Eraker (2001) MCMC Analysis of Diffusion Models With Application to Finance

Summary

Proposes a Markov chain Monte Carlo (MCMC) method for estimating parameters of continuous-time Itô diffusions from discrete observations by introducing m−1 auxiliary latent data points between each observed pair (Δt = 1/m), eliminating discretization bias while enabling Gibbs sampling from the augmented posterior. Applied to the constant-elasticity-of-variance (CEV) one-factor interest-rate model and a two-factor CEV + stochastic-volatility (SV) model; the SV model dramatically outperforms CEV on both tail fit and conditional variance dynamics for weekly US 3-month Treasury-bill (T-bill) data 1954–1997.

Key Claims

• Data augmentation eliminates discretization bias: m−1 latent points between observations set Δt = 1/m; as m → ∞ the Euler Gaussian transition density $p(\widehat{Y}_i|\widehat{Y}_{i-1};\theta)$ converges to the exact diffusion density; unobserved paths are integrated out of the augmented likelihood $\pi(\widehat{Y},\theta) \propto \prod_{i=1}^n p(\widehat{Y}_i|\widehat{Y}_{i-1};\theta)\,p(\theta)$ by Gibbs sampling.
• Proposition 1 (exact): For scalar constant drift μ and diffusion σ, $\widehat{Y}_i | \widehat{Y}_{i-1}, \widehat{Y}_{i+1}, \theta \sim \mathcal{N}(\tfrac{1}{2}(\widehat{Y}_{i-1}+\widehat{Y}_{i+1}),\, \tfrac{1}{2}\sigma^2\Delta t)$ exactly — the Brownian bridge midpoint distribution.
• Proposition 2 (approximate): For general Lipschitz/linear-growth drift and diffusion, $\Delta t^{-1/2}[\widehat{Y}_i - \tfrac{1}{2}(\widehat{Y}_{i-1}+\widehat{Y}_{i+1})] \to \mathcal{N}(0,\, \tfrac{1}{2}\sigma^2_{i-1})$ as Δt → 0; the Brownian bridge normal approximation improves at finer discretizations.
• Hybrid accept-reject Metropolis-Hastings (AR-MH) algorithm: Proposal $q(x|Y_{i-1},Y_{i+1},\theta) = \mathcal{N}(m_i, \tfrac{1}{2}[\sigma^2_{i-1}]^2\Delta t)$ with scale $C = p(m_i|\cdot)/q(m_i|\cdot)$ ensures supp$(q) \subseteq$ supp$(p)$; acceptance rates 0.90–0.98 for one-factor CEV models.
• CEV parameter estimates vs. competing methods: β ≈ 0.75 (MCMC, m=8) vs. ~1.5 (generalized method of moments, GMM; Chan et al. 1992) vs. ~0.67 (efficient method of moments, EMM; Andersen-Lund 1997a); CEV κ_r has only ~8% posterior mass below 0 for m=8 (weak mean reversion evidence); CEV fails quantile-quantile (Q-Q) test (leptokurtic residuals) and exhibits persistent ARCH-like autocorrelations in squared residuals — both symptoms of missing SV component.
• CEV+SV two-factor model: $dr_t = (\theta_r + \kappa_r r_t)dt + \sigma_r \exp(\tfrac{1}{2}Z_t) r_t^\beta dW_{1,t}$, $dZ_t = (\theta_z + \kappa_z Z_t)dt + \sigma_z dW_{2,t}$; estimated log-volatility persistence κ_z → first-order autocorrelation (AC) ≈ 0.976 (in line with GARCH/SV literature); σ_z ≈ 0.27 (vs. Andersen-Lund 1997a EMM: 0.176); κ_r entirely negative in SV model — strong mean reversion absent from one-factor CEV.
• Reparameterization to fix κ_z bias (Appendix D): High serial dependence in the volatility chain mean $\bar{Z}^{(g)}$ induces downward bias in κ_z; fix: subtract the running mean $k = \bar{Z}^{(g)}$ at each Gibbs step ($Z_t^* \leftarrow Z_t^{(g)} - k$), restoring unbiased conditionals without changing the stationary distribution.
• Discretization bias vs. convergence trade-off: As Δt → 0 Gibbs sampler converges more slowly (conditionals collapse toward Dirac measures); as Δt grows, discretization bias worsens; practical strategy: start with m=1, increase until posterior moments stabilize (convergence occurs around m=2–4 for CEV parameters; β requires highest m).
• Small-sample accuracy (200 experiments, T=500 and T=2,000, 20,000 Gibbs iterations, Δt=1/20): At T=2,000 root mean squared error (RMSE) is small for all parameters; at T=500, downward bias in κ_r and κ_z due to near-unit-root behavior and downward sample-autocorrelation bias — a known finite-sample phenomenon.

Concepts Introduced or Extended

• Diffusion Process
• Stochastic Volatility
• Gibbs Sampler

Entities Mentioned

• Bjørn Eraker
• Neil Shephard
• Siddhartha Chib
• Sangjoon Kim

Quotes

"The essential idea is to substitute the missing data, $Y_{t_i}$, with simulations, $\widehat{Y}_i$." "MCMC simulations are conditioned on more information than the Euler simulations on which Pedersen's approach was based."

My Take

The central contribution is clean: the Brownian bridge AR-MH proposal (Propositions 1–2) is not approximate in the limit — it becomes exact as Δt → 0 — and it enables standard Gibbs machinery to handle arbitrary Itô diffusions. The reparameterization trick in Appendix D is underappreciated: it quietly solves a simulation-induced bias problem that would corrupt κ_z estimates in any highly persistent latent-factor model. The CEV vs. SV evidence is compelling — not just better Q-Q, but qualitatively different mean-reversion conclusions — making the paper a strong argument for the SV approach to interest-rate modeling. Main limitation: computational cost at m=8 was ~8.5 hours in 2001; the one-column-at-a-time missing-data draw is slower than the block Independence Metropolis of Elerian-Chib-Shephard (1999).