Interest Rate Modelling Papers


Risk managing long-dated smile risk with SABR formula

Paper by Claudio Moni

In this paper, we show that the sensitivities to the SABR parameters can be materially wrong when the SABR formula is used, in particular for long expiries and in high volatility environments. For example, we obtain positive sensitivities to the spot-vol correlation parameter for low strike options, which is the opposite of what we expect from the SABR model. We discuss possible solutions.


An arbitrage-free method for smile extrapolation

Paper by Shalom Benaim, Matthew Dodgson and Dherminder Kainth

We introduce a method for extrapolating smiles beyond an "observable" region that is consistent with no arbitrage. The extrapolation is not unique, but can be tuned e.g., to different power-law decays. This method has important applications in various areas such as the calculation of CMS rates, inverse FX options etc.


Implementations of the LIBOR market model

Presentation given at ICBI 2005 by Mark Joshi and Alan Stacey

Two new appproaches to approximating the drift in LIBOR market model are introduced and shown to be superior to known methodologies.

Enhancements to the Longstaff-Schwartz Algorithm for bounding Callable Libor Exotics

Presentation given at ICBI 2005 by Mark Joshi and Dherminder Kainth
Discuss a variety of enhancements to the Longstaff-Schwartz algorithm for the pricing of callable Libor exotics. Non parametric regression and other tricks for lower bounds. Comments on the primal dual algorithm to enhance effciency of the upper bound (Note Broadie and Cao have some results in common independently.)

Term-Structure Models: a Review

Paper by Riccardo Rebonato (2003).

A survey and critique of approaches to modelling the term structure of interest rates. It is argued that new methodologies are needed for the pricing of complicated exotic products.

Real world evolution of yield curves presentation

Presentation given by Riccardo Rebonato in Geneva 2002.
Paper by Riccardo Rebonato, Sukhdeep Mahal, Mark Joshi and Lars-Dierk Bucholz (2003).
Figures to the paper (in powerpoint).

A presentation and paper on how to build a model for the evolution of yield curves in the real world rather than a risk-neutral simulation. In this paper we show how to evolve a yield curve over time horizons of the order of years using a simple but effective semiparametric method.The proposed technique preserves in the limit all the eigenvalues and eigenvectors of the observed changes in yields. It also recovers in a satisfactory way several important statistical features (unconditional variance, serial autocorrelation, distrubution of curvatures) of the real-world data. A simple financial explanation can be provided for the methodology. The possible financial applications are discussed.

A two-regime stochastic-volatility extension of the LMM

Presentation by Riccardo Rebonato and Dherminder Kainth (2002)
Paper by Riccardo Rebonato and Dherminder Kainth (2003)
Figures to the paper (in powerpoint).

A presentation and paper on an extension of the Joshi-Rebonato SVBGM model to allow two regimes of stochastic volatility. We propose a two-regime stochastic volatility extension of the LIBOR market model that preserves the positive features of the recently introduced (Joshi and Rebonato 2001) stochastic-volatility LIBOR market model (ease of calibration to caplets and swaptions, efficient pricing of complex derivatives, etc.) and overcomes most of its shortcomings. We show the improvements by analysing empirically and theoretically the real and the model-produced changes in swaption implied volatility

Bounding Bermudan swaptions in a swap-rate market model

Paper by Mark Joshi and Jochen Theis, published in Quantitative Finance Vol. 2 (2002) pages 370-377

We develop a new method for finding upper bounds for Bermudan swaptions in a swap-rate market model. By comparing with lower bounds found by exercise boundary parametrization, we find that the bounds are well within bid-offer spread. As an application, we study the dependence of Bermudan swaption prices on the number of instantaneous factors used in the model. We also establish an equivalence with LIBOR market models and show that virtually identical lower bounds for Bermudan swaptions are obtained.

A stochastic volatility displaced-diffusion extension of the LIBOR market model

Paper by Mark Joshi and Riccardo Rebonato (2001) published in Quantitative Finance Vol. 3 (2003) pages 458-469

We present an extension of the LIBOR market model which allows for stochastic instantaneous volatilities of the forward rates in a displaced diffusion setting. We show that virtually all the powerful and important approximations that apply in the deterministic setting can be successfully and naturally extended to the stochastic volatility case. In particular we show that i) the caplet market can still be efficiently and accurately fit; ii) that the drift approximations that allow the evolution of the forward rates over time steps as long as several years are still valid; iii) that in the new setting the European swaption matrix implied by a given choice of volatility parameters can be efficiently approximated with a closed-form expression without having to carry out a Monte Carlo simulation for the forward-rate process; and iv) that it is still possible to calibrate the model virtually perfectly via simply matrix manipulations so that the prices of the co-terminal swaptions underlying a given Bermudan swaption will be exactly recovered, while retaining a desirable behaviour for the evolution of the term structure of volatilities. We also show that, even after reducing the number of the possible fitting parameters, the market caplet surface across strikes and maturities can be well recovered. We notice the existence of what appears to be a systematic discrepancy for very low strikes, but we have refrained from attempting to recover this feature.

Drift Approximations in a LIBOR market model

Paper by Christopher Hunter, Peter Jäckel and Mark Joshi, published in Risk Magazine as Getting the Drift, July 2001

In a market model of forward interest rates, a specification of the volatility structure of the forward rates uniquely determines their instantaneous drifts via the no-arbitrage condition. The resulting drifts are state-dependent and are sufficiently complicated that an explicit solution to the forward rate stochastic differential equations cannot be obtained. The lack of an analytic solution could be a major obstacle when pricing derivatives using Monte Carlo if it implied that the market could only be accurately evolved using small time steps. In this paper we use a predictor-corrector method to approximate the solutions to the forward rate SDEs and demonstrate that the market can be accurately evolved as far as twenty years in one step.

A joint empircal and theoretical investigation of the modes of deformation of swaptions matrices: implications for model choice

Paper by Riccardo Rebonato and Mark Joshi (2001) published in International Journal of Theoretical and Applied Finance, Vol. 5, No. 7 (2002) .


Accurate and optimal calibration to co-terminal swaptions European swaptions in a FRA-based BGM framework

Paper by Riccardo Rebonato

This paper provides a calibration methodology that recovers (almost) exactly the prices of all the co-terminal swaptions underlying a given Bermudan swaption, and, at the same time, ensures optimal recovery of user-specified portions of the forward-rate covariance matrix (for instance, the attention could be focussed on caplet prices). The ability to obtain at the same time a virtually perfect fit to European swaptions and an optimal calibration to a user-specified forward-rate covariance matrix is extremely important, since virtually no exotic product depends on the dynamics of one set of state variables only (forward rates or swap rates).

On the simultaneous calibration of multi-factor log-normal interest-rate models to Black volatilities and to the correlation matrix

Paper by Riccardo Rebonato

It is shown in this paper that it is not only possible, but indeed expedient and advisable, to perform a simultaneous calibration of a log-normal BGM interest-rate model to the percentage volatilities of the individual rates and to the correlation surface. One of the contributions of the paper it to show that the task can be accomplished in two separate and independent steps: the first part of the calibration (i.e. to cap volatilities) can always be accomplished exactly thanks to straightforward geometrical relationships; the fitting to the correlation surface, thanks to a simple theorem, can then be carried out in a numerically efficient way so that the calibration to the volatilities is not spoiled by the second part of the procedure. The ability to carry out the two tasks separately greatly simplifies the overall task. Actual calculations are shown for a 3- and 4-factor implementation of the approach, and the quality of the overall agreement between the target and model correlation surfaces is commented upon. Finally, the dangers of overparametrization, i.e. of forcing (near) exact fitting to certain portions of the correlation matrix, are analysed by looking at the cases of a trigger swap, a Bermudan swaption and a one-way floater (resettable cap).

Linking caplet and swaption volatilities in a BGM/J framework: approximate solutions

Paper by Peter Jäckel and Riccardo Rebonato (2000)

We present an approximation for the volatility of European swaptions in a forward-rate based BGM/J framework which enables us to calculate prices for swaptions without the need for Monte Carlo simulations. Also, we explain the mechanism behind the remarkable accuracy of these approximate prices. For cases, where the yield curve varies noticeably as a function of maturity, a second and even more accurate formula is derived.

Using a non-recombining tree to design a new pricing method for Bermudan swaptions

Paper by Peter Jäckel (2000)

In the Libor market model framework of Brace-Gatarek-Musiela and Jamshidian (BGM/J) for the pricing of interest rate derivatives, the drifts of the underlying forward rates are state dependent. This poses a major difficulty for the construction of recombining lattice methods which are usually employed whenever exercise strategy dependent options such as Bermudans are to be evaluated. Monte Carlo methods, on the other hand, are (effectively) unaffected by the intrinsic high-dimensionality of the embedding space necessary to represent the dynamics as Markovian. However, accounting for the early exercise opportunities is a non-trivial issue when Monte Carlo techniques are used. In this article, we explain a new Monte Carlo method that can be used to price Bermudan swaptions with remarkable accuracy. The new technique is based on the functional parametrisation of the exercise boundary in a suitable coordinate system carefully selected by the aid of a non-recombining tree.

Non-recombining trees for the pricing of interest rate derivatives in the BGM/J framework

Paper by Peter Jäckel (2000)

In the Libor market model framework of Brace-Gatarek-Musiela and Jamshidian (BGM/J) for the pricing of interest rate derivatives, the drifts of the underlying forward rates are state dependent. Wherever Monte Carlo methods can be used for the numerical calculation of discounted expectations, this poses no major difficulty since the computational effort necessary in order to achieve acceptable accuracy depends only weakly on the dimensionality of the sampling space. For the valuation of options that involve finding the optimal exercise strategy, however, this means that any finite-differencing method enabling us to make the pointwise comparison directly between intrinsic value and discounted expectation will have to cope with "the curse of dimensionality" whereby the number of evaluations explodes exponentially. This document is about the implementation of a non-recombining multi-factor tree algorithm with a minimal number of branches out of each node for the representation of the desired number of factors. This method can serve as a benchmark for simple test cases for the development of other approximations such as exercise-strategy parametrisations in a Monte Carlo setup.