at any time. As an example, we analyse the dynamics given by a linear combination of geometric Brownian motions with perfectly correlated (decorrelated) returns. We finally construct a specific model in the class that reproduces exactly the market caplet volatilities given in input. Examples of the implied-volatility curves produced by the considered models are also shown.

In the more classical models for equities and interest rates evolution, constant volatility is usually assumed. However, in practice the volatilities are not constant in financial markets and different models allowing a varying local or stochastic volatility also appear in the literature. Particularly, here we consider the SABR model that has been first introduced in a paper by Hagan and coworkers, where an asymptotic closed-form formula for the implied volatility of European plain-vanilla options with short maturities is proposed. More recently, different works (Mercurio and Morini, Modeling Interest Rates: Advances in Derivatives Pricing, Risk Books 2009; Hagan and Lesniewski, LIBOR market model with SABR style stochastic volatility. Working Paper. http://lesniewski.us/papers/working/SABRLMM.pdf, 2008; Rebonato, A time-homogeneous SABR-consistent extension of the LMM. Risk, 2008) have extended the use of SABR model in the context of LIBOR market models for the evolution of forward rates (SABR-LMM). One drawback of these models in practice comes from the increase of computational cost, mainly due to the growth of model parameters to be calibrated. Additionally, sometimes either it is not always possible to compute an analytical approximation for the implied volatility or its expression results to be very complex, so that numerical methods (for example, Monte Carlo in the calibration process) have to be used. In this work we mainly review some recently proposed global optimization techniques based on Simulated Annealing (SA) algorithms and its implementation on Graphics Processing Units (GPUs) in order to highly speed up the calibration and pricing of different kinds of options and interest rate derivatives. Finally, we present some examples corresponding to real market data.KeywordsSABR volatility modelsSABR/LIBOR market modelsParallel simulated annealingGPUs

In this paper an efficient calibration method for the multi-factor LIBOR Market Model (LMM) is proposed and is applied for the Japanese interest rate market. At first the joint calibration method in the cap and swaption market is demonstrated using a new parameterization for the correlation matrix in the LMM. Then we implement the proposed methodology for calibrating the Japanese cap and swaption markets, where the computational procedure is shown to be tractable and provides a practical estimation for the implied correlation matrix in the LMM. The empirical analysis also illustrates that Black's swaption volatilities through our calibration fit the market data almost exactly and that the estimated implied correlation matrix is smooth and stable.

One key driver of a bank’s total interest rate risk is the position of non-maturing deposits. Several papers such as [6], [8], and [7] value non-maturing deposits in an arbitrage-free framework and analyze their risk profile. All these models consist of three major components: first, the short rate process, i.e. the dynamics of the default-free interest rate term structure; second, the interest rate pass-through, i.e. the link between the development of the deposit rates and the development of default-free interest rates, in general the short rate; third, the development of the deposit volume over time. In this paper, we concentrate on the interest rate pass-through. We provide some term structure model-free results on the valuation of deposits, when the deposit rates are linearly linked to some long-term swap rate (rather than a short-term interest rate) as the reference rate with an unnatural time lag.

A number of models have been successfully developed wherein the theoretical prices for caps, floors, and swaptions produced by the model are of the Black-76 form. In these models, discrete market rates are modelled like LIBOR rates in the LIBOR market models or forward swap rates in the swap market models; and under a suitable choice of numeraires, market rates can be modelled log normally. LIBOR caps and the market practice for pricing and quoting these instruments are discussed. It is shown that given a swap market model, the LIBOR rates will not be lognormal; thus, LIBOR market models and swap models are generally incompatible. Practice exercises are included.

Dynamic term structure models explain the yield curve variation well but perform poorly in pricing and hedging interest rate options. Most existing option pricing practices take the yield curve as given, thus having little to say about the fair valuation of the underlying interest rates. This paper proposes an m + n model structure that bridges the gap in the literature by successfully pricing both interest rates and interest rate options. The first m factors capture the yield curve variation, whereas the latter n factors capture the interest rate options movements that cannot be effectively identified from the yield curve. We propose a sequential estimation procedure that identifies the m yield curve factors from the LIBOR and swap rates in the first step and the n options factors from interest rate caps in the second step. The three yield curve factors explain over 99% of the variation in the yield curve but account for less than 50% of the implied volatility variation for the caps. Incorporating three additional options factors improves the explained variation in implied volatilities to over 99%.

The Co-Terminal Swap Market Model with Bergomi Stochastic Volatility

LIBOR Market Model with Stochastic Volatility

Valuation of Convexity Related Derivatives

We propose an elementary model in multi-curve setting that allows to price with simple exact closed formulas European swaptions. Swaptions can be both physical delivery and cash-settled ones. The proposed model is very parsimonious: it is a three-parameter multi-curve extension of the two-parameter J. Hull & A. White (1990) [Pricing interest-rate-derivative securities. Review of Financial Studies 3(4), 573–592] model. The model allows also to obtain simple formulas for all other plain vanilla Interest Rate derivatives and convexity adjustments. Calibration issues are discussed in detail.