Topics in Financial Economics:
Theories and International Finance
Question 1: (CAPM)
In this question we look at the properties of the minimum variance portfolio. Suppose you have the opportunity to invest in three stocks that A, B and C where the standard deviation of returns are given by: s A = 0.15, s B = 0.20, sC = 0.25 and s A,B = s A,C = sB ,C = 0
What is the minimum variance portfolio and what is its variance and covariance with each of the three assets?
Redo part a) for arbitrary standard deviations s A , s B ,sC
Prove that the covariance between the minimum variance portfolio (MV) and any other portfolio equals the variance of the minimum variance portfolio. While one can rely on the specific expression for the minimum variance portfolio show this in by way of contradiction. That is, assume that there is an asset for which the above is not true and consider a portfolio that consists of x in the minimum variance portfolio and 1-x in the other portfolio.
Question 2:
We are interested in pricing a 1-year put option on cisco with a strike price of . Currently CISCO is trading at 80 and the risk free rate is 5% per annum. In addition we assume that (i) CISCO will not pay any dividend and (ii) there are no other derivatives that are trading including other call or put options.
Suppose that we follow a Binomial model and CISCO may either appreciate by 50% or depreciate by 25%.
What should be the price of the option based on replication?
What should be the price of the option based on risk neutral pricing?
In this part we no longer assume a Binomial structure and future stock price can take arbitrary non-negative values. For each of the following possible prices for the put option determine whether it admits an arbitrage strategy.: (i) 100 (ii) 60 (iii) 10
In case there is an arbitrage, describe explicitly such strategy and calculate arbitrage profit today as well as payoff diagram in 1 year; in case there is not you are not required to prove it.
Suppose now that there is also a call option with a strike of that costs $10. For each of the following possible prices for the put option determine whether it admits an opportunity strategy: (i) 20 (ii) 30
Again as in part b), in case there is an arbitrage, describe explicitly such strategy and calculate arbitrage profit today as well as payoff diagram in 1 year; in case there is not you are not required to prove it.