Problem 1: Properties of Options (8 marks)

The price of a European put that expires in six months and has a strike price of $100 is $3.59. The underlying stock price is $102, and a dividend of $1.50 is expected in four months. The term structure is flat, with all risk-free interest rates being 8% (cont. comp.).

a.What is the price of a European call option on the same stock that expires in six months and has a strike price of $100? [1 marks]

b.Explain in detail the arbitrage opportunities if the European call price is $6.1. How much will be the arbitrage profit? [3.5 marks]

c.Explain in detail the arbitrage opportunities if the European call price is $8.8. How much will be the arbitrage profit? [3.5 marks]

Problem 2: Option Valuation (18 marks)

In this question, you need to price options with various approaches. You will consider puts and calls on a share. Please read following instructions carefully:

The spot price of this share will be determined by your student number. You need to use the last digit of your student number. The spot price of the share will be (last digit of your student number*10+6). For example, if the last digit of your student number is 5, the spot share price will be 5*10+6=56. If the last digit of your student number is 0, please replace it with 4, i.e. the spot price will be 4*10+6=46.

The strike price of the options will be the share price you just calculated +2. For example, if the share price you calculated based on your student number is 56, the strike price of the options will be (56+2)=58.

Based on this spot price and this strike price as well as the fact that the risk-free interest rate is 6% per annum with continuous compounding, please undertake option valuations and answer related questions according to following instructions:

Binomial trees:

Additionally, assume that over each of the next two four-month periods, the share price is expected to go up by 11% or down by 10%.

a.Use a two-step binomial tree to calculate the value of an eight-month European call option using the no-arbitrage approach. [2.5 marks]

b.Use a two-step binomial tree to calculate the value of an eight-month European put option using the no-arbitrage approach. [2.5 marks]

c.Show whether the put-call-parity holds for the European call and the European put prices you calculated in a. and b. [1 mark]

d.Use a two-step binomial tree to calculate the value of an eight-month European call option using risk-neutral valuation. [1 mark]

e.Use a two-step binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation. [1 mark]

f.Verify whether the no-arbitrage approach and the risk-neutral valuation lead to the same results. [1 mark]

g.Use a two-step binomial tree to calculate the value of an eight-month American put option. [1 mark]

h.Calculate the deltas of the European put and the European call at the different nodes of the binomial three. [1 mark]

Note: When you use no-arbitrage arguments, you need to show in detail how to set up the riskless portfolios at the different nodes of the binomial tree.

Black-Scholes-Merton model:

Using the information given above regarding the spot and strike price, risk-free rate of return and the fact that the volatility of the share price is 18%, answer following questions:

i.What is the price of an eight-month European call? [1 mark]

j.What is the price of an eight-month American call? [1 mark]

k.What is the price of an eight-month European put? [1 mark]

l.How would your result from k. change if a dividend of $1 is expected in three months? How would your result from k. change if a dividend of $1 is expected in ten months? [2 marks]

Note for calculations with the BSM model: Keep four decimal points for d1 and d2. Use the Table for N(x) with interpolation in calculating N(d1) and N(d2).

Finally,

m.Compare the results you obtained for the prices of European puts and calls using binomial trees and Black-Scholes-Merton model. How large are the differences when expressed as a percentage of the spot price of the share? Provide a possible explanation for these differences. [2 marks]

Problem 3: Derivatives Valuation (6 marks)

A stock price is currently $36. During each three-month period for the next six months it is expected to increase by 9% or decrease by 8%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off (max[(40-S_T),0])² where is the stock price in six months.

a.What are the payoffs at the final nodes of the tree? [1 mark]

b.Use no-arbitrage arguments (you need to show how to set up the riskless portfolios at the different nodes of the binomial tree). [2 mark]

c.Use risk-neutral valuation. [1 mark]

d.Verify whether both approaches lead to the same result. [1 mark]

e.If the derivative is of American style (S_T in the payoff function refers to the stock price when the option is exercised), should it be exercised early? [1 mark]

Problem 4. Value at Risk [18 marks]

This is a Bloomberg-based exercise.

Suppose you hold a portfolio consisting of a $350,000 investment in company A stock and a $650,000 investment in company B stock. Companies A and B are ASX50 constituent companies (see https://www.asx50list.com/).

Rules for the choice of companies A and B. Company A is ranked as the sum of all the digits in your student number, and company B is ranked as the sum of the last two digits of your student number. The ranking is based on market cap, with 1 indicating the largest company at the ASX and 50 the fiftieth largest company. If the sum of all the digits, in case of A, is more than 50, you will choose the company with the rank of (the sum - 50). In case of B, if the sum is 0, you then choose the rank of 1. For example, if your student number is s9999888, company A should be ranked as (9+9+9+9+8+8+8)=60-50=10; company B should be ranked as 8+8=16. If your student number is s1234500, you should choose company A with the rank of (1+2+3+4+5+0+0)=15, and company B with the rank of (0+0)=0+1=1. Please provide a screenshot of the ranking list.

Once you identify the two companies, you can retrieve relevant data from Bloomberg. Please use the instructions available under L@G/Course contents/Bloomberg remote access and timetable. In your course site, there is a section called "Bloomberg Recordings", where you can find some instruction videos which may be helpful to you in navigating the Bloomberg system. If you cannot access Bloomberg terminal, alternatively you can go to Yahoo! Finance to retrieve the relevant data.

You can now start performing the following tasks:

a.Search for the stocks of these two companies. Download historical daily price data over the last 501 trading days (approx. 2 years). [1 mark]

b.Calculate with Excel the daily returns of the stocks of companies A and B. [1 mark]

Now you need to estimate VaR with the two approaches you learned in class.

Historical simulation:

c.Based on the 500 returns for each stock calculated in b., calculate 500 alternative scenarios for the $ value of the $350,000 investment in company A stock and the $650,000 investment in company B stock, respectively. Sum these two for each scenario to obtain 500 simulations for the $ value of your portfolio consisting of stock A and stock B. [2 marks]

d.Based on the 500 scenarios for the $ value of your portfolio consisting of stock A and stock B, calculate the 500 alternative gains/losses for your portfolio. [1 marks]

e.Calculate the 5-day 99% VaR for this portfolio. What does it mean? [2 marks]

f.Briefly discuss the advantages/disadvantages of this approach. [1 mark]

Model-building approach:

g.Calculate the standard deviations of the stocks' returns over the last two years. [1 mark]

h.Calculate the coefficient of correlation between the stocks' returns. [1 mark]

i.Compute the 5-day 99% VaR for this portfolio. What does it mean? [2 marks]

j.By how much does diversification reduce the VaR? Also provide a brief comment on the reduction. [2 marks]

k.Briefly discuss the advantages/disadvantages of this approach. [1 mark]

Finally,

l.Compare the results of both approaches. Provide possible explanations for the differences. [2 marks]

m.Briefly discuss the usefulness of VaR. [1 mark]