Asset-backed stable numéraire approach for sustainable valuation

ABSTRACT

Interest rates underpin almost every instrument/transaction in conventional financial markets. Valuation of the instruments in relation to interest rates remains meaningful only if cash can be attributed a worth of its own (which is generally assumed to accumulate over time). The relevant concepts such as the stochastic short rate and the conventional numéraire (i.e. the money market account) not only become restrictive when one attempts to build more realistic models in quantitative finance, but also – as we demonstrate in this work – their application has de-stabilizing effects on asset valuations. This paper presents a detailed critique of the conventional numéraire and proposes an asset-backed stable numéraire for sustainable valuation of assets and/or transactions. In particular, we demonstrate how some of the key benchmarks of a sustainable footprint may be used as a numéraire currency. We also unveil the implicit assumption underlying the common practice of straightforward factorization in conventional relative pricing to be false.

KEYWORDS:

• Asset-backed numéraire
• interest-rate benchmarks
• fair valuation
• conventional pricing
• risk-neutral valuation

Acknowledgments

This paper has been admitted with a revision to the Journal of Sustainable Finance and Investment (Manuscript ID: 19-JSFI397-MP.R1) and is currently under a final review.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The ecological cost of crypto-currencies is beyond the scope of this work and assessed in Stoll, Klaassen, and Gallersdörfer (2019)

2 The latter point also leads to second-round effects.

3 The latter two properties imply that $\left\{{\pi }_t\right\}$ is a supermartingale, which ensures that the asset (e.g. bond) price is a decreasing function of time to maturity $T$.

4 The same argument would also hold if the asset and numéraire were instead complementary, such as, tea and sugar, respectively.

5 See, e.g. Armknecht and Silver (2012) for a discussion of arithmetic versus geometric weighting in constructing price indices.

6 Note that $\rho =\pm 1$ does not necessarily imply $S=\pm kM$.

7 It should be very small or zero because $C_t$ in $S_t={\mathbb{E}}_t^{\mathbb{Q}}\left[S_T/\right(C_T/C_t\left)\right]$ should reflect all possible information about future value of sustainable footprint $C_T$, thus making $C_T/C_t$ as close as possible to 1.

8 See Aydın and Rainer (2015) for a more detailed discussion.

9 Note that $\mathrm{d}{e}^X=\mathcal{E}\left(X\right)\mathrm{d}X$, where $\mathcal{E}\left(X\right)$ is the stochastic (Doléans-Dade) exponential (Protter 2004).

10 We use carbon emissions as shorthand for greenhouse gas emissions such as ${\mathrm{C}\mathrm{O}}_2$, ${\mathrm{C}\mathrm{H}}_4$ and ${\mathrm{N}}_2\mathrm{O}$.