Riccardo Rebonato

On the simultaneous calibration of multifactor lognormal interest rate models to Black volatilities and to the correlation matrix

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 3and 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 oneway floater (resettable cap). 1 – Introduction and motivation for the present study Until relatively recently, the calibration to market quantities of any interestrate option model was one of the most arduous parts of its implementation. Users of early short-rate-based models (such as the Black-Derman and Toy (1990), the Hull and White or the Black-Karasinsky (1991)) are too well aware of the difficulties one encounters when attempting to calibrate the model parameters so as to reproduce the prices of caps or swaptions. Also the more recent Heath-Jarrow-Morton (HJM) approach is, in its more general form, hardly more user-friendly when it comes to calibration of the model to market data. The common features of all these models was the fact that, explicitly or implicitly, within these traditional frameworks the stochastic behaviour was specified of unobservable financial quantities, such as, for instance, the 1 The ideas expressed in this article have evolved over time thanks to discussions with several colleagues, to whom I am indebted; in particular, I would like to remember Mike Sherring and Soraya Kazziha. instantaneous forward rates, the instantaneous short rate or its variance. The calibration of a model to a set of market quantities therefore required transforming, via the black box provided by the model itself, the dynamics of these unobservable quantities into the dynamics of observable quantities. The recently introduced Brace-Gatrek-Musiela (BGM) approach (1995), germane to the HJM (1989) model, has radically changed this picture: now directly observable market quantities, such as discrete (Libor) forward rates or swap rates, are evolved. Given the availability from the market of the volatilities of caplets and European swaptions, calibration to either set of variables has become, at least for onefactor implementations, virtually immediate. Pre-empting the more precise treatment to be found later on, given a set of (forward of swap) rates, the market gives the traded prices of series of caplets and European swaptions. From these one can directly impute, via inversion of the Black formula, the value of the average variance of the appropriate log-normal rate. In turn, this quantity determines the (integral of the square of) the time-dependent instantaneous volatility that must be assigned to the corresponding model rate in order to price the market instrument exactly (at least within numerical noise). See Equation (8) below. In general, the realization at time T of the k-th log-normal forward rate, of value fk(t0) today, in terms of its timedependent instantaneous volatility, sk(u), is given by (1) fk(T) = fk(t0) exp [ ∫t0 μk(u) 1⁄2 sk(u) du ] exp [∫t0 sk(u) dz(u)] In the expression above, μk(u) is the value at time u of the time-dependent drift that ensures that the model is arbitrage free. More generally, in the case of r orthogonal driving factors (2) fk(T) = fk(t0) exp [ ∫t0 μk(u) 1⁄2 sk(u) du ] exp [∫t0 ∑m=1,r skm(u) dzm(u)] 2 ‘Smile’ effects are not taken into account in this discussion. In most interest-rate markets they tend to be of much smaller magnitude than in the FX or equity markets. 3 The requirement of orthgonality for the driving factors entails no real loss of generality, as any nonorthogonal Brownian motions can be always transformed into an equivalent orthogonal set. Since working with orthogonal processes is computationally much simpler, the assumption will always be made in the following that E[dzi,dzj] = δij dt. under the constraint that (3) ∑m=1,r skm(u) = sk(u) As long as Equations (2) and (3) are satisfied, the variance of the forward rates, and hence the caplet prices, will always be correctly recovered, irrespective of the number of driving factors. Similarly, one can write for the realization at time T of the k-th lognormal swap rate (using similar notation), SRk(T), (4) SRk(T) = SRk(t0) exp [ ∫t0 μSRk(u) 1⁄2 σSRk(u) du ] exp [∫t0σSRk(u) dz(u)] Once again, in the case of r orthogonal driving factors, Equation (4) becomes (5) SRk(T) = SRk(t0) exp [ ∫t0 μSRk(u) 1⁄2 sSRk(u) ds ] exp [∫t0 ∑m=1,r sSRk m(u) dzm(u)] with the constraint that (6) ∑m=1,r sSRk m(u) = sSRk(u) As far as the pricing of either caplets or European swaptions is concerned any number of factors can be used to obtain exactly their market (Black) prices. In particular, one single factor (used with the appropriate state variables) is perfectly adequate, since, as shown in greater detail below, the quantities to be matched are (7 ) σBlack (T-t0) =∫t0 s (u) du (In Equation (7) σBlack is the market implied Black volatility for the forward or swap rate expiring at time T, and s(u) is the instantaneous volatility of the same rate from today (t0) to expiry). If one wants, however, to imply the dynamics of swap rates from the assigned process for forward rates, or vice versa, the issue of the dimensionality of the approach is quite subtle. More strongly, one will want, in most practical applications, to use the BGM approach to value exotic interest options; in this case the model price for the exotic product can strongly depend on the dimensionality of the underlying cap(or swaption-) fitting model. In particular, for instruments like spread options or trigger swaps the choice of an appropriate number of driving factor becomes all-important. (See Sidenius (1998) on this point). If one were to choose for the number of driving factors as many independent Brownian motions as forward rates in a given instrument, one would, of course, be giving the most general description of the problem at hand. Given the well-known results of Principal Component Analysis, however, such an extravagant increase in the dimensionality of the problem is generally deemed to be unnecessary, and computationally detrimental. With fewer factors than the number of forwards, any possible choice for the apportioning of the variance (Equations (3) or (6)), and for the time-dependence of the instantaneous volatilities will give rise to different terminal correlations, covariance elements, and, ultimately, exotic option prices. It is therefore unavoidable to optimize a very-highdimensional problem to a target time-dependent covariance matrix. Given the complexity of the task, it is not surprising to find in the literature statements along the lines of the following: “With such a large number of variables a straightforward optimization is impractical at best. The problem is that finding the global best fit[...] is very difficult in high dimensions. [...] The question of calibrating the correlation matrix is very interesting, but it seems impractical to undertake this calibration in parallel with the volatility calibration.” (Sidenius (1998)) I propose in this paper a calibration methodology that, contrary to the statement above, does allow a simple and computationally efficient calibration designed to recover exactly and in the most general way the instantaneous volatilities of all the forward rates in the problem, and, at the same time, to fit in the ‘best’ possible way, given the dimensionality of the approach, the correlation matrix. Looked at in this light, the 4 The issues of the consistency between the simultaneous joint log-normal assumptions for forward and swap rates and their impact on pricing are addressed in Rebonato (1998). essential problem of calibration within the framework of the BGM approach therefore boils down to i) choosing the most suitable number of factors given the problem at hand; ii) choosing a suitable time-dependence for the instantaneous volatility functions; iii) apportioning the weights necessary for the exact recovery of the desired total instantaneous volatility amongst the time-dependent volatilities of the different factors (see Equations(3) and (6)); in other words, choosing how large the contribution of the m-th factor, σkm(s) , towards σk(s) should be. The present work shows how the task can be considerably simplified by means of general relationships that must hold true for any model of the BGM family, irrespective of the choice of instantaneous volatility; therefore the joint fitting to volatilities and correlations not only can, but arguably should, be carried out in concert. This article also illustrates important restrictions on the resulting dynamics of the state variables if a truncation of dimensionality is carried out and provides a simple and effective means towards achieving the ‘best’ possible simultaneous calibration to volatilities and to the correlation matrix obtainable given the constraints alluded to above. 2 Definitions and results Let σBlack(Ti) be implied Black volatility of forward rate i of maturity Ti, fi(Ti), (often abbreviated in the following as fi), and let si(u,Ti) be the

A displaced-diffusion stochastic volatility LIBOR market model: motivation, definition and implementation

Abstract 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.

A Critical Assessment of Libertarian Paternalism

This paper tries to assess to what extent libertarian paternalism lives up to its libertarian credentials, and whether this “softer” version of paternalism is more or less desirable than the traditional, more coercive (but also more transparent) form. Since much is made in the libertarian paternalistic programme of the ease of reversibility of “nudges,” it is argued that the distinction between effective and nominal ability to reverse a nudge is more important than its theoretical ease of reversibility—the more so, if anchoring, framing and status quo bias are as powerful as the libertarian paternalists maintain. If the libertarian paternalistic nudges are effective, but not always transparent, it is argued that this raises some questions (which do not seem to have been adequately addressed in the current literature) about the legitimacy of the interventions; about how the true preferences of the “consumer” can be guessed by the choice architect (and the role played by rationality in this process) and about the effective respect of her autonomy. Finally, this paper highlights some alternatives to “nudging” which place a greater emphasis on the full process of choice—rather than just on its outcomes—and can therefore better preserve true autonomy of choice.

A JOINT EMPIRICAL AND THEORETICAL INVESTIGATION OF THE MODES OF DEFORMATION OF SWAPTION MATRICES: IMPLICATIONS FOR MODEL CHOICE

We present a joint empirical/theoretical analysis of the changes in the implied volatility swaption matrix for two currencies (USD and DEM/EUR). We recognize the existence of a small number of recognizable shape patterns, and comment about the speed of transition between them. By Principal/Component/Analyzing the associated correlation and covariance matrices we highlight a non/trivial interpretation for the leading eigenvectors. We also compare the empirically obtained eigenvectors and eigenvalues with the corresponding quantities produced by the stochastic/volatility LIBOR market model of Joshi and Rebonato[10]. This allows us to perform a measure-independent comparison that is of intrinsic interest, and that can also provide a general blueprint for analyzing the realism of and choosing among similarly-fitting stochastic models. We find that mean reversion of the instantaneous volatility is a necessary condition in order to obatin the market-observed shape of the first eigenvector associated with the covariance matrix.

A TWO-REGIME, STOCHASTIC-VOLATILITY EXTENSION OF THE LIBOR MARKET MODEL

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 analyzing empirically and theoretically the real and the model-produced change sin swaption implied volatility.

The nature of the dependence of the magnitude of rate moves on the rates levels: a universal relationship

We look at the dependence of the magnitude of rate moves on the level of rates, and we find a universal relationship that holds across currencies and over a very extended period of time (almost 50 years). For the very low level of rates, we find a proportional behaviour; for rates of an intermediate level we find that the magnitude of moves becomes independent of the level. The linear dependence resumes, however, for very high rates. We find the results to be very robust across currencies, tenors and time periods. Even the data we have collected for the UK Consol yields going back to the XIX century conforms closely to the same pattern. We discuss the importance of these findings for several theoretical and practical applications.

Linking caplets and swaptions prices in the LMM-SABR model

We use (and improve upon) a recent time-homogeneous extension of the stochastic alpha beta rho (SABR)-LIBOR market model (LMM) approach described in Rebonato (2007) to develop a quick and accurate analytical approximation to the implied swaption prices given the forward-rate SABR– LMM parameters. This approximation can be used for studies of calibration of a forward-rate-based LMM–SABR model to (portions of) the swaption matrix, to determine the constant maturity swap (CMS) drift corrections (see Hagan (2003)) and to study the congruence between the caplet and swaption markets.

WHICH PROCESS GIVES RISE TO THE OBSERVED DEPENDENCE OF SWAPTION IMPLIED VOLATILITY ON THE UNDERLYING

In this paper we investigate whether a CEV model can account for the observed variation in the at-the-money implied volatility as a function of the level of the at-the-money forward rate. We also determine which exponent β in the CEV process for the swap rate best accounts for the observed behaviour of the implied volatilities.

The nature of the dependence of the magnitude of rate moves on the rates levels

We look at the dependence of the magnitude of rate moves on the level of rates, and we find a universal relationship that holds across currencies and over a very extended period of time (almost 50 years). For the very low level of rates, we find a proportional behaviour; for rates of an intermediate level we find that the magnitude of moves becomes independent of the level. The linear dependence resumes, however, for very high rates. We find the results to be very robust across currencies, tenors and time periods. Even the data we have collected for the UK Consol yields going back to the XIX century conform closely to the same pattern. We discuss the importance of these findings for several theoretical and practical applications.

Forward-Rate Volatilities And The Swaption Matrix: Why Neither Time-Homogeneity Nor Time-Dependence Are Enough

This work presents the first systematic analysis of the whole swaption matrix by fitting a parsimonious, nonlinear, financially-inspired volatility model to market data. The study uses several years of data spanning period of major market volatility. We find that the quality of the fits is good (on average of the same magnitude as the bid-offer spread), and better when a displaced-diffusion approach is chosen, but some systematic shortcomings are observed and discussed. The analysis suggests that a two-regime Markov chain approach may be more successful and better financially motivated.More generally, the present study highlights the shortcomings of purely time-dependent or time-homogenous approaches. These findings should be applicable to other option markets as well.Finally, we find that the present (nonlinear) model vastly outperforms PCA-based approaches when in comes to predicting moves in implied volatilities.