Questions of the kind that follow will appear on the Midterm Exam: they require the use of simple no-arbitrage arguments which the problems that follow will teach you to use. You will find MM Lectures 9 and 10 useful. We also solve for a contingent market equilibrium of an economy with log utility functions.
The other securities traded are the riskless bond and options on the stock expiring at dateAssignment Project Exam Help T.
(b) Prove the put-call parity relation for European options. Give a geometric interpretation of this result using the figure in (a).
(c) Show that the following put-call inequality holds for American options
(d) Show that if K1 ≤ K2 then Qct1 ≥ Qct2.
(e) Show that Qbt(K2 − K1) ≥ Qct1 − Qct2.
(f) Show that the price of a call option is a convex function of its striking price, i.e. if K1 and K2 are the striking prices of two different options and if a third option has a striking price Kλ = λK1 + (1 − λ)K2 then Qctλ ≤ λQtc1 + (1 − λ)Qct2 with obvious notation. [Hint: find a portfolio which gives a larger payoff than the option with striking price Kλ]. Explain the intuition for the result.
Assignment Project Exam Help
(a) Show that there are no arbitrage opportunities if and only if d < R < u. Interpret this condition.
(b) Exhibit an arbitrage opportunity when (a) is violated.
(e) Show that the formula in (d) can be written as
, with µu > 0, µd > 0, and µu + µd = 1
How do you interpret this expression?
(f) Here’s another way of pricing the call. Find the portfolioAssignment Project Exam Help(∆,B) of the stock and the bond
3. Suppose the equity price is = 40 and is expected to go up by 10% or down by 10% for each of the next two three-month periods. Suppose the interest rate is known to be 12% per annum with continuous compounding for each period. Find the value of a six-month European put option with strike price K = 42, and the value of a six-month American put option with the same strike price.
4. In question 5 below we will calculate a contingent market equilibrium: to this end we begin by deriving a preliminary result on demand functions of agents with log utility functions vi. There is a simple piece of gymnastics that all of you should know and that only needs to be done once: for ever after life is simple. Consider a one period economy E(|RL,u,ω) with L goods and with spot markets for these goods. Suppose that an agent has a so-called Cobb Douglas utility function for bundles of the L goods i.e. v(x1,…,xL) = γ1 log(x1) + … + γL log(xL). Suppose also that the agent has a vector of initial endowments of the L goods, and faces spot prices for the goods 0. Show that the agent’s demand function (i.e. the solution of his utility maximizing problem over his budget set—it should be clear what his budget set is in this context) is given by f(p,pω) = (f1(p,pω),…,fL(p,pω)) with
Give a simple interpretation of this result and explain the proportion of his income that the agent spends on each good.
5. Now lets calculate the contingent market equilibrium of a stochastic economy in which agents have log preferences. Consider a one-good two period-economy E(|RS+1,u,ω) in which agents have log Bernouilli utility functions
S
ui(xi) = log(αi + xi0) + δ Xρs log(αi + xis), αi ∈ |R, i = 1,…,I
s=1
where δ is the common discount factor of the agents, and 0 < δ ≤ 1. Suppose agents can buy and sell on contingent marketsAssignment Project Exam Help: let π = (1,π1,…,πS) denote the vector of present-value prices. Define
ρ˜ = (ρ˜0,ρ˜1,…,ρ˜S) by ˜ρ0 = 1,ρ˜s = δρs, s = 1,…,S, and ∆ = . Let 1˜ = (1,1,…,1) ∈
and assume that the admissible consumption for agent i in any state is the set of ξ ∈ |R such that αi + ξ > 0.
i’s demand function is given by
Interpret.
(b) Derive the aggregate excess demand function defined by
I
Z(π,ω) = X(fi(π,πωi) − ωi)
i=1
(d) Find the equilibrium prices for the case (c) and show that they can be expressed as simplefunctions of the aggregate output , the aggregate coefficient , the discount factor δ, and the vector of probabilities ρ.
(e) Show that the equilibrium consumption of agent i is of the form
x¯i = ¯biw + ¯ai1˜ where
Find ¯ai. Clearly Pi¯bi = 1. Check that Pi a¯i = 0.
(f) Let Ti(ξ) denote the risk tolerance of agent i defined by
where Ai(ξ) is the risk aversion of the agent (defined in Problem Set#1). Show that for the
log Bernouilli functions
Assignment Project Exam HelpTi(ξ) = αi + ξ
