proof of the swaption formula in the black 1976 model

... each of the terms in the trace formula In the second paper [II], we established a key reduction in the proof of one of the global theorems In this paper, we shall complete the proof of the theorems ... the relevant coefficients in the trace formula The proof of Global Theorem is indirect It will be a consequence of a parallel set of theorems for all the other terms in the trace formula, together ... culminate in Section with the proof of the global theorems We have fixed the integers dder and rder in (1.1) In this section, we shall apply the induction hypotheses (1.2)–(1.5) to the terms in the...

Ngày tải lên: 05/03/2014, 23:20

... journal of combinatorics 13 (2006), #R110 In Sect 2, we derive the main formula for the entries of the ground state of the O(1) loop model corresponding to link patterns with three sets of nested ... ANR-05BLAN-0029-01 Appendix A Proof of symmetry under interchange of spectral parameters The purpose of this appendix is to provide an explicit proof of the symmetry of the partition function of weighted plane ... polynomial of the {αi } of degree at most a in each, of the {βi } of degree at most b in each, and of the {γi } of degree at most c in each We now rewrite Eq (2.3) in the case when zi is the last parameter...

Ngày tải lên: 07/08/2014, 13:21

... (5) The proof of (3) is completely combinatorial, the proof of (4) uses nothing more than the fundamental theorem of algebra, and the proof of (5) is a simple geometric argument Proof of (3) The ... combinatorics of Young diagrams Other proofs have the disadvantage of not being particularly combinatorial We offer the proof below as a simple combinatorial approach for the non-specialist In ... Garsia for pointing out the q, t-analog of the hook formula in [GT96] This suggested that the recursion for the hook formula should have an expression in terms of a rational function involving content...

Ngày tải lên: 07/08/2014, 15:23

... vary the argument substantially), and it is the purpose of the present note to try to lay bare the basic mechanism of Kouba’s proof Proof of Theorem The proof of the Theorem will be based on the ... paper The present author wishes to emphasize that the proof in the present paper differs from Kouba’s proof only in the setting and the way in which the coefficients for Kouba’s linear form are obtained ... direct proof of Theorem was given by Michalek in [5] who explicitly remarks that the proof works for integral domains as well Moreover, the differences ± (s − s′ ) : {s, s′} ∈ S2k in the proof below...

Ngày tải lên: 08/08/2014, 11:20

... turning to these, it is worth pointing out to the reader that the numerical solution of partial differential equations is something of an art form, containing many hidden pitfalls We explain the ... δT 2 σ This is precisely the Jarrow–Rudd version of the binomial model, summed up in equation (7.6) The binomial model and the explicit finite difference solution of the Black Scholes equation are ... from a seemingly unrelated line of reasoning This should of course be of no great surprise: r The binomial model is a graphical way of approximating the probability density function of a stock...

Ngày tải lên: 25/10/2013, 19:20

... available since February 2004 In the summer 2005 an alternative proof of Theorem 2.6 appeared in the preprint [19], and a proof of Conjecture 2.5 was announced by C Schultz Proof of Theorem 1.3 ... Completion of the sketch of the proof of the Lov´sz conjecture Cona sider one of the two maps ι : K2 → C2r+1 mapping the edge to the Z2 -invariant edge of C2r+1 Clearly, ι is Z2 -equivariant Since ... s ∈ S, let ← be the leftmost element of the first gap s → to the left of the gap containing s, and let − be the leftmost element of the s first gap to the right of the gap containing s For x, y...

Ngày tải lên: 06/03/2014, 08:21

... proof of the main theorem of the chapter A PROOF OF THE KEPLER CONJECTURE 1089 Proof of Theorem 2.9 The rules defining the Q-system specify a uniquely determined set of simplices The proof that they ... structure of the proof Statement of theorems Basic concepts in the proof Logical skeleton of the proof Proofs of the central claims Construction of the Q-system 2.1 Description of the Q-system ... the origin, the simplices of the Q-system having a vertex at the origin, the V -cell V C(D) at the origin, the Voronoi cell Ω(D) at the origin, and so forth In fact, the indexing sets in the definition...

Ngày tải lên: 22/03/2014, 20:20

... stress-coping model Despite the exclusion of some of the aspects of the model, the power to detect a model without a good fit remained small; < 0.20 [27] As a consequence, the low values of the test ... compare results of these studies to the results found in the present study In the present study, tentative support was found for the notion that the stress-coping model reflects most of the relationships ... appraisal of the disease, coping, and quality of life are all significantly related to each other This part of the extended stress-coping model might be seen as a representation of the theory of Lazarus...

Ngày tải lên: 18/06/2014, 22:20

Ngày tải lên: 05/08/2014, 09:46

... take each other in the final configuration By inclusion-exclusion, we see that the resulting configurations in which the set S of rooks on B is nonempty cancel out of the above sum, because they are ... positive integer Add x extra rows to [d] × [d] Then rk (x + d − k)d−k is the number of ways of first placing k rooks on B and then placing d − k more rooks anywhere (i.e., on B, B or on the extra ...  
      is, and the purpose of this note is to provide such a proof The knowledgeable reader will recognize that the main idea is borrowed from [4] Proof Observe that d (−1) R(B;...

Ngày tải lên: 07/08/2014, 06:20

... the resulting polytopes will meet each of the original sets Proof of Lemma If we can find a set of f(d, g) points that meets the convex hull of every subset of g|Q| of the elements of Q, it will ... dimensions by induction on dimension, using the following procedure: Find a hyperplane H such that half the points of Q are on each of its sides Consider the |Q|2 /4 points of intersection of line segments ... every one of our original polytopes will have to contain a proportion g of our points This concludes our proof of the second lemma Proof of Lemma If one of our polytopes has hM clones, then the fact...

Ngày tải lên: 07/08/2014, 06:22

... two latin squares L, M of order n G-isotopic if there exists g ∈ G such that Lg = M The orbit LG of L under G is the electronic journal of combinatorics (1998) #R28 the G-isotopy class of L The ... Although the truth of the extended conjecture is still unknown for n = 9, the first even value of n which is not of the form given in Corollary is 50, whereas the previous first unknown case of the ... the electronic journal of combinatorics (1998) #R28 Since any latin square can be transformed into a fixed diagonal latin square by a permutation of rows, and since permuting rows does...

Ngày tải lên: 07/08/2014, 06:22

... s ≥ The approach above no longer works, since now the points of intersection need not occur in the same order on X and Y Suppose first that the intersection contains an endpoint of one of the ... path of length n For example, the complete graph K4 is an edge-disjoint union of two paths of length 3, each path meeting the other in four vertices The cycle C6 is the edge-disjoint union of two ... union of two paths of length with common endpoints In the first case, the graph has twelve paths of length 3; in the second there are six such paths The two-path conjecture arose in a problem on...

Ngày tải lên: 07/08/2014, 06:23

... denote the tableau obtained from T by applying the Bender-Knuth involution σk to the subtableau T

Ngày tải lên: 07/08/2014, 06:23

... maintains some of the combinatorial properties of the classical formula, one might hope for a simpler formula For this reason it may be of interest to investigate other q-analogues of the sum of ... that the set S consists of pairs of partitions with no primed parts In the definition of φ−1 when there is a primed integer in the image, we consider only the value of the integer and ignore the ... about a proof described to him by N.M Ferrers The Ferrers shape of a partition is an array of boxes, left justified, in which the number of boxes in the first row is equal to the size of the first...

Ngày tải lên: 07/08/2014, 08:20

... tableaux In several of these rules, the set being counted is the lattice points in a polytope (and in fact the polytopes from the different rules are linearly equivalent) The first was in the unpublished ... given in [KT2] the electronic journal of combinatorics 11 (2004), #R61 In this paper we give a new self-contained proof of this lattice-point-counting rule, in its incarnation as counting the hives ... is either a or a + 1, and that binary choice determines the rest of the strip Proposition In the hive ring, bλbωi = bλ+π π∈{0,1} n , π=i λ+π ∈ Zn dec In other words, the hive ring obeys the Pieri...

Ngày tải lên: 07/08/2014, 08:20

... complement of T By the reasoning above, the average of the the electronic journal of combinatorics 11 (2004), #R62 operators R(g) on the space N is zero and the proof may proceed by applying the tail ... satisfying pi (ρ) ≤ ½ Recalling that R contains a single copy of the trivial representation and observing that the operator g R(g) has rank (indeed, in the basis above, each entry in the corresponding ... below, the elements of graph theory, representation theory, and probability theory required for the statement of the theorem and the subsequent proof The exposition here is primarily for the purposes...

Ngày tải lên: 07/08/2014, 08:20

... whose length is quadratic in terms of the length of the input objects is shown to be log-concave The proof of our claim 2.1 The outline of the proof It is easy to see that the sequence (ak )0≤k≤m ... , and therefore, it implies the existence of the maps fn,k,l for k < l − 2 That will complete the induction step of our proof 2.2 The details of the proof It is time that we carried out the strategy ... is an injection Proof: First, it is clear that fn,k,k+2 maps into Ik+1 × Ik+1 since both rules increase the number of inversions of the first permutation by one, and decrease the number of inversions...

Ngày tải lên: 07/08/2014, 08:22

... center of the right-hand edge of the ith row Namely, each step of the path is from the center of a domino edge (where a domino is regarded as having six edges of unit length) to the center of another ... is the number of inversions of σ the electronic journal of combinatorics 12 (2005), #R18 Using the technique of a sign-reversing involution over a signed set, we prove that the (1) (1) cardinalities ... reduced to the computation of the determinant of a Hankel matrix of order n that involves large Schr¨der numbers In this o note we give a proof by means of Hankel determinants of the large and...

Ngày tải lên: 07/08/2014, 08:22

... ], and achieves the proof of Lemma Before proceeding with the proof of Theorem 1, let us recall that the total degree of a polynomial P from K[X1 , , Xm ] is the largest value of d1 + d2 + · ... coefficients in P Proof of Theorem For each j in {1, , m}, we may assume that card (Sj ) = nj + (by discarding the extra elements if necessary,) then, using Lemma 2, we find a familly S of scalars ... (d1 ,d2 , ,dm )∈D Finally, the conclusion of the theorem follows since m λS11 λS22 · · · λSm P (t1 , t2 , , tm ) = Φ(P ) = t t t (t1 , ,tm )∈S1 ×···×Sm This ends the proof of Theorem References...

Ngày tải lên: 07/08/2014, 21:21