Posted: July 21st, 2023

Stochastic evolution of the stochastic system

Stochastic evolution of the stochastic system

This thesis focuses on the stochastic evolution of the stochastic system represented by the oil futures term structure (FTS). In particular, it focuses on its correlation structure, given its crucial role in a wide range of financial applications, such as in derivatives pricing or in risk management applications.

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Three modeling frameworks are presented. First, the Heath-Jarrow-Morton (HJM) approach and its generalization through Random Fields (RF) and/or Hilbert-space-valued Stochastic Partial Differential Equations (H-SPDE) are introduced: the different constituents of the oil FTS, the futures contracts, are the main focus of analysis. Based on these two frameworks, we specify two log-normal multi-factor models, the 'Traditional Term Structure' model (TTSM) and the 'Parsimonious Discrete Stochastic String' model (PDSSM). A dimension-reduction strategy, based on the eigenvalue/eigenvector decomposition from a 'Principal Component Analysis' (PCA) carried out on the historical correlation matrix of futures log-returns (FLR), is performed on the TTSM, allowing for its efficient estimation (to historical data) and implementation. It's found that, although two factors (the 'Principal Components', PC) are enough to explain most of the system's variance, a two-factor TTSM fails to account for the empirically observed correlation structure at the Front-end of the oil FTS.

Qualitative and statistical tests on these eigenvalues and eigenvectors, suggest that the assumption of homoskedastic risk factors implicit in the log-normal TTSMis inappropriate: while the eigenvector structure seems to be quite stable over time, the eigenvalues do fluctuate considerably. This motivates the modeling of the main risk factors through GARCH-type processes, in order to account for their time-varying conditional volatilities.

As a result, a second modeling procedure is put forward, the 'Orthogonal GARCH' model (O-GARCH), which, in contrast to the TTSM, it allows for a time-varying and mean-reverting covariance matrix for the oil FLRs. However, this model is only designed to be applied under the physical measure, in contrast to the previous model which could, in principle, be applied under the physical or an equivalent martingale (risk-neutral) measure. Also, the O-GARCH model with just two factors shares the same shortcomings as the two-factor TTSM in terms of the correlation structures that is able to generate.

Finally, a well-known 'Spot-driven' model (SDM) is analyzed: the 'Short-Term/Long-Term' model (STLT) of Schwartz and Smith. As in all SDMs, the focus is placed on the spot price dynamics (as opposed to the whole FTS, as in the HJM- or RF-type of models). Not surprisingly, being a two-factor model, it also fails to account for the exponentially-decaying correlation behavior at the Front-end of the oil FTS. The dynamics of the first differences of the two latent factors of the STLT model are found to be closely linked to the first two PCs of the correlation matrix of FLRs.

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These three modeling approaches should not be viewed as competitors, but complementary to each other: while the TTSM the STLT are better suited for risk-neutral pricing of contingent claims, the O-GARCH is more appropriate for actuarial-type applications. Given the diffusive nature of these models, comparisons of the covariance structures implied by them under the physical measure are appropriate and consistent. As main conclusion, two-factor models should not be employed if a realistic correlation structure is sought after.

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