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17-01-2022 | Research Article

Equilibrium pricing of commodity spot and forward under incomplete markets with implications on convenience yield

Author: Katsushi Nakajima

Published in: Annals of Finance | Issue 1/2022

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Abstract

This paper analyzes the relation between commodity spot, forward prices, and convenience yield under incomplete markets. Since production is a necessary process for commodity markets, we include firms that use inputs and produce outputs in our model. Thus, we show a financial pricing model of spot and forward commodity in an explicit fashion with production under incomplete markets. One of the most important results of this paper is the difference between commodity spot and forward equilibrium price can be explained by the discounted shadow price of storage constraint minus the discounted marginal storage cost and it can be interpreted as the net convenience yield in the existing literature. Here the discounted factor is affected by the incompleteness of the markets. We prove the generic existence of the equilibrium and thus the obtained spot forward price relation is the equilibrium price formula. We also derive the firm’s optimal production plan and trading strategy.

previous article Constrained dynamic futures portfolios with stochastic basis

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For a nice introduction to the differential topological methods applied to general equilibrium analysis, see Villanacci et al. (2002).

Most of our notation and definition follows Nakajima (2020).

\(L^1(\Omega , {\mathcal {F}}, P)\) is a set of measurable and integrable functions.

Notations are summarized in Table 1 in the Appendix.

For \(x = (x_{n}), y = (y_{n}) \in {\mathbf {R}}^{N}\), \((x_{n}) \ge (y_{n})\) means \(x_{n} \ge y_{n}\) for all n and \((x_{n}) \gg (y_{n})\) means \(x_{n} > y_{n}\) for all n.

See Magill and Quinzii (1996), Section 26 for the equivalence between the two representations.

\({\mathbf {R}}_-^M\) is a nonpositive set \(\{ (x_n)_{n=1,...,M} : x_n \le 0, n=1,...,M \}\).

A set S has zero Lebesgue measure if for each \(\epsilon > 0\) there exists a countable family \(\{I_{i}\}_{i \in {\mathbb {N}}}\) of n-intervals such that \(S \subset \bigcup _{i=1}^{\infty } I_{i}\) and \(\sum _{i=1}^{\infty } \Vert I_{i} \Vert < \epsilon \) where \(\Vert \cdot \Vert \) is the product of the side lengths of the points of the vertices of the interval.

See Morris (1976), pp. 68–69.

For the Grassmannian manifold and its properties, see Villanacci et al. (2002), Chapter 13, Appendix or Duffie and Shafer (1985), Section 4.

This part is taken from Magill and Quinzii (1991), step 1 in the proof of Lemma 2.

The proof is similar to step 3 and step 4 of the proof of Lemma 2 of Magill and Quinzii (1991).

For Transversality Theorem, see Theorem 26 and Corollary 27 in Chapter 6 (pp. 151–153) of Villanacci et al. (2002).

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Title: Equilibrium pricing of commodity spot and forward under incomplete markets with implications on convenience yield
Author: Katsushi Nakajima
Publication date: 17-01-2022
Publisher: Springer Berlin Heidelberg
Published in: Annals of Finance / Issue 1/2022
Print ISSN: 1614-2446
Electronic ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-021-00402-7

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