Laws and Likelihoods for Ornstein Uhlenbeck-Gamma and other BNS OU Stochastic Volatilty models with extensions

Lancelot F. James

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Laws and Likelihoods for Ornstein Uhlenbeck-Gamma and other BNS OU Stochastic Volatilty models with extensions

Lancelot F. James

Statistics Theory · Submitted 5 Apr 2006 · 2 references

Pith's verdict: The Gamma/Dirichlet factorization is a real contribution, but the paper's 'perfect sampling' claim is approximate CFTP, not exact sampling.

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arXiv IDmath/0604086 (version 2)

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CitationLancelot F. James. "Laws and Likelihoods for Ornstein Uhlenbeck-Gamma and other BNS OU Stochastic Volatilty models with extensions." arXiv:math/0604086 (2006). https://pith.science/paper/XU5MEVXJ

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Editor's note

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Quick take: the OU-Gamma factorization in Proposition 3.1 is a genuine idea and I think it is correct; but the paper's "perfect sampling" claim is not. As written, the GHW scheme in §2.3.2 stops when the upper/lower chains are within ε, not when they meet. The width is (b−a)∏(1−B_{n,θ}), with B_{n,θ} ~ Beta(1,θ) in (0,1), so exact coalescence has probability zero. That is approximate CFTP with bias of order ε, and no unbiased correction is supplied. This matters because §3.1 and §6.1 lean on that sampler for the headline "exact sampling" result. The bounded-support qualification the reader flagged is a separate, second layer; the OU-Gamma case has bounded support [e^{-a},1] and still fails the exactness test.

What is actually new and good: the representation (G_θ(a), Y_{θa}) = (T_{θa}, T_{θa} M_{θa}) with independence, derived via the Cifarelli-Regazzini identity, is a smart and, as far as I can tell, correct way to collapse the infinite-dimensional OU-Gamma integrated volatility to finite-dimensional variables. That is the kind of structural observation that could save people a lot of pain. The extension to FGGC models (Theorem 6.1) and the augmentation for the likelihood via Laplace transforms are also constructive and potentially useful. The paper is honest about the provenance of Proposition 4.2, which appeared in an earlier unpublished version; that disclosure is to its credit.

Where it is soft: the explicit dilogarithm densities in Proposition 3.3 are stated without derivation and without numerical checks. A referee should ask for the calculation, or at least a reference that goes through it. The abstract and introduction also overstate perfect sampling for all FGGC models; the finite-support assumption on H is buried in Proposition 6.1(iv). No code or simulation is provided, so the practical message — Monte Carlo likelihood inference at scale — remains a promise. None of this kills the central representation, but it does weaken the paper's stronger claims.

Who this is for: anyone working on likelihood inference for BNS models or on Dirichlet process mean functionals. It deserves a serious referee; the errors are fixable and the core contribution is worth preserving. I would send it to peer review with a request for derivation/numerical checking of the densities, a corrected or qualified statement about exactness, and a small simulation.

Referee report

Significance

If the corrections described below are made, the Gamma/Dirichlet factorization of Proposition 3.1 is a genuinely elegant and potentially influential result: it gives, for the first time, an explicit finite-dimensional representation of the OU-Gamma integrated volatility that could enable exact or near-exact Monte Carlo likelihood inference. The use of the Cifarelli-Regazzini identity and the connection between GGC laws and Dirichlet mean functionals are conceptually strong, and the paper is honest about the special nature of the Gamma case. The Fourier-cosine route to general BNS likelihoods is also a useful contribution. However, the current version contains load-bearing gaps: the advertised "perfect sampling" is only epsilon-coalescent as described, the main Bessel likelihood formula appears to have a parameter error, and the explicit density formulas in Proposition 3.3 do not survive a direct consistency check. These issues affect the central claims and require substantive revision.

Major objections (4)

[§2.3.2, §3.1, §6.1] The perfect-simulation procedure is not exact as described. The coupled chains (17)–(18) satisfy D_{n+1}=(1-B_{n,\theta})D_n with B_{n,\theta}\sim\mathrm{Beta}(1,\theta), so D_n>0 for every finite n almost surely and exact coalescence has probability zero. Declaring coalescence when D<\varepsilon is an approximate CFTP rule with a bias of order \varepsilon; no unbiased correction is supplied. This matters because §3.1 uses this algorithm for M_{\theta a} for all \theta a, and §6.1 reuses it for FGGC components. The paper's claim that OU-Gamma integrated volatility is "exactly samplable" therefore needs either a genuinely exact coalescence argument, a different exact sampler for the parameter ranges claimed, or a qualified statement that the GHW-based method is approximate. The bounded-support condition of §2.3.2 does not repair this, since exact coalescence still fails under that condition.
[Theorem 3.1 and Proposition 3.6] The Bessel/GIG reduction in Theorem 3.1 is not consistent with the likelihood (10). Conditional on S, writing y=T, the y-integrand coming from (10) is proportional to y^{\kappa-n/2-1}e^{-y}\exp(-\sum_i A_i^2/(2yS_i) - \beta^2 y\sum_i S_i/2). With the paper's definition \delta^2=\sum_i A_i^2/(2S_i), the exponent is -\delta^2/y - (\gamma^2/2)y, not -\frac12(\delta^2/y+\gamma^2 y) as presupposed by the identity 2K_\nu(\delta\gamma)/(\gamma/\delta)^\nu = \int y^{\nu-1}e^{-\frac12(\delta^2/y+\gamma^2 y)}dy. The correct GIG parameter is \delta_*^2=\sum_i A_i^2/S_i, equivalently a factor 2^{\nu/2} and a rescaling of the Bessel argument appear. As written, the displayed likelihood and the importance estimator (28) do not integrate the target density; the special half-integer cases (ii)–(iii) inherit the same error.
[Proposition 3.3, equations (21)–(23)] The explicit density formulas in Proposition 3.3 appear to contain algebraic errors. For \theta a=1, direct evaluation of the Cifarelli-Regazzini formula (15) with H=F_a gives f_{V_a}(v)=\frac{a}{\pi}\sin(\pi v)\exp\Big(\frac{\pi^2}{3a}-\frac{a v^2}{2}-\frac{1}{a}[\mathrm{Li}_2(e^{-av})+\mathrm{Li}_2(e^{-a(1-v)})]\Big), which for a=1 integrates to one. Equation (22), by contrast, has e^{-[v+v^2]} in place of e^{-av^2/2} and lacks the factor a; it is not the density of V_a=-\log(M_1)/a. Correspondingly, equation (21) has a different power of x than the change of variables from (15) produces. These formulas are used for the exact rejection sampling described in §3.1 and in Proposition 3.4, so they need to be corrected or rederived.
[Abstract and §6.1(iv)] The abstract claims that the FGGC class allows perfect sampling, but this is only stated in the body for the case where H has finite support (Proposition 6.1(iv)), equivalently bounded support as required by §2.3.2. For a general FGGC with unbounded H, such as a lognormal H, the distribution Q_a has unbounded support and the upper/lower dominating chains cannot be initialized from finite bounds. The perfect-sampling claim should be restricted to FGGC with bounded-support H, and the abstract and introduction should reflect this qualification.

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The paper

The paper claims

The paper proves that the OU-Gamma stochastic volatility model's likelihood, previously intractable, can be calculated by sampling independent Gamma and Dirichlet mean components.

The reading

This paper tackles the intractability of likelihood inference for a popular class of continuous-time stochastic volatility models in which volatility follows a non-Gaussian Ornstein-Uhlenbeck (OU) process driven by a Lévy process. The central discovery is that when the driving process is a Gamma process, the infinite-dimensional random objects entering the likelihood become finite-dimensional: the pair made of a Gamma increment and an integrated OU term is, in joint distribution, equal to an independent Gamma variable times a Dirichlet process mean functional. Because Dirichlet process mean functionals can be sampled exactly, the joint law of the latent integrated variances is determined by a small set of independent variables and Monte Carlo likelihood inference becomes possible. The same Gamma-Dirichlet machinery extends to the wider class of finite Generalized Gamma Convolutions, where perfect sampling of option-pricing and likelihood densities is available when the mixing measure has bounded support. For general OU models the paper supplies Fourier-cosine integral representations of the likelihood that can be evaluated by Monte Carlo even when the Lévy density is unknown.

What carries the argument

The load-bearing object is the Gamma-Dirichlet factorization of Gamma process mean functionals: for shape measure $\theta H$, the total $G_{\theta H}(X)=T_\theta$ is independent of the normalized Dirichlet process $P_{\theta H}=G_{\theta H}/T_\theta$, so any mean functional $G_{\theta H}(g)$ equals $T_\theta \int g\,dP_{\theta H}$ with $T_\theta$ independent of the mean. In the OU-Gamma model a change of variables puts the pair $(G_\theta(a),Y_{\theta a})$ into exactly this form, with the Dirichlet mean $M_{\theta a}$ supported on $[e^{-a},1]$. For FGGC models every quantity in the likelihood is a generalized gamma convolution, meaning a Lévy law with Lévy exponent $\int d_\theta(\omega x)H(dx)$, and hence a Gamma-process mean functional, so the same factorization supplies the independent variables augmented in Theorem 6.1. The explicit densities of these means come from the classical formula for Dirichlet process mean densities; in the OU-Gamma case the formula evaluates in terms of the dilogarithm function $\mathrm{Li}_2(x)=-\int_0^x \log(1-u)/u\,du$, giving densities that can be rejection-sampled.

Extended reading notes

Core claim

For each fixed $\Delta>0$ and $a=\lambda\Delta$, the paper proves the joint distributional identity $(G_\theta(a), Y_{\theta a}) = (T_{\theta a}, T_{\theta a}M_{\theta a})$, where $G_\theta(a)$ is the Gamma-process increment, $Y_{\theta a}=e^{-a}\int_0^\Delta e^{\lambda y}G_\theta(d\lambda y)$ is the integrated OU term, $T_{\theta a}$ is Gamma($\theta a,1$), and $M_{\theta a}$ is a Dirichlet process mean functional independent of $T_{\theta a}$. The identity is established by checking joint Laplace transforms and using the classical Dirichlet mean identity. Its force is that the dependent coordinates of the original pair are replaced by independent components, so the latent volatilities $(\tau_1,\ldots,\tau_n)$ in the OU-Gamma likelihood have a representation as functions of $2n+2$ independent variables. For the FGGC extension, the paper shows that the joint Laplace transform of the likelihood can be augmented by Gamma and Dirichlet mean components, yielding explicit Monte Carlo densities; perfect sampling of these components is guaranteed when the mixing measure $H$ has finite support, as in the arcsine example.

Load-bearing premise

For the FGGC extension, the perfect-sampling claim assumes the mixing measure $H$ has finite support, so that the Dirichlet mean functional is bounded and the coupled chains can be started at finite upper and lower values.

Editorial extensions

If this is right

OU-Gamma likelihood inference can be implemented by iid sampling of $n$ independent pairs $(T_i,M_i)$ together with $v(0)$, instead of simulating infinite-dimensional process paths or truncating infinite series.
For $\theta a=1$ the density of $M_{\theta a}$ is explicit in terms of the dilogarithm function; for integer $\theta a=m$, $M$ is a finite Dirichlet mixture of $m$ iid copies of $M_1$, so rejection sampling is straightforward.
For OU-FGGC models with finite-support $H$, the perfect sampler of Dirichlet means applies to every component entering the option-pricing and likelihood densities, and for the arcsine example all mean functionals are Beta variables.
The general Fourier-cosine representation expresses the OU likelihood as an $n$-dimensional integral of the joint Laplace transform of the integrated variances, which Monte Carlo can handle in high dimensions.
Superpositions of independent OU-Gamma processes and time-changed integrated OU processes preserve the Gamma-Dirichlet representation, with only the shape and scale parameters in the likelihood formula changing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the joint identity is accepted, the same argument should apply to any moving-average model driven by a Gamma process where the kernel can be written as a function of one coordinate; the paper shows a two-coordinate kernel where this fails, so the boundary of the method is likely tied to one-dimensional mean functionals.
The finite-support condition on $H$ is a practical check for claiming perfect sampling in FGGC models: for unbounded $H$ such as lognormal, the coupled-chain sampler lacks finite starting bounds, though approximate sampling from the Dirichlet mean density may still be feasible.
The appearance of dilogarithm functions in the OU-Gamma densities suggests that exact option-pricing quantities may be expressed through dilogarithm and Clausen-function identities, a route the paper leaves mostly implicit.
The scale-mixture form $X_i=\mu\Delta+\beta\zeta T S_i+(\zeta T S_i)^{1/2}\varepsilon_i$ indicates that making $\zeta$ random yields a wider class of return distributions for which the Theorem 3.1 likelihood formula remains valid with reinterpreted $\kappa,\gamma,\delta$; this is a direct but untested extension.

Original abstract

In recent years there have been many proposals as flexible alternatives to Gaussian based continuous time stochastic volatility models. A great deal of these models employ positive L\'evy processes. Among these are the attractive non-Gaussian positive Ornstein-Uhlenbeck (OU) processes proposed by Barndorff-Nielsen and Shephard (BNS) in a series of papers. One current problem of these approaches is the unavailability of a tractable likelihood based statistical analysis for the returns of financial assets. This paper, while focusing on the BNS models, develops general theory for the implementation of statistical inference for a host of models. Specifically we show how to reduce the infinite-dimensional process based models to finite, albeit high, dimensional ones. Inference can then be based on Monte Carlo methods. As highlights, specific to BNS we show that an OU process driven by an infinite activity Gamma process, that is an OU-$\Gamma$, exhibits unique features which allows one to exactly sample from relevant joint distributions. We show that this is a consequence of the OU structure and the unique calculus of Gamma and Dirichlet processes. Owing to another connection between Gamma/Dirichlet processes and the theory of Generalized Gamma Convolutions (GGC) we identify a large class of models, we call (FGGC), where one can perfectly sample marginal distributions relevant to option pricing and Monte Carlo likelihood analysis. This involves a curious result, we establish as Theorem 6.1. We also discuss analytic techniques and candidate densities for Monte-Carlo procedures which apply more generally.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

In recent years there have been many proposals as flexible alternatives to Gaussian based continuous time stochastic volatility models. A great deal of these models employ positive L\'evy processes. Among these are the attractive non-Gaussian positive Ornstein-Uhlenbeck (OU) processes proposed by Barndorff-Nielsen and Shephard (BNS) in a series of papers. One current problem of these approaches is the unavailability of a tractable likelihood based statistical analysis for the returns of financial assets. This paper, while focusing on the BNS models, develops general theory for the implementation of statistical inference for a host of models. Specifically we show how to reduce the infinite-dimensional process based models to finite, albeit high, dimensional ones. Inference can then be based on Monte Carlo methods. As highlights, specific to BNS we show that an OU process driven by an infinite activity Gamma process, that is an OU-$\Gamma$, exhibits unique features which allows one to exactly sample from relevant joint distributions. We show that this is a consequence of the OU structure and the unique calculus of Gamma and Dirichlet processes. Owing to another connection between Gamma/Dirichlet processes and the theory of Generalized Gamma Convolutions (GGC) we identify a large class of models, we call (FGGC), where one can perfectly sample marginal distributions relevant to option pricing and Monte Carlo likelihood analysis. This involves a curious result, we establish as Theorem 6.1. We also discuss analytic techniques and candidate densities for Monte-Carlo procedures which apply more generally.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In recent years there have been many proposals as flexible alternatives to Gaussian based continuous time stochastic volatility models.	background 1.00	0.04	0.02	asserted 1.00
s1	A great deal of these models employ positive L\'evy processes.	background 1.00	0.04	0.02	asserted 0.99
s2	Among these are the attractive non-Gaussian positive Ornstein-Uhlenbeck (OU) processes proposed by Barndorff-Nielsen and Shephard (BNS) in a series of papers.	background 1.00	0.04	0.03	asserted 1.00
s3	One current problem of these approaches is the unavailability of a tractable likelihood based statistical analysis for the returns of financial assets.	gap 0.95	0.05	0.03	asserted 1.00
s4	This paper, while focusing on the BNS models, develops general theory for the implementation of statistical inference for a host of models.	contribution 1.00	0.89	0.17	asserted 0.99
s5	Specifically we show how to reduce the infinite-dimensional process based models to finite, albeit high, dimensional ones.	method 0.80	0.91	0.14	asserted 1.00
s6	Inference can then be based on Monte Carlo methods.	method 0.98	0.41	0.08	asserted 0.82
s7	As highlights, specific to BNS we show that an OU process driven by an infinite activity Gamma process, that is an OU-$\Gamma$, exhibits unique features which allows one to exactly sample from relevant joint distributions.	result 0.76	0.93	0.24	asserted 1.00
s8	We show that this is a consequence of the OU structure and the unique calculus of Gamma and Dirichlet processes.	interpretation 0.85	0.86	0.09	asserted 1.00
s9	Owing to another connection between Gamma/Dirichlet processes and the theory of Generalized Gamma Convolutions (GGC) we identify a large class of models, we call (FGGC), where one can perfectly sample marginal distributions relevant to option pricing and Monte Carlo likelihood analysis.	contribution 0.52	0.93	0.21	asserted 1.00
s10	This involves a curious result, we establish as Theorem 6.1.	result 0.76	0.94	0.07	asserted 0.97
s11	We also discuss analytic techniques and candidate densities for Monte-Carlo procedures which apply more generally.	contribution 0.78	0.84	0.06	asserted 0.70

What kind of paper the classifier reports this is: theoretical result 0.77 · evidence from proof or derivation 1.00.

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References

Works cited (2)

[1]

A¨it-Sahalia, Y., Mykland, P. A. (2003). The eﬀects of random and discrete sampling when estimating continuous-time diﬀusions. Econometrica 71 483-549. A¨it-Sahalia, Y., Mykland, P. A. (2004). Estimators of diﬀusions with randomly spaced discrete observations: a general theory. Ann. Statist. 32 2186-2222. Andrews, G., Askey, R. and Roy, R. (1999). Special...

View on Pith 2003
[76]

Springer-Verlag, New Y ork. Carr, P., Geman, H., Madan, D.B. and Yor, M. (2003). Stochastic volatility for L´ evy pro- cesses. Math. Finance 13 345-382. Carr, P., Geman, H., Madan, D.B. and Yor, M. (2005). Self-Decomposability and Option Pricing. Math. Finance to appear. Carr, P. and Schr ¨oder, M. (2004). Bessel processes, the integral of geometric Brown...

View on Pith arXiv 2003

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Uses its method or tool 1 paper

The papers [ref] exploit the analytical tractability to develop maximum likelihood estimation using the results of [ref] for Dirichlet processes.

Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models · Introduction · 0.87

Discussion

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[1] [1]

A¨it-Sahalia, Y., Mykland, P. A. (2003). The eﬀects of random and discrete sampling when estimating continuous-time diﬀusions. Econometrica 71 483-549. A¨it-Sahalia, Y., Mykland, P. A. (2004). Estimators of diﬀusions with randomly spaced discrete observations: a general theory. Ann. Statist. 32 2186-2222. Andrews, G., Askey, R. and Roy, R. (1999). Special...

View on Pith 2003

[2] [76]

Springer-Verlag, New Y ork. Carr, P., Geman, H., Madan, D.B. and Yor, M. (2003). Stochastic volatility for L´ evy pro- cesses. Math. Finance 13 345-382. Carr, P., Geman, H., Madan, D.B. and Yor, M. (2005). Self-Decomposability and Option Pricing. Math. Finance to appear. Carr, P. and Schr ¨oder, M. (2004). Bessel processes, the integral of geometric Brown...

View on Pith arXiv 2003