Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
Module Code MF3053
Module Name Financial Modelling for Actuarial Science 2
Paper Number 1 (for a total of 30 marks)
External Examiner Prof. L. Sodha FIA
Head of School Dr. K. Hayes
Internal Examiner(s) Dr. T. Carroll
Instructions to
Candidates
Answer all questions.
For each question separately, perform all calculations relevant to the question in the Excel workbook provided. Rename your Excel workbook as myname.MF3053.xlsx Carry out calculations for each question in a new sheet. Save your work regularly in your assigned examination folder.
Duration of Paper 1.5 hours.
Special Requirements You may not access the internet during this examination.
1. The price of an asset, currently e100, is modelled by a 4-period binomial model. The price either increases by 4% or decreases by 6% in each period, while the risk-free bond increases by 1% per period.
(i) Determine the arbitrage-free prices of European call options on the asset with strike prices e95, e100, e105, respectively. [3 marks]
(ii) Determine the arbitrage-free prices of European put options on the asset with strike prices e95, e100, e105, respectively. [2 marks]
(iii) Determine the arbitrage-free price of an American put option on the asset with strike price e100. [3 marks]
[Total marks 8]
2. Historical share prices for the company Kingspan are given in Sheet 2 of the Excel file in your exam folder. The prices are for consecutive trading days and are denominated in pounds sterling.
(i) Calculate the standard deviation of the daily returns on Kingspan’s shares, based on this data set. Give your answer correct to 6 decimal places. [2 marks]
(ii) Calculate the annualised historical volatility σˆ of Kingspan. Give your answer as a percentage correct to 2 decimal places. [2 marks]
[Total marks 4]
3. Daily returns for the company CRH from 18 February 2022 to 29 July 2022 are listed in the sheet labelled ‘Qu3 Data’. You are asked to produce a QQ-Plot of the daily returns to test the assumption that asset returns are normally distributed.
(i) Sort the data in Column A. [1 mark]
(ii) Compute the percentile corresponding to each data point. [1 mark]
(iii) Compute the Z-Score for each percentile [2 marks]
(iv) Produce a QQ-Plot of the daily returns. [2 marks]
(v) Interpret the QQ-Plot in the context of the assumption that asset returns are normally distributed. [2 marks]
[Total marks 8]
4. A non dividend-paying Black-Scholes stock has volatility 23%. Its share price at time t is St with S0 = e11.63.
The risk-free rate of interest is 4% per annum compounded continuously.
A Black-Scholes calculator is provided in Sheet ‘Qu4’ of the Excel file.
(i) Calculate the price at time 0 of a call option on the stock that expires in 3 years’ time and has a strike price of e10. [1 mark]
(ii) The market price for the call option in Part (i) is e3.10. Calculate the implied volatility. [1 mark]
A special European option on the stock is available. It has payoff P3 in terms of the share price S3 after three years where the graph of P3 is shown below.
(iii) Write the payoff P3 in terms of the payoffs from standard options. [3 marks]
(iv) Compute the value of the option whose payoff is P3. [2 marks]
(v) An investor has written 100 such options. How many shares should she hold in order to hedge her exposure? [3 marks]
[Total marks 10]