By comparing three different approaches to interest rate modelling, namely spot, forward and market models, Han Lee of Commerzbank Securities finds that each has a different method of constructing the effective volatility function, which determines its use

The modelling of the term structure of interest rates has produced a variety of approaches since the advent of arbitrage-free pricing theory and it continues to occupy the efforts of both academics and practitioners.

Unlike for other asset classes (equities, foreign exchange), where the lognormal Black-Scholes framework is universally accepted, no such agreement exists with regard to interest rate modelling. One reason for this is that the phenomenon we are attempting to model – the random fluctuation of the whole yield curve – is much more complex than the movements of a single stock or index price. One can intuitively relate this to the difference in the dynamics of a scalar variable (in the case of an index) and a vector (representing the yield curve).

A second reason, that is perhaps more fundamental from a market perspective, relates to the nature of the vanilla market in interest rate derivatives. This consists of caps/floors and swaptions, which the market prices using the Black framework where the respective forward Libor and swap rate underlyings are lognormal but the discount factors are non-stochastic. Thus the market standard for the purposes of hedging must regard these vanilla instruments to be independent, where the volatility matrix for swaption prices has for the most part no bearing on the volatility curve associated with the cap/floor market. Moreover, the assumption of simultaneous lognormal behaviour in the Libor and swap rates is not mathematically easy to reconcile. Nevertheless, the goal of interest rate modelling is to provide a framework under which a large class of interest rate sensitive securities can be priced in a consistent manner.

The term structure of interest rates or the yield curve can be described in a variety of different ways, which are equivalent: l Zero Coupon (or discount) bond prices P(t,T): P(t,T) is the value at time t of a discount bond paying one unit at time T

However, beyond these basic identities there remains the freedom to choose from both in terms of ‘microscopic’ detail (eg, specifying dynamics, volatilities and number of factors) and more importantly in the choice of the bond pricing ‘framework’. The latter is determined partly by the actual variable used to describe the model, and can be categorised into three families: spot rate, forward rate and market models. Although all three of these prescriptions are mathematically consistent (by definition of a term structure model), each approach leads to distinct development, implementation and calibration issues. Moreover, the resulting intuition gained from using the model, especially important for relating the pricing and hedging of products based on the model, is different in each case. Before the characteristic of each framework is elaborated and comparisons made between them, it is useful to first discuss what is required in any term structure model.

In this article we will restrict the discussion to default-free term structures, and interest rates driven purely by Brownian motion (ie, not including jump processes). We can insist that the interest rate market (or equivalently the bond market) from an economic point of view must follow the principles below:
•The zero-coupon bond price is strictly positive;
•As no default is possible, P(T,T) =1;
•There is no arbitrage in the market. Added to the above list of principles from a practitioner’s perspective, the following abilities are also desirable in any model:
•To match the initial term structure;
•To capture a wide range of realistic future term structures;
•To have simple inputs (preferably straightforward to associate with market parameters);
•To be numerically efficient and practical to use within risk management systems.

Spot models (pioneered by Vasicek) attempt to describe the bond dynamics through directly modelling the short-term interest rate. Heath Jarrow and Morton (HJM) established the general framework where these principles are satisfied and formulated the interest rate dynamics explicitly in terms of the continuously compounded forward rate. Market models are a class of models within the HJM framework that describe variables directly observed in the market, such as the discretely compounding Libor and swap rates.

Why is it beneficial and relevant to compare the different categories of spot, forward and market modelling? Besides being a useful pedagogical exercise, these approaches have historically developed according to this order. Consequently, not only are the models evolving along this path of increasing sophistication (from spot models initially to market models), it means that many institutions will have a combination of different approaches running within their research, trading and risk management areas. Hence for effective understanding of typical ‘front office’ pricing and hedging tool kits and libraries (with a mixture of models), an appreciation of model error and an overall appreciation of the academic literature requires knowledge of all three approaches. We will concentrate on exploring the similarities and differences in terms of formulation of each model category. The important topic of comparing numerical and implementation methodologies is too large to be covered in this article.

Common themes
In all the model categories (see box), we are describing the yield curve through stochastic differential equations driven by a diffusion term and a drift term. Based on the arbitrage-free principle, the market price of risk is removed by the choice of the drift (this occurs in the portfolio replication argument for stock options in Black-Scholes, where the drift is equal to the risk free rate). This is performed in different ways. Spot rate models have to match the initial yield curve that implicitly holds information on investor choice and hence market price of risk, through the drift function. Models formulated with instantaneous forward rates explicitly relate the choice of volatility function to the form of the drift (imposed through the HJM condition), in order for the no-arbitrage principle to hold. Similarly, for market models the drift is adjusted to ensure that the model remains arbitrage-free.

An important consideration in specifying models within each category is based on finding the ‘natural’ choice of asset on which to base the dynamics and the pricing of contingent claims. This idea exists in all areas of derivatives pricing, but within term structure modelling the yield curve, as well as evolving through time, has a maturity index (along the vector), and hence this ‘relative asset’ is time-dependent. Technically, this is the ‘choice of numeraire’, the set of which are related through ‘changes in measure’. Intuitively, the idea is to base pricing of contingent claims relative to some asset, and to change this ‘relative measure’ when it is convenient. This choice of numeraire plays an important role in the intuitive understanding of a model. In particular, the actual formulation of models within each category is determined by this choice. For example, spot models are in the standard formulation based on the ‘money market measure’, in which the numeraire is simply the bank account accruing at the risk-free rate.

However, in the implementation of the model there is no restriction on which numeraire to use. They are usually selected to ease the numerical implementation or in some cases may even simplify the pricing problem to allow analytic tractability. For example, the Gaussian spot and HJM models allow for simplification in pricing European contingent claims when written down in the ‘T-forward’ measure, where the numeraire is the zero-coupon bond with maturity at time T. In the Libor market model, the formulation is explicitly measure-dependent; each Libor process is driftless in its own ‘terminal’ measure. The processes can be bought to any particular measure through changing the drift. It is in this way that the mathematical consistency of the term structure model is maintained.

To use any model for pricing of contingent claims, it must be calibrated to the market. Besides matching the initial yield curve, the prices of caps/floors and swaptions are required. Ideally the model is capable of providing an analytical formula for these vanilla instruments, but otherwise a very efficient numerical algorithm is necessary. Spot and forward models must derive the appropriate quantities from the underlying state variables to construct the equivalent of the option pricing formulae. By construction, market models are based on observable rates in the market and hence (in some measure) readily price-standard instruments. The process of calibrating any model must start with making the choice of distribution or volatility function.

Spot rate models require a specification of the dynamics, examples of which include a normal or Gaussian distribution (Hull-White), lognormal (Black-Karasinski) or something in between (eg, the ‘square root’ type model equivalent of the Cox-Ingersoll-Ross model). Variables derived from the spot rate, such as the zero-coupon and Libor or swap rates, will have a distribution dependent on that of the short rate; for example the discount bond is lognormal for Gaussian spot rate models such as Hull-White. For forward rate models, the critical factor in determining the behaviour of a model is the form of the (HJM) volatility function.

For reasons of analytic tractability, the most common models in this category are the Gaussian forward rate models, so called when the volatility function is independent of the forward rate itself. In market models there is a choice in both the distribution of the underlying market variable, or perhaps a function of that variable, and in the functional form of the volatility.

For example, a Libor market model may have a lognormal volatility for the forward Libor rate (the original BGM), or we may specify instead the lognormal distribution of one plus the Libor rate times the period of accrual. Market models have the advantage when calibrating to their associated vanilla product (ie, a Libor model for cap products) in allowing a separate fitting to volatility and correlation, since the formulation of this category of model allows a decoupling between the two. More effort is required when calibrating to products that are not based on the associated rate or have combinations – eg, callable reverse floaters, which have swap and cap components within the Libor model. In every case, the volatility specification for a model and the covariance property is measure-independent; only the drift changes.

For use in exotic derivatives, models are required that price and hedge tradeable products. These should capture all risks associated with the product. Usually no single model can satisfactorily price and risk manage all exotic trades, hence traders like to keep a selection of different models available. Risk managers also benefit by having a spread of model evaluations to keep a check on model error.

The choice of distribution possible in all models is not crucial in distinguishing the pricing or hedging properties for exotic products. Although the standard market variables are assumed to be lognormal in the pricing of vanilla instruments, they clearly do not, in reality, fit this distribution. One obvious manifestation of this is the cap and swaption volatility smile. The most important constituent in determining what impact a particular model has on the pricing and risk sensitivity for an exotic option is the form of the model’s effective volatility, or more generally the form of the covariance structure. This has different effects dependent on the exotic product.

A simple product that requires a model to properly handle the risk is a swap where the coupon payment is dependent on the constant maturity swap (or CMS) index. To value the CMS accurately requires a ‘convexity adjustment’ that is volatility dependent. Assuming liquid market data, all models should price this to within the bid/offer spread of each other. The reason is the required calibration to the market, which in this case means the relevant swaption volatility. Although the ends (the valuation of the product) are the same, the means (model usage) differ. The ease of the calculation varies considerably depending on the model, or more specifically on the numeraire used. Here, the market swap model is the most natural choice.

A more complex example is the ‘Libor ratchet’ – a path-dependent product sensitive to the serial or auto-correlation of successive Libors. If we price this with a typical spot model, say BDT, the effective volatility for the lognormal distribution is a function of the mean reversion of the spot rate, which in turn is determined by the time derivative of the logarithm of the short rate volatility. The larger the mean reversion, the more decorrelated the Libors. This means that even after the appropriate caps are matched, there is still freedom in size of mean reversion to alter the price of the ratchet. Similarly, a Gaussian forward rate model that has an exponentially decaying volatility function with a decay constant has the same freedom; in fact this decay constant plays the role of a mean reversion in a Hull-White spot rate model. Using a BGM Libor model, there is even more freedom to adjust the auto-correlation structure between Libor rates and for the lognormal version, calibration to the total (cap) volatility is trivial as this model reproduces the Black formula for caps by construction.

Another product that illustrates this theme is the Bermudan or multi-callable swaption. In this product, the holder of the option may call the swap on any of the multiple exercise dates which usually coincide with the periods of the swap. All models that price Bermudans must calibrate to the sequence of underlying swaption market volatilities. Where the models will differ lies primarily in the ‘relative’ specification of the successive volatilities, sometimes referred to as forward (forward) volatility structure (an analogy with the usage in compound options, of which the Bermudan swaption may be viewed as a more involved form). This may be equivalently viewed as dependent on, respectively, the mean reversion of a spot rate model, the time dependence of the volatility function in the HJM model, or the serial-correlation in a Libor market model. Unlike the CMS swap, where calibration to the appropriate swaption uniquely determines the volatility and hence the value of the product, there is no obvious constraint from the vanilla market which specifies the forward volatilities. Thus the effective volatility of different models, and therefore the option evaluation arising from them, will differ.

The same arguments apply to multi-factor versions of all these models. Although instantaneous correlation is not the critical feature determining the risk of any Libor ratchet or Bermudan swaption products, a model driven by multifactors provides a richer covariance structure or effective volatility. In this context, the market models offer a more flexible approach to investigate the separation of risk between volatility and correlation.

Whichever model is used, there will be freedom to adjust the parameters governing the ‘indeterminate’ parts of the effective volatility for the pricing of particular products, and in practice this adjustment is made depending on the needs of the user (eg, whether buying or selling the option and how aggressive or conservative the trader is), and by indirect means such as by calibrating to other quoted prices in the market when possible.

Conclusion
We have described three categories of interest rate term structure modelling approaches, namely spot, forward and market models, and compared the respective ingredients that make up their construction and formulation. Although all these approaches, by virtue of being arbitrage-free term structure models, are equivalent mathematically and are all within the general HJM framework, each is distinguished by different methods of constructing the effective volatility function which determines its use in practice. The implementation as well as the formulation of the models allows a freedom of pricing measure, and this demonstrates to a great extent the different intuition that accompanies each model. It is clear that the wide scope of interest rate modelling will undoubtedly spur further developments from both the theoretical and practical sides for many years to come.

Han H Lee is head of fixed-income derivatives research at Commerzbank Securities in London