Recovering Volatility from an Integrated Covariance in the Nelson–Siegel Model

Summary

The document poses a parameter-recovery problem in an arbitrage-free Nelson–Siegel term-structure model. A known integrated covariance matrix is expressed as an integral involving a matrix exponential, the state-dynamics matrix, and an unknown volatility matrix. The goal is to recover the volatility matrix when it is constrained to be lower triangular and the known covariance and dynamics matrices are general rather than diagonal.

The central difficulty is that the volatility matrix does not commute with the matrix exponential, so element-by-element simplifications available in diagonal cases do not carry over. The question allows numerical methods as well as an algebraic derivation, but supplies no proposed solution, algorithm, numerical experiment, or evidence that a particular recovery method works. Its value is therefore primarily in identifying a structured inverse problem relevant to covariance estimation in fixed-income modeling; uniqueness, conditioning, and practical estimation limits remain unspecified.

Key ideas

• The target is a lower-triangular volatility matrix in an integrated covariance equation for an arbitrage-free Nelson–Siegel model.
• The state-dynamics matrix may be nonsymmetric, while the observed integrated covariance can be non-diagonal.
• Noncommutation between volatility and matrix exponentials prevents simple diagonal-case algebra from generalizing directly.
• Numerical recovery is raised as a possible route, but the document does not provide a method or results.
• Uniqueness and numerical conditioning are not addressed in the posed problem.

Tags

# Risk-Neutral covariance matrix of arbitrage-free Nelson Siegel


# Risk-Neutral covariance matrix of arbitrage-free Nelson Siegel












For my thesis on a Bayesian sampling routine for a modification on arbitrage-free Nelson-Siegel I came across an equation that involves a matrix exponential within an integral, i.e.

$\int_{0}^{\Delta t} e^{-K*s}\Sigma \Sigma' e^{-(K)'*s}ds = Q$

where I need to extract $\Sigma$ $(3 \times 3)$ from the integral. Here, $s$ is a scalar and $K$ $(3 \times 3)$ and $Q$ $(3 \times 3)$ are known.

In one model the off-diagonal elements of all are set to zero and algebraic expressions may easily be obtained. I however seek to derive the elements of $\Sigma$ for a more general case, where $\Sigma$ is a lower triangular matrix, K is a non-symmetric matrix of state dynamics with positive real eigenvalues and Q is a non-diagonal covariance matrix. Additionally, $\Sigma$ and $e^{-K*s}$ do not commute. Numerical methods would also be more than welcome.

Kind regards,

Gert

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