Static hedging with options. Consider a parametrised family of contingent claims with the payoff Y (α) at time T given by the following expression Y (α) = min 􏰀α, β + 2|β − ST | − ST 􏰁 where a real number β > 0 is fixed and the parameter α is an arbitrary real number such that α ≥ 0. (a) Foranyfixedα≥0,sketchtheprofileofthepayoffY(α)asafunctionofST ≥0 and find a decomposition of Y (α) in terms of the payoffs of standard call and put options with maturity date T (do not use a constant payoff). Notice that a decomposition of Y (α) may depend on the value of the parameter α. (b) Assume that call and put options with all strikes are traded at time 0 at some finite prices. For each value of α ≥ 0, compute the arbitrage price π0(Y (α)) at time t = 0 for the claim Y (α) using the prices at time 0 of call and put options and a suitable decomposition obtained in part (a). (c) For any α > 0, examine the sign of an arbitrage price of the claim Y (α) in any (not necessarily complete) arbitrage-free market model M = (B,S) with a finite state space Ω. Justify your answer. (d) Consider a complete arbitrage-free market model M = (B, S) defined on some finite sample space Ω. Show that the arbitrage price of Y (α) at time t = 0 is a monotone function of the variable α ≥ 0 and find the limits limα→0 π0(Y (α)), limα→∞ π0(Y (α)) and limα→3β π0(Y (α)).

e place ourselves withinthe setup of the Black-Scholes market model M = (B, S) with a unique martingalemeasure ˜P. Consider a European contingent claim Y with maturity T and thefollowing payoffY = γST − max (ST , L)where L = erT S0 and γ > 0 is a real number. We take for granted the Black-Scholespricing formulae for the call and put options.(a) Sketch the profile of the payoff Y as a function of the stock price ST at time Tand show that Y admits the representation Y = γST − CT (L) − L where CT (L)denotes the payoff at time T of the European call option with strike L.

Consider a contingent claim economy with two consumers (A and B), two states tomor-row and one good in each state. Agents have the following utility functions:uA(x0A, x1A, x2A) = ln x0A + ln x1A + ln x2Aand uB (x0B , x1B , x2B ) = ln x0B + ln x1B + ln x2BAs for endowments,ωA = (1, 1, 2) and ωB = (1, 2, 1).(a) (1 point) Write down the state-by-state budgent constraints for A and B. Use bsAand bsB to denote the units of contingent claims consumers buy for each state s.(b) (1 point) State the utility maximization problem for A and B without the use ofbsA and bsB mentioned above.(c) (3 points) Derive consumers’ demand functions with the Lagrangian method.(d) (3 points) Find the market equilibrium for this contingent claim economy.(e) (1 point) Verify that the equilibrium allocation is Pareto efficient.

The underwriting function has increasingly included the _____ dealMultiple choice question.best effortsbought

Question 1Joe’s preferences are described by the following utility functionU (x, y) = xαyβwith α > 0 and β > 0.(a) Let I denote Joe’s income, and px and py denote the prices of good x and y, respectively.Find Joe’s optimal consumption bundle.

Firms A and B choose how much of a a homogenous good to produce at a marginal cost of1. The inverse demand function is p = 25 − qA − qB where qA is the output choice of firm A and qBis the output choice of firm B. A firm’s action is its output choice and can be any number greaterthan or equal to zero.Eliminating all strongly dominated actions by the two firms in the first round. In the gamethat remains eliminate all strongly dominated actions in a second round. What is the set of actionsthat are still available to each of the two firms after the second elimination round?[Write your answer as an interval; e.g.: [0.2, 0.4] or (0.3, 0.5], etc.