On the probability density function of baskets

The state price density of a basket, even under uncorrelated Black-Scholes dynamics, does not allow for a closed from density. (This may be rephrased as statement on the sum of lognormals and is especially annoying for such are used most frequently in Financial and Actuarial Mathematics.) In this note we discuss short time and small volatility expansions, respectively. The method works for general multi-factor models with correlations and leads to the analysis of a system of ordinary (Hamiltonian) differential equations. Surprisingly perhaps, even in two asset Black-Scholes situation (with its flat geometry), the expansion can degenerate at a critical (basket) strike level; a phenomena which seems to have gone unnoticed in the literature to date. Explicit computations relate this to a phase transition from a unique to more than one "most-likely" paths (along which the diffusion, if suitably conditioned, concentrates in the afore-mentioned regimes). This also provides a (quantifiable) understanding of how precisely a presently out-of-money basket option may still end up in-the-money.Comment: Appeared in: Large Deviations and Asymptotic Methods in Finance, Springer proceedings in Mathematics & Statistics, Editors: Friz, P.K., Gatheral, J., Gulisashvili, A., Jacquier, A., Teichmann, J., 2015, with minor typos remove

The Heston Riemannian distance function

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometrical approach is based on the study of the Heston geodesics, while the analytical approach exploits the links between the Heston distance function and the sub-Riemannian distance function in the Grushin plane. For the Grushin plane, we establish an explicit formula for the Legendre-Fenchel transform of the limiting cumulant generating function and prove a partial large deviation principle that is true only inside a special set

Near infrared spectrophotometry and the 2.2 micron galactic survey

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An HDTV encoder for PALplus transmissions, supplemented with a digital residual signal

This paper discusses the design of a video encoder which will enable PALplus compatible HDTV transmission, by embedding a digital residual component within the standard analogue PALplus television signal. The design strategy includes a 2-D diagonal prefilter, together with a QMF band-splitting filter pair, from which the residual is initially derived. The performance of these filters is reviewed, especially in respect of the errors introduced by aliasing and crosstalk distortion. A number of different digital modulation techniques are then examined for the transmission of the residual signal and a qualitative appraisal of their effectiveness presente