Abstract
In this paper we consider a general class of diffusion-based models and show that, even in the absence of an Equivalent Local Martingale Measure, the financial market may still be viable, in the sense that strong forms of arbitrage are excluded and portfolio optimisation problems can be meaningfully solved. Relying partly on the recent literature, we provide necessary and sufficient conditions for market viability in terms of the market price of risk process and martingale deflators. Regardless of the existence of a martingale measure, we show that the financial market may still be complete and contingent claims can be valued under the original (real-world) probability measure, provided that we use as numéraire the Growth-Optimal Portfolio.
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Notes
- 1.
It is worth pointing out that, if Assumption 4.2.1 does not hold but condition (4.6) is satisfied, i.e. we have \(\mu_{t}-r_{t}\mathbf{1}\in\mathcal {R} (\sigma_{t} )\) P⊗ℓ-a.e., then the market price of risk process θ can still be defined by replacing \(\sigma_{t}' (\sigma_{t}\,\sigma_{t}' )^{-1}\) with the Moore–Penrose pseudoinverse of the matrix σ t .
- 2.
The (NUPBR) condition has been introduced under that name in Karatzas and Kardaras (2007). However, the condition that the set \(\{ \bar{V}^{\,\pi}_{T}:\pi\in\mathcal{A} \}\) be bounded in probability also plays a key role in the seminal work Delbaen and Schachermayer (1994), and its implications have been systematically studied in Kabanov (1997), where the same condition is denoted as “property BK”.
- 3.
We want to remark that an analogous result has already been given in Theorem 2 of Loewenstein and Willard (2000) under the assumption of a complete financial market.
- 4.
More precisely, note that the process \((\int_{0}^{t}\theta_{u}^{2}\,du )_{0\leq t \leq T}= (\int_{0}^{t}\frac{1}{u}\,du )_{0\leq t \leq T}\) jumps to infinity instantaneously at t=0. Hence, as explained in Remark 4.3.12, the model considered in the present example allows not only for arbitrages of the first kind, but also for strong arbitrage opportunities. Of course, there are instances where strong arbitrage opportunities are precluded, but still there exist arbitrages of the first kind. We refer the interested reader to Ball and Torous (1983) for an example of such a model, where the price of a risky asset is modelled as the exponential of a Brownian bridge (see also Loewenstein and Willard 2000, Example 3.1).
- 5.
Alternatively, one can show that the probability measures Q and P fail to be equivalent by arguing as follows. Let us define the stopping time τ:=inf{t∈[0,T]:S t =0}. The process S=(S t )0≤t≤T is a Bessel process of dimension three under P, and, hence, we have P(τ<∞)=0. However, since the process S=(S t )0≤t≤T is a Q-Brownian motion, we clearly have Q(τ<∞)>0. This contradicts the assumption that Q and P are equivalent.
- 6.
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Acknowledgements
Part of this work has been inspired by a series of research seminars organised by the second author at the Department of Mathematics of the Ludwig-Maximilians-Universität München during the Fall Semester 2009. The first author gratefully acknowledges financial support from the “Nicola Bruti-Liberati” scholarship for studies in Quantitative Finance. We thank an anonymous referee for the careful reading and for several comments that contributed to improve the paper.
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Fontana, C., Runggaldier, W.J. (2013). Diffusion-Based Models for Financial Markets Without Martingale Measures. In: Biagini, F., Richter, A., Schlesinger, H. (eds) Risk Measures and Attitudes. EAA Series. Springer, London. https://doi.org/10.1007/978-1-4471-4926-2_4
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