Celebratio Mathematica

M. Atlan, H. Geman, and M. Yor: Options on hedge funds under the high water mark rule. Preprint, October 2005. ArXiv math/0510497 techreport

The rapidly growing hedge fund industry has provided individual and institutional investors with new investment vehicles and styles of management. It has also brought forward a new form of performance contract: hedge fund managers receive incentive fees which are typically a fraction of the fund net asset value (NAV) above its starting level–a rule known as high water mark. Options on hedge funds are becoming increasingly popular, in particular because they allow investors with limited capital to get exposure to this new asset class. The goal of the paper is to propose a valuation of plain-vanilla options on hedge funds which accounts for the high water market rule. Mathematically, this valuation leads to an interesting use of local times of Brownian motion. Option prices are numerically computed by inversion of their Laplace transforms.

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M. Atlan, H. Geman, D. B. Madan, and M. Yor: “Correlation and the pricing of risks,” Ann. Finance 3 : 4 (October 2007), pp. 411–453. Zbl 1233.91320 article

Given a pricing kernel we investigate the class of risks that are not priced by this kernel. Risks are random payoffs written on underlying uncertainties that may themselves either be random variables, processes, events or information filtrations. A risk is said to be not priced by a kernel if all derivatives on this risk always earn a zero excess return, or equivalently the derivatives may be priced without a change of measure. We say that such risks are not kernel priced. It is shown that reliance on direct correlation between the risk and the pricing kernel as an indicator for the kernel pricing of a risk can be misleading. Examples are given of risks that are uncorrelated with the pricing kernel but are kernel priced. These examples lead to new definitions for risks that are not kernel priced in correlation terms. Additionally we show that the pricing kernel itself viewed as a random variable is strongly negatively kernel priced implying in particular that all monotone increasing functions of the kernel receive a negative risk premium. Moreover the equivalence class of the kernel under increasing monotone transformations is unique in possessing this property.

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M. Atlan, D. Madan, and H. Geman: “Marc Yor and mathematical finance,” pp. 79–90 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

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