First Year Interest Group Seminar - Assignment 4 Problems | N 1, Assignments of Health sciences

Download First Year Interest Group Seminar - Assignment 4 Problems | N 1 and more Assignments Health sciences in PDF only on Docsity! HW: 4 Course: Introduction to Financial Economics Page: 1 of 3 University of Texas at Austin HW Assignment 4 Problem 4.1. Show that the set Q(D̄) of arbitrage-free prices of the contract with dividend process D̄ is given by Q(D̄) = {Eπ[D̄|·] : π ∈M}, where M denotes the set of all present-value vectors. Conclude that a financial market is complete if and only if each dividend process admits exactly one arbitrage-free price. Problem 4.2 (Prices as discounted expectations). Let F be an arbitrage free financial market, and let π be a process of present-value prices. For each non-terminal node ξ, define δ(ξ; π) = 1 π(ξ) ∑ ξ′>cξ π(ξ′) (where the sum is taken over all children of ξ), and set r(ξ; π) = 1/δ(ξ; π)− 1. The (not-necessarily-positive) quantity r(ξ; π) is called the π-implied short rate at ξ. (1) Suppose that there exists a short-lived bond for the node ξ with price q(ξ). Then δ(ξ; π) = q(ξ), for all π ∈M. (2) Give an example of a situation where r(ξ; π) depends on π. (3) A process X is called an N -martingale density, if X(ξ) > 0 for all ξ ∈ N and X(ξ) = ∑ ξ′>cξ X(ξ′), for all non-terminal nodes ξ. An N -martingale density X is said to be normalized if X(ξ0) = 1. Show that for each normalized N -martingale density there exists a unique probability - denoted by QX - on (Ω,AT ) with the property that for A ∈ At, t = 0, . . . , T , we have QX(A) = ∑ X(ξ), where the sum is taken over all nodes ξ such that (t, ω) ∈ ξ for some ω ∈ A. (4) A process Y is called N -predictable if Y (ξ) = X(ξ−) for some (adapted) process X and all non-initial nodes ξ (where ξ− denotes the parent of the non-terminal node ξ). Characterize all processes X which are both N -predictable and N -martingale densities. (5) Show that, for each π ∈M, there exists a unique normalized martingale density Xπ and a predictable process β with βπ(ξ0) = 1 such that π(ξ) = β(ξ)Xπ(ξ). The process β is called the π-implied discount process, and the measure Qπ = QXπ is called the martingale measure (corresponding to π). What is the relation between β and r? (6) For a dividend process D, let D̂ be the π-discounted version of D, which is given by D̂(ξ) = β(ξ)D(ξ). Let the random variable C on (Ω,AT ) be defined by C(ω) =∑ D̂(ξ), where the sum is taken over all ξ such that (t, ω) ∼ ξ for some t = 0, . . . , T . Show that Eπ[D|ξ0] = EQπ [C], Instructor: Gordan Žitković Semester: Summer 2009 HW: 4 Course: Introduction to Financial Economics Page: 2 of 3 where the right-hand side is the (proper) expectation of the random variable C with respect to the measure Qπ. Problem 4.3 (The Cox-Ross-Rubinstein Model). In this model T ≥ 1 and Ω = {(x1, . . . , xT ) : xi ∈ {1− b, 1 + a}, i = 1, . . . , T}, where a, b > 0, b < 1. Set X0(ω) = q0, and Xt(ω) = Πti=1xi, where ω = (x1, . . . , xT ), for t = 1, . . . , T . Let At be the algebra generated by X0, X1, . . . , Xt (since X is, by construc- tion, adapted to (A0, . . . ,AT ) we will interpret it as a function on the nodes without explicit mention). The financial market F (which defines the Cox-Ross-Rubinstein model) is based on Ω, the filtration (A0, . . . ,AT ), and 2T contracts: • A “stock”: with the dividend process D given by Dstock(ξ) = { X(ξ), ξ is terminal 0, otherwise, • A system of short-lived bonds: for each non-terminal node ξ, there is a contract Dbond−ξ, issued at ξ, with the dividend process Dbond−ξ(ξ ′) = { 1, ξ′ >c ξ, 0, otherwise, The prices of these contracts are fixed and given by • qstock(ξ) = { X(ξ), ξ is not terminal 0, otherwise, • A system of “short-lived bonds” . For each non-terminal node ξ, there is a contract Dbond−ξ, issued at ξ, with the dividend process qbond−ξ(ξ ′) = { 1/(1 + r), ξ′ = ξ, 0, otherwise, for some r > 0. Test your understanding of the whole theory by answering the following questions: (1) For which values of a, b and r is the market arbitrage-free? (2) Assume from now on that a, b and r are such that the market is arbitrage-free. Show that there is a unique process of present-value prices, and that the market is complete. (3) Decompose the (unique) π into a product of a normalized martingale density and the implied discount process. In particular, what is the value of Qπ(ω) for ω ∈ Ω? (4) Let D̄ be a divided process such that (and we assume this only for simplicity) D̄(ξ) = 0, unless ξ is terminal. Assume, further, that D̄T is AT -measurable, i.e., that there exists a function φ : R → R such that D̄(ξ) = φ(X(ξ)), for non-terminal ξ. Show Instructor: Gordan Žitković Semester: Summer 2009