Robust statistical arbitrage strategies

Abstract

We investigate statistical arbitrage strategies when there is ambiguity about the underlying time-discrete financial model. Pricing measures are assumed to be martingale measures calibrated to prices of liquidly traded options, whereas the set of admissible physical measures is not necessarily implied from market data. Our investigations rely on the mathematical characterization of statistical arbitrage, which was originally introduced by Bondarenko [Statistical arbitrage and securities prices. Rev. Financ. Stud., 2003, 16, 875–919]. In contrast to pure arbitrage strategies, statistical arbitrage strategies are not entirely risk-free, but the notion allows one to identify strategies which are profitable on average, given the outcome of a specific σ-algebra. Besides a characterization of robust statistical arbitrage, we also provide a super-/sub-replication theorem for the construction of statistical arbitrage strategies for path-dependent options. In particular, we show that the range of statistical arbitrage-free prices is, in general, much tighter than the range of arbitrage-free prices.

Keywords:

• Statistical arbitrage
• Robust valuation
• Trading strategies
• Super-replication duality

JEL Classification:

• G11
• G13
• C22

Acknowledgements

We are grateful to Ludger Rüschendorf for various insightful remarks and discussions. Further, we are thankful to two anonymous referees for carefully reading the manuscript and for several comments that helped to significantly improve our paper. Julian Sester acknowledges the financial support of the Carl-Zeiss Stiftung.

Disclosure statement

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

Notes

1 This can always be achieved by considering ${\varphi }_i-p_i$ instead of ${\varphi }_i$.

2 In the case $\mathrm{\Phi }=\mathrm{\varnothing }$, we have ${\mathcal{Q}}_{\mathrm{\varnothing }}^{\mathcal{P}}=\{\mathbb{Q}\in {\mathcal{Q}}_{\mathrm{\varnothing }}:\mathbb{Q}\ll \mathbb{P}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\mathbb{P}\in \mathcal{P}\}$.

3 For this result, the authors assume additionally a specific product structure of $\mathbb{P}\in \mathcal{P}$ as well as convexity of $\mathcal{P}$. Moreover, the considered trading strategies are assumed to be universally measurable. The corresponding framework is described in further detail within Bouchard and Nutz (2015, Section 1.2.). In particular, the extreme cases $\mathcal{P}=\left\{\mathbb{P}\right\}$ and $\mathcal{P}=\mathcal{M}\left({\mathbb{R}}^n\right)$ are covered in this framework.

4 By the theorem we technically obtain a measure defined only on $\mathbb{R}$, which by Kolmogorov's extension theorem can however be considered as a measure on ${\mathbb{R}}^n$. Moreover, the result does not include traded options. But in the one-period model, we may simply consider a finite amount of traded options ${\varphi }_i$ as additional tradable assets. The martingale property then writes for these options as ${\mathbb{E}}_{{\tilde{\mathbb{Q}}}_{\epsilon }}\left[{\varphi }_i\right]=0$.

Additional information

Funding

Julian Sester acknowledges the financial support of the Carl-Zeiss Stiftung.