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By S H Man Th May and Or Aω

Abstract

Lets fix some notation for the rest of this article. We assume that we have a binomial tree and the branching is given by the following diagram. The tree starts at (0, 0). If we are at node (t, j), then it can go to node (t + 1, j + 1) or (t + 1, j). The following notation will be used throughout this article. • D(1), D(2),..., be the (default-free) discount factors over [0, 1], [0, 2],... respectively. We can think of D(t) as the time 0 value of a default-free zero-coupond bond with maturity t and face value of $1. • r(t, j) denotes the (risk free) spot rate at (t, j) over [t, t + 1]. • D(t, j) be the discount factor at (t, j) over [t, t + 1]. • B(t) be the time t value of a deposit account, earning risk free interest, with B(0) = 1. • For n ≥ 1, let Ωn = {(ω1, ω2,..., ωn) | ωi = H or T} Also if 0 ≤ t ≤ n and ω ∈ Ωn, we define #t(ω) to be the number of H in the first t entries of ω. 2 Some results on measureable function over a finite set Through out this section, Ω is a finite set. Let A1, A2,..., Ak be a partition of Ω. Clearly, {∪i∈IAi | I ⊆ {1, 2,..., k}} (1) is a σ-algebra of Ω. We now show that every σ-algebra on Ω could be constructed in this way. Lemma 2.1 Suppose Ω is a finite set and F is a σ-algebra. For any ω ∈ Ω, defin

Topics: is the smallest element in F that contains ω1
Year: 2014
OAI identifier: oai:CiteSeerX.psu:10.1.1.416.4095
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