To price contingent claims in a multidimensional frictionless security market it is sufficient that the volatility of the security process is a known function of price and time. In this note we introduce optimal and risk-free strategies for intermediaries in such markets to meet their obligations when the volatility is unknown, and is only assumed to lie in some convex region depending on the prices of the underlying securities and time. Our approach is underpinned by the theory of totally non-linear parabolic partial differential equations (Krylov and Safanov, 1979; Wang, 1992) and the non-stochastic approach to Ito's formation first introduced by Follmer (1981a,b). In these more general conditions of unknown volatility, the optimal risk-free trading strategy will, necessarily, produce an unpredictable surplus over the minimum assets required at any time to meet the liabilities. This surplus, which could be released to the intermediary or to the client, is not required to meet the contingent claim. One sees that the effect of unknown volatility is the creation of a 'with profits' policy, where a premium is paid at the beginning, the contingent claim is collected at the terminal time, but that in addition an unpredictable surplus available as well. The risk-free initial premium required to meet the contingent claim is given by the solution to the Dirichlet problem for a totally non-linear parabolic equation of the Pucci-Bellman type. The existence of a risk-free strategy starting with this minimum sum is dependent upon theorems ensuring the regularity of the solution and upon a non-probabilistic understanding of Ito's change of variable formulae. To illustrate the ideas we give a very simple example of a one-dimensional barrier option where the maximum Black-Scholes price of the option over different fixed values for the volatility lying in an interval always underestimates the risk-free 'price' under the assumption that the volatility can vary within the same interval. This paper puts together rather standard mathematical ideas. However, the author hopes that the overall result is more than the sum of its parts. The ability to hedge under conditions of uncertain volatility seems to be of considerable practical importance. In addition it would be interesting if these ideas explained some features in the design of existing contracts.volatility, derivative contract, random volatility, Pucci-Bellman equation, Black-Scholes Formula,

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IntroductionDiscovery in science is the generation of novel, interesting, plausible, and intelligible knowledge about the objects of study. Literature-related discovery (LRD) is the linking of two or more concepts that have heretofore not been linked (i.e., disjoint), in order to produce new knowledge (i.e., potential discovery). Two major variants of LRD are: open discovery systems (ODS), where one starts with a problem and generates a potential solution (or vice versa), and closed discovery systems (CDS), where one starts with a problem and a potential solution and generates linking mechanism(s).This chapter reviews the state-of-the-art in ODS LRD only. It examines the major LRD concepts, evaluates each concept in detail from the perspective of discovery capability, and examines the level of potential discovery reported in the literature from each concept's implementation. In the evaluation of potential discovery claimed in the published literature, a vetting process is used that requires both characteristics of ODS LRD to be present in order for potential discovery to be affirmed: concepts are linked that have not been linked previously and novel, interesting, plausible, and intelligible knowledge is produced.The major conclusions are that, until recently, most of the reported ODS LRD techniques 5-3 have not generated discovery and this lack of discovery has hampered the growth of ODS LRD substantially. However, ODS LRD techniques have been developed that allow significantly greater amounts of potential discovery to be generated systematically.Discovery is ascertaining something previously unknown or unrecognized. More formally, "Discovery in science is the generation of novel, interesting, plausible, and intelligible knowledge about the objects of study" (Valdes-Perez, 1999, p. 336). It can result from uncovering previously unknown information, from synthesis of publicly available knowledge whose independent segments have never been combined, and/or through invention. In turn, the discovery could derive from logical exploitation of a knowledge base and/or from spontaneous creativity (e.g., Edisonian discoveries from trial and error) (Kostoff, 2003). Innovation reflects the metamorphosis from present practice to some new, ideally "better" practice. It can be based on existing non-implemented knowledge. It can follow discovery directly or resuscitate dormant discovery that has languished for decades.Literature-related discovery (LRD) is a systematic approach to bridging unconnected disciplines based on text mining procedures. LRD allows potentially radical discovery to be hypothesized using either the technical literature alone, or the literature and its authors.In the LRD context, discovery is linking two or more literature concepts that have heretofore not been linked (i.e., disjoint), in order to produce novel, interesting, plausible, and intelligible knowledge. Thus, simply linking two or more disparate concepts is a necessary, but not sufficient, condition for LRD. In particular, concepts may be disjoint b...

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