The First Yield Curve Models - the second period sees the introduction of yield curve models to the pricing methodologies, such as Vasicek and Cox, Ingersoll and Ross frameworks. These are equilibrium models (endogenous term structure models) because the term structure of interest rates is an output of, rather than an input to, these models.
The Second-Generation Yield Curve Models - the first yield curve models lay the groundwork for the next wave of yield curve models that introduced a deterministic component to the drift of the short rates, such as Hull and White (1990). These are the no-arbitrage models, designed to replicate the current term structure of interest rates. However, it is not possible to arbitrage using simple interest rate instruments in this type of no-arbitrage models.
The Modern Pricing Approach - more recently, as mathematics advanced, the short rate models evolved into models calibrated with traded market instruments. The Heath-Jarrow-Morton (HJM) heralded this most recent period which led to the development of market models such as the LIBOR Market Model and the Swap Market Model.
Fisher Black, Myron Scholes and Robert Merton pioneered the way into financial engineering by coming up with a closed form analytical solution to the pricing of stock options. Their work showed that options could characterise the capital structure of a company. This changed the whole aspect of trading, which at that time was predominantly a speculative game. The Black-Scholes (1973), Black (1976) and Merton (1973) models assume that the underlying variable has a lognormal distribution. However, these models oversimplify many of the market behaviours and because of their pull-to-par behaviour, (they assume that the underlying derivative's volatility is constant - this is not true for coupon or discount bonds, since the price converges to par at their maturity). This remained a trivial matter as long as the expiry of the bond option was much shorter than the maturity of the underlying bond; but it posed problems when the expiry and maturity times of the option and the underlying bond, respectively, were comparable. The immediate solution considered a non-traded quantity (the bond yield) as the underlying lognormal variable. This offered an advantage as the yield has no deterministic pull-to-par characteristics. However, this approach suffered as the yield is not a traded instrument and therefore, the Black and Scholes concepts regarding a self-financing dynamic trading strategy to reproduce the final option payoff are difficult to implement. This lognormal yield model did not gain widespread use among the trading community who stuck to the Black formula for caplets and European swaptions despite the lack of their theoretical justification at that time. Nevertheless, the blind trust in these formulae paved the way for the development of the later general approaches (such as the LIBOR market model) to price complex interest-rate derivatives consistent with the (Black) caplet market.
The solution to the pull-to-par problem was the Black, rather than the Black and Scholes, formula that uses the forward (as opposed to the spot) price and the volatility of the forward as inputs. This is expedient as no pull-to-par effect is associated to a forward bond price, as the forward bond price remains the same throughout the time to maturity of the bond. However, the drawback is that although different bond prices correspond to different assets, they correlate significantly, but the Black formula does not incorporate this joint dynamics. This approach enjoyed widespread approval among the trading populace despite its disadvantages as it offered the convenience of avoiding the pull-to-par phenomenon.
SHORT RATE MODELS
Vasicek and Cox, Ingersoll and Ross (CIR) (Cox, et al. 1985) first developed the concept of the instantaneous short rate, which is the shortest tenured yield of the term structure. They proposed that the instantaneous short rate could drive the dynamics of the whole yield curve. The evolution of the model depends on a stochastic differential equation made up of a deterministic component and a stochastic part. The model is overly simple in that it assumes that a single source of uncertainty - the short rate - models the evolution of the market behaviour. Nevertheless, this simple structure helps develop an intuitive understanding of the pricing issues.
Black, Derman and Toy (1990), and Hull and White (1990) extended the Vasicek and CIR models by adding a purely deterministic (time-dependent) term to the mean-reverting component in the drift of the short rate. This meant that whatever shape the market yield curve has, these models could always reproduce the market prices by adding a deterministic 'correction' term to the mean-reverting drift. Although these models are able to reproduce any yield curve, they do not automatically account for the prices of all plain vanilla options (caplets and European swaptions). Due to their failure to account exactly for option prices if implemented without constant volatility parameters, the explanatory mandate of these models is assessing the validity of the market-volatility term structures, rather than accounting for the yield curve shape. Therefore, these serve as a complement and sanity check for the more sophisticated models developed later. These models failed to satisfy exotic-options traders who demanded a model to price at least the required option hedges for each trade consistently with the plain-vanilla market.
Short rate models start by defining a stochastic differential equation (SDE) for the short rate. Assume that the short rate follows the SDE:
where is a Wiener process in a risk neutral measure. The discounting process is:
In a short rate model, the zero-coupon bond, which pays 1 unit at maturity T, is the numeraire. From the risk neutral pricing formula:
The yield is:
is a Markov process (a memory-less process in which the present state is independent of the past and future states) as it is the solution to an SDE.
EQUILIBRIUM MODELS
As stated above, the evolution of interest rate modelling during the second and third periods is classified into equilibrium modelling and no-arbitrage modelling streams. The equilibrium models take off by specifying conditions on macroeconomic variables, which then define the SDE. The Vasicek, Rendleman and Bartter and the Cox-Ingersoll-Ross models are popular equilibrium models. The Rendleman and Bartter model is similar to the Black-Scholes model except that the interest rates are instantaneous in the former. The model assumes lognormal short rates, which might be good as a starting point, but does not accommodate for the mean reverting behaviour of the rates to a long-term average. Vasicek accounts for the mean reversion through the following short rate SDE:
This is an Ornstein-Uhlenbeck process. The SDE is solved by taking the differential of, using Ito's lemma and Ito's isometry and integrating the SDE:
Knowing that the Ito integral of a deterministic function, is normally distribute with zero mean and variance, the bond or a derivative security can be priced. The Vasicek is the first model to incorporate mean reversion. The mean reversion property is seen by examining the behaviour of the drift term:
If, the drift is negative and reverts towards.
If, the drift is positive and damps out to.
If, then.
If, then.
Therefore, the process reverts to the mean for all possible scenarios. The main advantage of the Vasicek model, however, is that the short rate can become negative with positive probability. Nevertheless, the analytical tractability of the model from its Gaussian distribution is the main advantage.
The Cox-Ingersoll-Ross (CIR) model provides the first correction. The SDE is:
The bond price is found by solving for an affine form:
Substituting this into the partial differential equation (PDE) and establishing the boundary conditions, bond prices are calculated. Options are priced similarly by using numerical methods. The key point is the presence of the radical term, which signifies that the short term will never be negative.
The CIR model is an epitome of equilibrium models; it captures mean reversion, the rates are positive and pricing is relatively easy. However, the equilibrium models fail to capture the current behaviour of the yield curve and calibration of these models never emulates the real world data entirely. Some authors argue that sometimes even reasonable approximations are unavailable. Hull, for example writes, "By choosing parameters judiciously, they [equilibrium models] can be made to provide an approximate fit to many of the term structures that are encountered in practice. But the fit is not usually an exact one and, in some cases, no reasonable fit can be found."
NO-ARBITRAGE MODELS
The insensitivity of the equilibrium models led to the development of the no-arbitrage models. Contrary to the equilibrium models, the SDE parameters defining the interest rate dynamics vary with time for no-arbitrage models. The starting point is the current term structure rather than the macroeconomic parameters. The input term structure fits the time dependant SDE parameters and hence, captures the market dynamics exactly. Ho and Lee proposed the fist no-arbitrage method by modelling a binomial tree with two parameters: the short rate standard deviation and the market price of risk of the short rate. The continuous-time limit of the model is:
Other no-arbitrage models are Hull-White (both one- and two-factor models), Black-Karasinski and the Black-Derman-Toy models. The Hull-White model is also referred to as the extended-Vasicek model. The Hull-White one-factor SDE is:
Thus, the Hull-White model is a Vasicek model with time-dependant reversion level, which justifies why it is characterised as extended-Vasicek model. It is also the Ho-Lee model with the mean reversion at rate a.
The short-term interest rates in the Hull-White and Ho-Lee models can become negative. The Black-Karasinski model captures this phenomenon, at the expense of analytical tractability, by the following SDE:
The short-rate is normal in the Ho-Lee and Hull-White models, but it is lognormal in the Black-Karasinski model. The variable follows the same path as in the Hull-White model. Table shows some short rate models and their dynamics:
Even with the advantages that the no-arbitrage models provide, namely their analytical tractability, replicating the term structure dynamics and numerical efficiency, they are not the panacea for interest rate modelling. Their drawbacks include clumsy calibration techniques, because the model parameters need to be implied from the available market data for option prices. Numerical techniques are used to extract the parameters from the market data. This is computationally inefficient at the best and unstable. One approach of calibration is the Jamshidian (1989) approach in which a financial instrument like a swaption is treated analogous to an option on a coupon-bearing bond. The option is decomposed into several options on discounted zero-coupon bonds and a closed form solution obtained.
As seen from Table, short rate models predominately have a single stochastic driver, although two- and three-factor models sprung up as well. These include Longstaff-Schwartz (1992), Hull-White (1994), and Peter-Stapleton-Subrahmanyan (2003). The single factor models offer the advantage that recombining tree can efficiently price path-dependant options. The two- and three- factor models provide a richer term structure and richer volatility patterns of the short rate.
HEATH-JARROW-MORTON (HJM) FRAMEWORK
Markets are characterised by bonds and various interbank rates. Therefore, short rate models, even though they are simple to price pose a problem - they describe only one yield. Therefore, they oversimplify the market behaviour by describing only one macro-economic short rate. Thus, it is difficult for the short rate models to analytically price caplets and European swaptions. The calibration to caplets or swaptions to find the optimal volatility parameter σ is clumsy and computationally inefficient as the whole process reproduces itself every time for an iterated value of the volatility to calculate the option prices.
The HJM model is similar to the Ho-Lee short rate model, but it describes the instantaneous forward rate rather than the forward price. Thus, it is characterised by the following SDE:
The drift term and diffusion term are adapted processes. The corresponding stochastic integral for the forward rate is:
The term is the instantaneous forward rate at time t for a time T, where. The drift and diffusion terms corresponding to the SDE above at the functions and, respectively. The argument (.) suggests any dependence on other state variables. Typically, this is either on itself or on the instantaneous spot interest rate at time t. The absence of riskless arbitrage opportunities means that the drift term is a function of the volatility function and the market price of risk. The relationship is:
Applying the Girsanov's theorem, the stochastic integral reduces to the following:
Hence, the market price of risk is absorbed into an equivalent probability measure.
The equivalent probability measure generates the Wiener process. The spot interest rate is extracted from the forward rate as. Under the equivalent probability measure, this is:
and hence, the PDE is:
Here is the partial derivative of with respect to the second argument. While the HJM permits an arbitrary selection for the volatility term, this is cumbersome as there are no efficient numerical techniques to solve the model. Another problem associated with HJM model is the explosion of the instantaneous forward rates under lognormal assumptions. To go around this issue, the process is adjusted with discrete forward rates of finite-tenor. This leads to the development of the Brace-Gatarek-Musiela (BGM) model or the LIBOR market model, which uses forward rates.
MARKET MODELS
Difficulty in calibrating the models to market quantities has been characteristic of interest rate models. Even the HJM, in its general form is difficult to calibrate to market data. This is because the stochastic drivers for all of these models, explicitly or implicitly, depend upon unobservable market phenomena, namely the instantaneous short or instantaneous forward rates. The need then is to transform the dynamics of these abstract market quantities into dynamics of market observable quantities via a framework within the model itself. Brace, Gatarek and Musiela (1997), Miltersen, Sandmann and Sondermann (1997) and Jamshidian (1997) introduced a successful class of models that captured observable market quantities. These are the LIBOR Market Model (LMM) and Swap Market Model (SMM). Jamshidian (1997) introduced the SMM.
These models capture observable market phenomena and allow straightforward calibration to option prices. Another characterising attribute is that these models are equivalent to Black and Scholes (1973) with the underlying as interest rates. The reason for this is the flat implied volatilities for caps and swaptions produced by these models. The forward LIBOR rates are lognormal martingales, therefore, the interest rates cannot become negative for a deterministic volatility LMM. However, the swap rates in LMM are not lognormally distributed. Although this may imply that the LMM is not feasible in for swaption pricing, it is not the case. Jackel and Rebonato (2003b) and Hull and White (2000) provide accurate approximation formulae for swaption implied volatilities. As a result, LMM can be calibrated to swaption volatilities. This is the main contributor of the success of LMM among interest rate derivative traders.
LIBOR MARKET MODEL
LMM is implemented in three steps, namely, the calibration, pricer, and derivative steps. The first step adjusts the LMM parameters to the market data. This is a user-specific step - the practitioner specifies the market data for calibration. The pricer stage calculates the interest rate derivative prices. The time zero forward LIBOR rates and output from the first stage, are inputs to the pricer engine, which then uses either Monte-Carlo simulations or analytical formulae to calculate the derivative prices. The final stage is to return the derivative payoff given the specified market scenario.
The LMM implementation over time has primarily been done using Monte-Carlo simulations. Because of its complex dynamics and a state-dependant drift of the forward LIBOR rates, it was argued that recombining tree methodologies are incapable of capturing LMM dynamics completely. Other problems associated with the LMM were how to incorporate early-exercise features and how to discretize the LIBOR dynamics. Glasserman and Zhao (1999) provide simulation schemes free of arbitrage opportunities. Monte-Carlo simulation approaches are versatile because they can be applied to complex payoffs. This, however, is at the expense of computational efficiency. Hunter, Jackel and Joshi (2001) and Jackel (2002) introduced the predictor-corrector drift approximation to reduce the simulation to a single time-step version. Further, Pietersz, Pelsser and Van Regenmorte (2005) introduced the Brownian bridge method as a fast drift approximation technique.
London (2005) argues:
"However, due to the complex dynamics of LMMs and the fact that the forward rates have a fully state-dependent drift, recombining lattices/trees cannot be used to evolve interest rate dynamics and thus price interest rate derivatives, as they can for spot and forward rate models. Instead, Monte Carlo and other numerical techniques must be used to price caps, swaptions, and other (exotic) interest rate derivatives." However, a seminal paper by Ho, Stapleton and Subrahmanyam (1995) laid the groundwork into efficient pricing of a general multivariate lognormal distribution through a multivariate binomial process. The work of Ho, et al. (1995) extended models of Nelson and Ramaswamy (1990) and Amin (1991). They allowed the number of binomial stages to be greater than unity and vary between specified intervals. The model allows for general multivariate processes with the individual assets having individual variance and mean reversion. In contrast to the bushy tree approaches, this model ensured that the nodes recombined. Hence, computational times decreased exponentially as the computational time for bushy (non-recombining) trees followed an order, while the computational time for recombining tree follows an order. Based within this framework, Derrick Stapleton and Stapleton (2005) build a recombining tree method. The paper demonstrates how to build the recombining tree for LMM and use it to price interest rate derivatives.
