INVENTORY HEDGING AND OPTION MARKET MAKING

In this paper, we develop an inventory-based approach to analyze the option market making activity. Indeed, we formulate and analytically solve the price-setting problem of a monopolistic option market maker facing exogenous public supply and demand first on a single exercise price (the "single option economy") and next on multiple exercise prices (the "multi-options economy"). While in the "single option economy" the familiar result that market maker inventory and price level are inversely related holds, the same is not necessarily true in the "multi-options economy". Additionally, we examine under which theoretical condition hedging is totally effective (i.e., the variance of the market maker hedged position is zero). Last but not least, our model is fully consistent with actual option market making practices, which consist in trading hedge portfolios to reduce risk. As such, our approach can be considered as a bridge between market microstructure and standard option pricing literature.

Expansions of dense linear orders with the intermediate value property

Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of (R, *) definable in ℜ, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of (R, *) (Theorem 2.3).Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

Teaching Word Analysis with Math and Engineering Terms

When science and engineering students apply their analytic skills to the study of words, they make several discoveries: that words have components just as machines have components; that many of these word components are already known to them; and that world analysis frequently pays off in making sense out of familiar words and in making unknown words familiar. Result: greater comfort with their growing vocabularies and increased enthusiasm for language study.

The Dependence of Population Distribution on Location Costs

Part of a household's constrained budget is assumed to be comprised of a general cost of location. By using a simple generalisation of Muth's economic theory of the spatial pattern of urban housing, the population density distribution in a plane city is shown to be dependent on the distribution of this generalised cost of location. If housing demand is elastic, then this dependence is shown to be exponential, a conclusion which agrees with a familiar result obtained by maximising the entropy of a population system subject to a cost constraint. If the housing demand is less or more elastic, then the dependence is binomial. Implications of the theory are discussed and illustrations given.

A Synthetic Derivation of the Class of the Φ Conic

The theorem that a line cutting a pair of conies in four harmonically separated points envelopes a conic, called the Φ conic, is a familiar result which admits of a simple proof by analytical methods. A synthetic proof, however, if we exclude the use of (2, 2) correspondences, is rather elusive. I have not been able to find such a proof in any book, and the only one published as far as I am aware is that set as a question in the 1934 Mathematical Tripos, due to Mr F. P. White. The proof written out below is rather more direct and may therefore be worth recording.