- 1. A grocery company owes a wholesaler a debt of $1,800 which is 12 months overdue. The grocery company holds two zero-interest bonds, one for $800 due in 2 years and one for $1,200 due in 12 months, which it offers the wholesaler in settlement now. If money is worth 4% compounded annually, who is the loser in such a settlement, and by how much?
2. Consideradiscrete-timefinancialmarketconsistingofariskyassetandarisk-freeasset.Theprices of the risky and risk-free assets at time t (where t = 0,1,2,...) are denoted by S(t) and A(t) dollars, respectively. Let the prices A(0) = 100, A(1) = 105, S(0) = 100 and
{
110, with probability p, S(1)=
90, with probability 1 − p ,
where 0 < p < 1. Suppose that a European call option and a European put option are associated with
the risky asset. Applying the No-Arbitrage Principle,
(a) computethepricec(0)ofacalloptionwithstrikeprice$100andexercisetime1, (b) computethepricep(0)ofaputoptionwithstrikeprice$100andexercisetime1.
3. AEuropeancalloptionandaEuropeanputoptiononthesamestockbothexpirein12months,both have a strike price of 100 Euros, and both sell for the price of 5 Euros. If the nominal continuously compounding interest rate is 5%, and the current stock price is 100 Euros.
(a) Isthereanarbitrageopportunity?Explainbrieflyhowyouderiveyourconclusion. (b) Identifythearbitragestrategy,ifitexists.
4. In this question, we study daily close prices of the main stock market index in New Zealand (NZ50) ranging from 3 January 2019 up to 29 November 2019 from the Yahoo finance website https://finance.yahoo.com/quote/%5ENZ50?ltr=1.
(a) Download the data from the website. Then, import the data into SAS and plot the daily close price data.
(b) Calculate the daily returns of the NZ50 in SAS.
(c) Calculate the standard deviation of the daily returns of NZ50 in SAS.
(d) Calculate the historical volatility and the realized volatility of NZ50 daily returns in SAS.
5. ConsiderarandomvariableSthatfollowsthestochasticprocess
d S = μd t + σd z .
For the first three years, the parameter values are μ = 1 and σ = 3; for the next three years, the
parameter values are μ = 2 and σ = 1. The initial value of the variable S0 is 100.
(a) What is the probability distribution of the value of the variable S at the end of year 3?
(b) What is the probability distribution of the value of the variable S at the end of year 6?
(c) Write a SAS function to generate one sample path of S with △t = 1 week.
(d) Use the Monte Carlo simulation with SAS to estimate the mean and variance of S at the end of year 3.
(e) Use the Monte Carlo simulation with SAS to estimate the mean and variance of S at the end of year 6.
