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We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.
相似文献
相似文献
56.
We study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system. In that application x lies in R, but here we study primarily the periodic case × R S1. We establish existence, uniqueness, and regularity of solutions, locally in time, for positive initial data in H1(S1), and prove the existence of several families of Lyapunov functions for the evolution. From the latter we establish a sharp connection between existence globally in time and positivity preservation: if [0], T*) is a maximal half open interval of existence for a positive solution of the equation, with T* < ∞, then limtT* w(t,·) exists in C1(S1) but vanishes at some point. We show further that if T* > (1 + √3)/16π2 √3 then T* = ∞ and limt∞ w(t,.) exists and is constant. We discuss also some explicit solutions and propose a generalization to higher dimensions. © 1994 John Wiley & Sons, Inc. 相似文献
57.
Pavel Novotný 《Geometriae Dedicata》1994,50(1):53-55
This paper deals with the following kind of approximation of a convex bodyQ in Euclidean space E
n
by simplices: which is the smallest positive numberh
S(Q) such thatS
1 Q S
2 for a simplexS
1 and its homothetic copyS
2 of ratioh
S(Q). It is shown that ifS
0 is a simplex of maximal volume contained inQ, then a homothetic copy ofS
0 of ratio 13/3 containsQ. 相似文献
58.
Pavel Pyrih 《Potential Analysis》1994,3(3):273-281
We study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other.LetI denote the set of complex functionsf: for which there exists a quasi-analytic classC{M
n} containingf. Let denote the set of complex functionsf: for which there exist a fine domainU containing the real line and a function
finely holomorphic onU satisfyingf(x)=
(x) for allx . The power of unique continuation is incomparable in these two cases (I\ is non-empty, \I is non-empty).Research supported by the grant No. 201/93/2174 of Czech Grant Agency and by the grant No. 354 of Charles University. 相似文献
59.
We investigate the statistics of the numberN(R, S) of lattice pointsnZ
2, in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c
1
T,c
2,T],c
2>c
1>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area. 相似文献
60.
A. A. Klopotov Yu. A. Timoshnikov V. A. Plotnikov N. M. Matveeva Yu. A. Sazanov I. A. Lapsker 《Russian Physics Journal》1994,37(8):746-749
The influence of small doses of radiation on the structure-sensitive properties of CuTi amorphous metallic alloy (AMA) is investigated. The AMA exhibit brittle behavior. It is found that, in samples exposed to radiation, the nonlinear behavior of the load-elongation curve is more pronounced and begins sooner. A small change in the fracture morphology in these samples is observed by raster electron microscopy. The additional broadening of the first amorphous maximum and its shift on the x-ray diffraction patterns correlates with increase in the dose. The modification of the structural-relaxation processes on heating the initial and irradiated samples is traced by the acoustic-emission method. The activation energy of these processes is determined.Tomsk State Architectural and Building Academy. Translated from Izestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55–59, August, 1994. 相似文献
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