Two-Factor Models

A natural step in the theoretical development of interest rate models was to consider a more realistic approach by increasing the number of sources of randomness. The volatility of the interest rates observed over long periods of times indicated a possible stochastic nature that could be modelled by a process involving a separate Brownian motion. Also, the single-factor short rate models could be generalised to a stochastic mean or/and volatility, evolving this way into two- or three-factor models. These developments have coincided with the important Principal Component Analysis (PCA) of Litterman and Scheinkman (1991) who claimed that over 95% of the variability in the interest rates changes could be explained by three common factors - the level, the slope and the curvature, with 88% attributed solely to the first factor. As a result, many researchers have explored this idea and have considered various candidates for the second factor: Brennan and Schwartz (1979) chose the long-term rate, Schaefer and Schwartz (1984) preferred the spread between the short- and long-term rates; Heston (1986), Pearson and Sun (1994), Sun (1992), Cox et al. (1985) and Pennacchi (1991) considered the inflation, Balduzzi et al. (1997), Naik and Lee (1993) selected the mean level of the short-term interest rate, Schaefer and Schwartz (1987), Fong and Vasicek (1991) and Longstaff and Schwartz (1992) considered the volatility of the interest rate changes.

The Brennan and Schwartz Two-Factor Model (1979, 1982)

Derived in a partial equilibrium framework, the two-factor model proposed by Brennan and Schwartz (1979, 1982) is defined by two sources of uncertainty: the short rate r t( ) and a long-term consol rate L t( )¹⁴. Initially, the logarithms of these variables constitute the two factors that follow an Ito joint diffusion process, with a linear and a quadratic transformation of an OU process, respectively.

1 1

The model can be rewritten in terms of more complex processes for the short rate and the consol rate themselves, providing a useful financial interpretation with the two factors interpreted as the level and the steepness of the yield curve, respectively. Under no-arbitrage conditions Brennan and Shwartz (1979) derived the pricing equation for default-free pure-discount bonds which is also satisfied by any contingent claims that depend on

14 Roughly speaking the consol rate is the return on a claim that pays perpetually a constant dividend, providing a “synthesis of the whole term structure up to infinity” (Brigo and Mercurio, 2001).

32 r and L, such as bond and bond futures options. Thus, in their framework the yield curve is entirely specified by the joint stochastic evolution of its short and long-term extremities. However, this joint specification of the state variables has been questioned by Hogan (1993) and Duffie et al. (1995) who proved that there is no real-valued solution to their diffusion equations.

The Richard Model (1978)

A rather different affine two-factor model of the term structure of interest rates was developed by Richard (1978) who employed two independent stochastic factors: the expected real short-term rate q t( ) and the expected instantaneous inflation rate ( )t ,

The model is additive, in the sense that the short rate is modelled as a linear combination of the two factors, hence allowing for the decomposition of both bond prices and yields, into their real and inflationary components.

The Schaefer and Schwartz (1984)

Motivated by empirical evidence of orthogonality between the long-term rate and the spread, Schaefer and Schwartz (1984) proposed another affine two-factor model where the two uncorrelated state variables are the long-term rate l t( ) and the spread z t( )r t( )l t( ). While the spread follows a standard OU process, the long rate process is more complex with a non-arbitrary drift and CIR type diffusion function:

1 1 1 1

Fong and Vasicek (1991, 1992) considered two sources of uncertainty for explaining the term structure of the interest rate: the short rate and the instantaneous variance of the

33 changes in the short rate. The behaviour of these stochastic variables is described by the following diffusion processes:

Both processes incorporate mean reversion, the instantaneous volatility of the short rate has itself a volatility proportional to the current level of the short rate volatility and the two driving Brownian motions are assumed correlated. Under the condition of no-arbitrage Fong and Vasicek derived the closed formula for computing the price of pure discount bonds that involves complex algebra calculations.

The Longstaff and Schwartz Two-Factor (LS) Model (1992)

From the category of two-factor models, the Longstaff and Schwartz (1992) (LS) model evolves from a general equilibrium model of the economy and leads to a term structure model with a stochastic volatility. The model is both tractable and flexible, with closed formulae for the prices of pure discount bonds. Starting with two underlying state variables x_t and y_t that follow individual CIR standard processes the short rate and the volatility are additive functions of the two underlying economic state variables:

( ) ₂( ) ₂( )

In the LS model the two factors are the short-term interest rate and interest rate volatility.

An alternative interpretation is one in which the two factors are the short-term rate and a long-term rate, which is similar in spirit to the work of Brennan and Schwartz (1979).

The Hull and White Two-Factor Model (HW) (1994)

Following Brennan and Schwartz (1979), Hull and White (1994) propose a two-factor model where the additional state variable is a random long-term equilibrium rate.

The model is made arbitrage-free by including a time variant shift in the drift, allowing therefore for consistency with the currently observed term structure.

34 where the parameters ,a b are real constants and  ₁, ₂  are real constants and the two 0 separated Brownian motions W₁ and W₂ are correlated.

The Andersen and Lund Model (1997)

In line with the Dybvig (1988) and Longstaff and Schwartz (1992) theoretical specifications, Andersen and Lund (1997) developed a two-factor model that incorporates the main behavioural features observed in the evolution of interest rates: mean reversion and volatility heteroscedasticity. Their model can be seen as an extension of the CKLS model with the addition of a stochastic log-volatility factor:

1 1 1 two-factor formulation, with the second two-factor represented by a stochastic variance or standard deviation that is modelled within a diffusion-GARCH framework. In the original paper two alternatives are considered for the discrete-time GARCH effect - a linear symmetric GARCH model (Bollerslev (1986)) and a TS-GARCH model (Taylor (1986) and Schwert (1989)). The continuous-time model implies mean reversion for both the log-interest rate level and the instantaneous standard deviation of the log-interest rate changes:

1 1 1 equivalent model is obtained with the level of the short rate as the first state variable:

35 dynamics of the interest rates. The inclusion of extra factors brings more complexity to the mathematical formulae of reconstruction of bond and derivative prices, with the effect of reducing the tractability of the model. However, some three-factor models such as Fong and Vasicek (1991), Sorensen (1994) and Chen (1996) still possess explicit solutions. The most common choice for the three state variables is a natural one with the short rate, the long-term mean and the volatility of the changes in the interest rates being driven by separate Brownian motions that are assumed to be either independent or tractability, only in specific cases there exist analytical solutions for discount bonds and certain interest rate derivatives (see Chen, 1996).

The Balduzi, Das, Forezi and Sundaram Model (BDFS) (1996)

A popular model in the group of three-factor models was proposed by Balduzzi et al. (1996) (thereafter BDFS). In the BDFS model the mean _t and the volatility _t of