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Making no-arbitrage discounting-invariant: A new FTAP version beyond NFLVR and NUPBR

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  • What is absence of arbitrage for non-discounted prices? How can one define this so that it does not change meaning if one decides to discount after all?

    The answer to both questions is a new discounting-invariant no-arbitrage concept. As in earlier work, we define absence of arbitrage as the zero strategy or some basic strategies being maximal. The key novelty is that maximality of a strategy is defined in terms of share holdings instead of value. This allows us to generalise both NFLVR, by dynamic share efficienc, and NUPBR, by dynamic share viability. These new concepts are the same for discounted or undiscounted prices, and they can be used in general models under minimal assumptions on asset prices. We establish corresponding versions of the FTAP, i.e., dual characterisations in terms of martingale properties. As one expects, "properly anticipated prices fluctuate randomly", but with an endogenous discounting process which cannot be chosen a priori. An example with N geometric Brownian motions illustrates our results.

    Mathematics Subject Classification: 91G99, 91B02, 60G48.

    Citation:

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  • Figure 1.  Graphical summary of Theorem 4.1. Assumptions are $ {S}\ge0 $ and that $ \eta $ is a reference strategy (which is assumed to exist). The shorthand sm stands for share maximal, vm stands for value maximal. The equivalences on the left side need in addition that $ \eta $ and $ {{{S}}^{\eta}} $ are bounded (uniformly in $ (\omega,t) $)

    Table 1.  Overview of existing FTAP-type results. Note that we have $ {\rm NA1} = {\rm{NUPBR}} $ on $ [0,T] $

    price process $ {S} $ time condition dual condition
    KK [27] $ (1,X) \in \mathcal{S}^{1+d}_{++} $ $ [0,\infty) $ NUPBR $ \exists $ $ {S} $-tradable SMD $ D>0 $, $ \forall\ H{\cdot} X $ with $ H\in{L_{{\rm{adm}}}}(X) $, with $ D_\infty>0 $
    TS [39] $ (1,X) \in \mathcal{S}^{1+d} $ $ [0,T] $ NUPBR $ \exists \sigma \mathrm{MD}\ D>0 \text { for } X $
    K [31] $ (1,X) \in \mathcal{S}^{1+1} $ $ [0,T] $ NA1 $ \exists \operatorname{LMD} D>0 \text {, } \forall H \cdot X \text { with } H \in L_{\mathrm{adm}}(X) $
    KKS [21] $ (1,X) \in \mathcal{S}^{1+d} $ $ [0,T] $ NA1 $ \exists S\text {-tradable LMD } D>0, \forall H \cdot X \text { with } H \in L_{\mathrm{adm}}(X), \text { in any neighbourhood of }P $
    H [17] $ \text { in } \mathcal{S}^{N} $ $ [0,T] $ NINA $ \exists (\text discounter, \text {E} \sigma \mathrm{MM}) \text { pair for }S $
    here $ \text { in } \mathcal{S}^{N}_+ $ $ [0,\infty) $ DSV for $ \eta $ $ \exists \sigma \mathrm{MD}=\operatorname{LMD} D>0\text { for }S\text { with }\inf _{t \geq 0}\left(\eta_{t} \cdot\left(S_{t} / D_{t}\right)\right)>0 $
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