Abstract
This chapter explains how to describe the value of the portfolio in terms of “guaranteed tubular viability kernels of capture basins”.
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Notes
- 1.
Since the floor takes infinite values, it conceals tubular constraint: \((S, P) \in K(t)\) if and only if \(L(t, S, P) <+\infty \). This classical trick of epigraphical analysis allows us to simplify the notations, knowing that at the very end, formulas should be made explicit for involving the associated tubular constraint.
- 2.
An asset \(i\) is riskless at time \(t\) if its lower and upper bounds coincide: \(S^{\flat }_{i}(t)=S^{\sharp }_{i}(t)\). If we want to distinguish a riskless asset on exercise period, we assign to it the label \(0\) (actually, we shall not use the fact that an asset is risky or not).
- 3.
The terminology of impetus has been introduced in [24] for denoting the sum of the transaction value \( P'_{i}(t)S_{i}(t)\) of asset \(i\) and of the price impact \( S'_{i}(t)P_{i}(t)\) on asset \(i.\)
- 4.
This is not a problem for computing the minimum guaranteed investment and the VPPI management rule, but complicates the analytical formulas presented below (and which are not used in the viability algorithms). However, extrapolation of the lower and upper bounds of the price tubes, the only available information, is sufficient since it “preserves” or “conserves” past interdependency relations between the prices of the assets. If such is the case, it is safe to assume that the price tubes are the products of price interval tubes of each asset.
- 5.
See for instance, [16, 119, 126, 173].
- 6.
which can be chosen to take infinite values when no other bounds on the transactions than the ones derived from the price tube are imposed.
- 7.
However, for taking into account short selling, when it is (unfortunately) authorized, shares which are not owned can be regarded as negative shares, so that the lower bounds may be negative, but finite.
- 8.
This a situation analogous to the capital asset pricing types of models of Harry Markowitz in [135].
- 9.
For the supremum norm instead of the Euclidian norm which requires the convexity of the \(\mathcal {V}(S, P)[d, D]\). See more details in Sect. 10.2 of [11], and Proposition 12.4 of [18].
- 10.
The Greek \(\displaystyle {\varTheta }:= \frac{\partial W^{\heartsuit }[D](D-t, S, P)}{\partial t}\) measures the sensitivity with respect to the time to exercise \(\tau :=D-t\). We shall not use the other “Greeks” since, using an inverse approach, we are insensitive to sensitivity analysis (see Chap. 3).
- 11.
Yet to be nicknamed, to the best of our knowledge. We shall use \(\varOmega \) in this book.
- 12.
The excess demand is the right-hand side of the Walras tâtonnement governing the evolution of prices (through Adam Smith’s visible hand) without making transactions and waiting infinity for the market to be cleared. The scarcity constraints are not viable under the Walras tâtonnement.
- 13.
Both excess demand and excess prices can be involved in a “bilateral tâtonnement” regulating viable economic evolutions by using the two hands of Adam Smith’s invisible Man.
- 14.
- 15.
However, the value function of the portfolio is the solution to both the Black-Sholes partial differential equation and of an equivalent Hamilton-Jacobi partial differential equation thanks to the Stroock-Varadhan theorem (see [43, 99]).
- 16.
See for instance [44, 158].
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Aubin, JP., Chen, L., Dordan, O. (2014). General Viabilist Portfolio Performance and Insurance Problem. In: Tychastic Measure of Viability Risk. Springer, Cham. https://doi.org/10.1007/978-3-319-08129-8_5
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DOI: https://doi.org/10.1007/978-3-319-08129-8_5
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