Forward rate dependent Markovian transformations of the Heath-Jarrow-Morton term structure model

In this paper, a class of forward rate dependent Markovian transformations of the Heath-Jarrow-Morton [HJM92] term structure model are obtained by considering volatility processes that are solutions of linear ordinary differential equations. These transformations generalise the Markovian systems obtained by Carverhill [Car94], Ritchken and Sankarasubramanian [RS95], Bhar and Chiarella [BC97], and Inui and Kijima [IK98], and also generalise the bond price formulae obtained therein. Introduction In the risk neutral Heath-Jarrow-Morton [HJM92] term structure model, evolution of the forward rate process is completely determined by the forward rate volatilities. The HJM framework is very general and contains many of the earlier interest rate models as special cases, including [Vas77], [CIR85], [HW90], and [BK91], among others. One drawback of the generality, from a practical perspective, is that the model is non-Markovian in general and consequently does not readily lend itself to efficient solution techniques. Suitable restrictions on volatility processes led Carverhill [Car94], Ritchken and Sankarasubramanian [RS95], Bhar and Chiarella [BC97], and Inui and Kijima [IK98] to transform the HJM model to finite dimensional Markovian systems. In [RS95] and [BC97] only the one-factor HJM models are considered, while, under a more transparent framework, [IK98] generalise the [RS95] models to the multifactor case. In [BG99] and [BS99], a theoretical framework is introduced for obtaining necessary and sufficient conditions under which HJM models are Markovian, and for constructing minimal realisation in such cases. The [BC97] model has the feature that spot rate volatility may be an arbitrary function of the spot rate, and although the Markovian systems of [RS95] and [IK98] have the same feature, the bond price formulae obtained therein applies to a more general class of volatility processes, such as those which depend on a finite number of fixed tenor forward rates. In each case, the forward rate volatility processes are expressible as a product of the spot rate volatility and a path-independent function. It should be noted that although [RS95] and [BC97] both consider the one-factor HJM model, they overlap only for a small set of volatility processes. Date: First version January 6, 1998. Current revision March 30, 1999. Printed April 23, 1999. 1This fact does not appear to have been noted by the authors however. 1 2 CARL CHIARELLA AND OH KANG KWON In this paper, a common generalisation of the above models is obtained, in which the multifactor HJM model is transformed to a finite dimensional Markovian system, and in particular, a multifactor generalisation of the [BC97] model is obtained. Further, the volatility processes in the generalised models are allowed to be arbitrary functions of a finite number of fixed tenor forward rates. Consequently, they include finite dimensional forward rate dependent Markovian transformations of the multifactor HJM model. The key observation in [IK98] was that when volatility processes σi(t, T, ω), 1 ≤ i ≤ n, satisfy the condition ∂Tσi(t, T, ω) ∂T = κi(T )σi(t, T, ω), (1) and κi(T ) are path independent functions of T , then the n-factor HJM model can be transformed to a 2n-dimensional Markovian system, and the bond price can be obtained in terms of the 2n state variables. The condition (1) arises naturally from the desire to replace path dependent terms in the differential of the spot rate by an expression involving the spot rate itself, and is in fact sufficient to reduce the HJM model to a finite dimensional Markovian system, as described in [RS95] and [IK98]. Let Li = ∂/∂T − κi(T ). Then (1) can be rewritten Liσi(t, T, ω) = 0. That is, the [IK98] condition requires that, for each t, the volatility processes σi(t, T, ω) satisfy a first order, linear, homogeneous, ordinary differential equation in T . The essentially arbitrary initial condition for the differential equation then allows the spot volatility σi(t, t, ω) to be unrestricted. However, in order for the corresponding model to transform to a Markovian system with respect to state variables introduced in [RS95] and [IK98], the initial condition must be of the form σi(t, t, ω) = σi(t, t, r(t, ω)). (2) That is, the spot volatility must be a function of the time variable t, and the spot rate r(t, ω). As mentioned earlier, although (2) is required to transform to a Markovian system, the bond price formula obtained in [IK98] applies to a larger class of volatility processes. In this paper, the approach of [IK98] is generalised by requiring that each σi(t, T, ω) is a function of t, T , and m forward rates f(t, t+ς1, ω), . . . , f(t, t+ςm, ω), so that σi(t, T, ω) = σi(t, T, f(t, t+ ς1, ω), . . . , f(t, t+ ςm, ω)), (3) and satisfies a differential equation of the form Liσi(t, T, ω) = 0, where Li = ∂i ∂Tmi − mi−1 ∑ j=0 κi,j(T ) ∂ ∂T j (4) is an mi-th order linear differential operator and the coefficients κi,j(T ) are path independent functions of T . The corresponding n-factor HJM model can then be transformed to a finite dimensional Markovian system of dimension at most m ∑n i=1m 2 i (mi + 3)/2. Further, for each i, the mi arbitrary boundary conditions for the differential equation, Liσi(t, T, ω) = 0, allow σi(t, t + T, ω) to be arbitrary functions of the forward rates f(t, t+ ς1, ω), . . . , f(t, t+ ςm, ω). As in [RS95] and [IK98], although the bond price formulae given in §4 remains valid for a more general class of volatility processes, the restriction (3) is required to obtain a Markovian system. 2The ω in σi(t, T, ω) represents the dependence of the volatility process on the path followed by the underlying Wiener process. See §1. 3With respect to the state variables introduced in [IK98]. FORWARD RATE DEPENDENT MARKOVIAN HJM MODELS 3 The outline of the paper is as follows. It begins with a brief review of the HJM term structure model and the parametrisation, T = t+ ς, due to Brace and Musiela [BM94] in §1. The main results of this paper are then presented in §2, in which the transformation of the multifactor HJM model to finite dimensional Markovian systems is outlined. Natural extensions of the [IK98] model are considered in §4, and explicit expressions for the bond price as a function of the state variables are obtained for these models. Finally, the paper concludes in §5. 1. Risk Neutral Heath-Jarrow-Morton Model and the Brace-Musiela Parametrisation This section reviews in brief the HJM model and the parametrisation, T = t+ ς, introduced by Brace and Musiela [BM94]. For details, refer to [HJM92], [BM94], [MR97], or [Bjö96]. 1.1. Risk Neutral Heath-Jarrow-Morton Model. Fix a trading interval [0, τ ], τ > 0, and let (Ω,F ,P) be a probability space, where Ω is the set of states of the economy, F is the σ-algebra of measurable events, and P is a probability measure on (Ω,F). It is assumed that there are n independent standard P-Brownian motions W i t , 1 ≤ i ≤ n, that generate a complete right continuous filtration {Ft}0≤t≤τ on (Ω,F). For each maturity T ∈ [0, τ ], the time t instantaneous forward rate f(t, T, ω) in the risk-neutral n-factor HJM model is a stochastic process determined by suitably well-behaved volatility processes σi(t, T, ω), 1 ≤ i ≤ n, and evolves according to the stochastic integral equation f(t, T, ω) = f(0, T, ω) + n ∑