- expectation maximisation
- latent variable
- stable distributions
- mean reversion
- brownian motion
- formulas
- African Institute for Mathematical Sciences
- commodity market
- dynamics
- joint probability
- National Research Foundation of South Africa

Different from existing filtering methods for models with latent variables, we show that the commodity future price under a one factor model with a subordinated random source driver, can be expressed in terms of the subordinator which can be reduced to the latent regression models commonly used in population dynamics with their parameters estimated using the expectation maximisation method

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- Commodities exhibit distinctive features that a good model ought to capture
- Different from existing filtering methods for models with latent variables, we show that the commodity future price under a one factor model with a subordinated random source driver, can be expressed in terms of the subordinator which can be reduced to the latent regression models commonly used in population dynamics with their parameters estimated using the expectation maximisation method
- We show that the affine property is attainable and applicable to generalised commodity spot models, and as illustration, we consider a stochastic differential equation with subordinated Brownian motion as the source of randomness to derive the commodity future prices
- We show that the commodity future price under a one-factor model with a subordinated random source driver can be expressed in terms of the subordinator, which can be reduced to the latent regression models commonly used in population dynamics with their parameters estimated using the expectation maximisation method
- Our approach is adaptable to the latter approach through the Dynkin–Lamperti Theorem) where the unobserved variable follows a stable distribution defined on ðÀ1; 1Þ with α{ð0; 2 and the observable variable y represents the log-returns of the future prices
- We considered a stochastic differential equation with the source of randomness as subordinated Brownian motion as a specific example to derive the futures price
- The numerical implementation of the two factor model is left for future work

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