Finance and Economics Discussion Series: 2012-12 Screen Reader version ^{♣}

Keywords: Square-root diffusion, CIR process, multivariate gamma distribution, difference of gamma variates, Krishnamoorthy-Parthasarathy distribution, Kibble-Moran distribution, Bell polynomials

Abstract:

We derive the multivariate moment generating function for the stationary distribution of a discrete sample path of observations of a square-root diffusion (CIR) process, . The form of the mgf establishes that the stationary joint distribution of
for any fixed vector of observation times
is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. As a corollary, we obtain the mgf for the increment
, and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution. Simple closed-form solutions for the
moments of the increments are given.

*JEL Classification:* C46, C58

Let follow the Feller (1951) square-root diffusion process with stochastic differential equation

where is a Brownian motion. We assume that , and . This process is widely used in economics and finance, especially in modeling interest rates and corporate credit risk, where it is usually known as the CIR process after Cox, Ingersoll and Ross (1985). In this paper, we derive the moment generating function for the stationary multivariate distribution of a discrete sample path of this process.

Let be a discrete sample path for a given vector of ordered observation times . Let denote the vector of auxilliary variables and let be the diagonal matrix with diagonal entries . Let be the symmetric matrix with elements . is the identity matrix. Define the scale parameter . The central result of this paper is

The proof is set out in Section 1.The distribution of is a special case of the broader class of Krishnamoorthy and Parthasarathy (1951) multivariate gamma distributions with mgf of the form for nonsingular and (see also Kotz et al., 2000, §48.3.3). Series solutions for the density and cumulative distribution functions are given by Royen (1994) for the case in which the inverse of is tridiagonal (see also Kotz et al., 2000, §48.3.6), which applies for our matrix . These series solutions are computationally practical only for low dimension .

The stationary square-root process has exponential decay in the autocorrelation function (Cont and Tankov, 2004, §15.1.2), so for pairs in , the correlation is given by

From this relationship, the matrix is known as the accompanying correlation matrix.In the bivariate case, the mgf has a simple form

This is the Kibble-Moran bivariate gamma distribution (see Kotz et al., 2000, §48.2.3). In Section 2, we use this corollary to study the stationary distribution of the increment for fixed time-step . We show that this increment is equivalent in distribution to a scaled difference between two independent gamma variates, and provide a simple closed-form solution for the moments of this distribution. Applications are discussed in the concluding section.1 Moment generating function

It is well known that the transition distribution for
given is noncentral chi-squared.^{1} Letting denote the conditional mgf for
given , we have

(2) | ||

As the square-root diffusion is a Markov process, we have

To exploit the conditional mgf, we write in nested form:

(3) |

where Repeating this process times in total, we get where the modified auxilliary variables have the forward recursive relationship

for and where we fix (so ). The stationary distribution of is gamma with shape parameter and scale parameter , which has mgf

(5) |

so we arrive at

Equation (6) is computationally convenient but analytically cumbersome. Let be the expression inside the brackets, so that . We now simplify by writing it as a finite series in powers of .

Let be the set of subsequences of length from the sequence , so that if , then

For , define the functions(7) |

In Appendix A, we prove

To prove Theorem 1, we need to prove that has the same expansion as in Proposition 1. Recall that the characteristic polynomial of a square matrix is defined as . For a subsequence , let denote the order diagonal minor of with elements and let for be defined as

For notional convenience, we define . Then the characteristic polynomial of has the expansion (Gantmacher, 1959, §III.7)Substituting and in (8), we have

Since is a diagonal matrix, the diagonal minor is equal to the product . Thus, we have

For the case of , for all , so

For the case of , we make use of this lemma:The diagonal minor takes on the same form as , i.e., there is a vector such that has elements . Applying Lemma 1, for any and we have

Since , we haveWe substitute equation (11) into equation (9) and arrive at the same expansion as in Proposition 1. This completes the proof of Theorem 1.

2 Moments of the increments

Under stationarity, for all , so without loss of generality we examine the stationary distribution of . From Corollary 1,

where and is the univariate mgf for . An immediate implication of (12) is that is equivalent in distribution to times . Furthermore, is equivalent in distribution to the difference between two

Consider the general problem of the moments of the difference between two independent and identically distributed (iid) gamma variates. Let for shape parameter and scale parameter , and define . The cumulant of is

Central moments are obtained from the cumulants via the complete Bell polynomials, i.e., For any sequence , the Bell polynomials satisfy so(13) |

Furthermore, since the distribution is symmetric around zero, we know that the odd moments are zero.

In Appendix B, we prove a general identity on the complete Bell polynomials:

(14) |

and the odd central moments are zero. As kurtosis is often of particular interest, we note

Application to the moments of is direct. We substitute and get even moments

(15) |

The kurtosis of is , which is invariant with respect to the time increment .

3 Conclusion

Our main result is a simple closed-form expression for the moment generating function of the stationary multivariate distribution of a discrete sample path of a square-root diffusion process. We establish that the distribution is within the Krishnamoorthy-Parthasarathy class, and thereby draw a connection between a stochastic process and a multivariate distribution that each first appeared in the literature in 1951.

Our result has application to estimation of parameters of the continuous-time square-root process from a discrete sample. It gives a simple and computationally efficient way to generate moment conditions for the generalized method of moments estimator of Chan et al. (1992). The empirical characteristic function approach of Jiang and Knight (2002) can also be easily implemented. Indeed, Jiang and Knight consider the example of a square-root diffusion, but their solution to the characteristic function corresponds roughly to our intermediate equation (6), rather than to the simple form in our Theorem 1.

Three of our auxilliary results may have application elsewhere. First, Lemma 1 provides a simple solution to the determinant of the autocorrelation matrix for a discrete sample of any process with exponential decay in autocorrelation. This decay rate holds in a large class of stationary Markov processes, including Gaussian and non-Gaussian Ornstein-Uhlenbeck processes as well as the square-root process (Cont and Tankov, 2004, §15.1.2, §15.3.1). Second, our Bell polynomial identity in Lemma 2 generalizes a known relationship between Bell polynomials and the Gamma function (i.e., for the case of in our lemma). Finally, we provide a simple formula for the moments of the difference of two iid gamma variates. It complements existing results that allow the variates to differ in scale parameter (Johnson et al., 1994, §12.4.4), but which lead to more complicated expressions for the moments.

A. Proof of Proposition 1

Let us define and, for , recursively define

Since , we have . We similarly generalize the functions as(16) |

We now demonstrate

by induction. For the case , which satisfies equation (17). For , let us assume that (17) is satisfied for . Then

Since is linear in each argument,

Substituting into (18), we get

Collecting terms on , we can write where and and, for ,

The last term is

so

By equation (A),

so

where the last equality is by equation (A). Thus, for , so equation 17 is proved. Substituting and , we arrive at Proposition 1.

B. Proof of the Bell polynomial identity

For any sequence of scalars , the generating function of the complete Bell polynomials is

where we fix . When , we have where we introduce the change of variable .

Using identities from Comtet (1974, pp. 135, 136) and DLMF (2010, §26.8.7), we have

where denotes the Stirling number of the first kind. Restoring the original variable , we haveMatching terms to the right hand side of (21) with the same power of , we have Whenever is not a multiple of , the coefficient on in the right hand side of (22) is zero, so

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* I thank Yacine Aït-Sahalia, Luca Benzoni, Jens Christensen, Yang-Ho Park, Steven Shreve, Richard Sowers and David Zelinsky for helpful discussion. Bobak Moallemi provided excellent research assistance. The opinions expressed here are my own, and do not
reflect the views of the Board of Governors or its staff. Email: michael.gordy@frb.gov. Return to Text

1. See Alfonsi (2010) for a summary of basic properties of the square-root diffusion. Return to Text

2. I thank David Zelinsky for suggesting the proof of this lemma. Return to Text