Free Lunch

The concept of absence of opportunities for free lunches is one of the pillars in the economic theory of financial markets. This natural assumption has proved very fruitful and has lead to great mathematical, as well as economical, insights in Quantitative Finance. Formulating rigorously the exact definition of absence of opportunities for riskless profit turned out to be a highly non-trivial fact that troubled mathematicians and economists for at least two decades. The purpose of this note is to give a quick (and, necessarily, incomplete) account of the recent work aimed at providing a simple and intuitive no-free-lunch assumption that would suffice in formulating a version of the celebrated Fundamental Theorem of Asset Pricing.Comment: 3 pages; a version of this note will appear in the Encyclopaedia of Quantitative Finance, John Wiley and Sons In

Free Lunch for Optimisation under the Universal Distribution

Function optimisation is a major challenge in computer science. The No Free Lunch theorems state that if all functions with the same histogram are assumed to be equally probable then no algorithm outperforms any other in expectation. We argue against the uniform assumption and suggest a universal prior exists for which there is a free lunch, but where no particular class of functions is favoured over another. We also prove upper and lower bounds on the size of the free lunch

Nanotechnology, No Free Lunch

Nanotechnology is the new science and technology of the super small. Particles at the nano-scale, from one to one hundred billionths of a metre, exhibit novel properties. Nanotechnology is an active area of research and rapid commercialization. The food industry has been targeted as a potential recipient of this new technology and engineered nanoparticles are reportedly already in some super-market products. Nanotechnology is currently unregulated, and there are no requirements for mandatory labelling, this leaves consumers unprotected and uninformed. Consumers are largely unaware of nanotechnology, expect labelling on nano-products, are unclear of the cost/benefit balance, and express an unwillingness to purchase nanofood. The asymmetric information status of nanotechnology, together with its undetermined safety, raises issues, opportunities, and risks for food manufacturers and retailers. Some local organic food standards, including AUstralia and UK, have nanotechnology exclusions in place

Market free lunch and large financial markets

The main result of the paper is a version of the fundamental theorem of asset pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover, various notions of no asymptotic arbitrage are characterized in terms of no asymptotic market free lunch; the difference lies in the set of utilities. In particular, it is shown directly that no asymptotic market free lunch with respect to monotone concave utilities is equivalent to no asymptotic free lunch. In principle, the paper can be seen as the large financial market analogue of [Math. Finance 14 (2004) 351--357] and [Math. Finance 16 (2006) 583--588].Comment: Published at http://dx.doi.org/10.1214/105051606000000484 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Exact No Free Lunch Theorem for Community Detection

A precondition for a No Free Lunch theorem is evaluation with a loss function which does not assume a priori superiority of some outputs over others. A previous result for community detection by Peel et al. (2017) relies on a mismatch between the loss function and the problem domain. The loss function computes an expectation over only a subset of the universe of possible outputs; thus, it is only asymptotically appropriate with respect to the problem size. By using the correct random model for the problem domain, we provide a stronger, exact No Free Lunch theorem for community detection. The claim generalizes to other set-partitioning tasks including core/periphery separation, $k$-clustering, and graph partitioning. Finally, we review the literature of proposed evaluation functions and identify functions which (perhaps with slight modifications) are compatible with an exact No Free Lunch theorem