ETC4460/ETC5460 FINAL EXAM FOR 2019
INSTRUCTIONS TO STUDENTS
1. For full marks, all parts of Question 1, Question 2 and Question 3 must be answered.
2. Not all question are worth an equal number of marks.
3. There are 80 marks available in total on the exam paper.
4. There is one Appendix containing supplementary information for Question 1. This Ap- pendix is located on page 4, following the Önal part of Question 1.
5. A statistical table for the Normal distribution is located on page 9.
Question 1 (25 marks)
Consider the following two continuous time processes, denoted as ëModel Aíand ëModel Bí, respectively, with each proposed for modelling a monthly stock price of a given Önancial asset, Pt:
and
Model A: dPt = ?dt + ?dwt (1.1) Model B: dPt = ?Ptdt + ?Ptdwt: (1.2)
a. For each of the two models, Model A in (1.1) and Model B in (1.2), produce an appropriate description for the probability distribution of the random increment ?Pt = Pt ? Pt??t, where ?t > 0. In each case, provide a justiÖcation for your description, including when the description may be considered appropriate.
[5 marks]
b. Which model, Model A in (1.1) or Model B in (1.2), would be relatively more appropriate for the task of modelling the monthly stock price series? Provide two reasons to support your judgement.
[6 marks]
c. Using only Model B in (1.2), explain how one could use the sample information provided in Table 1 to estimate annual parameters ? and ?. (Note that you are not required to calculate the numerical estimates, just explain how to produce them.)
[6 marks]
d. Explain the concept of the Value at Risk (VaR) measure for an investment of $100,000 in the Önancial asset associated with the monthly price series shown in Figure 1 and summary statistics provided in Table 1. Again using only Model B in (1.2), detail the calculation required to report a 5% VaR for the investment associated with a single one-month-out- of-sample period. (Note that again here you are not required to calculate the numerical estimates, but you must explain in detail how to produce them.)
- Question 1 continues on next page -
[8 marks]
Question 1 (continued)
Appendix: Supplementary information for Question 1
Figure 1: Monthly share price Pt series (top panel), with the corresponding logarithmic returns rt (bottom left panel) and simple returns Rt (bottom right panel).
Series
sample size (T)
sample average
sample standard deviation value of series in Önal month T
Price 50 $24.10 $3.205 $29.54
log return 49 0.0072 0.0338 0.0238
simple return 49 0.0078 0.0338 0.0241
Table 1: Summary statistics for Pt, a monthly share price series (column two), and for the corresponding log price return series rt (column three) and simple price return series Rt (column four). For each price and returns series, the sample size (T ), sample average, sample standard deviation (with T ? 1 used in the denominator) and the value of the series in the Önal month T are each shown in rows two through four, respectively.
- End of Question 1 -
Question 2 (30 marks)
Consider an ItÙ process for the price of a Önancial asset, Pt, given by
dPt =?(Pt;t)dt+?(Pt;t)dwt; (2.1) where wt denotes the standard Wiener process.
a. Describe in detail how to obtain an appropriate representation for ct = c(Pt;t). In your answer, list any assumptions required of the function c (?; ?) in order for this representation to be valid.
[6 marks]
DeÖne as ct the price of a European call option having strike price K and maturity date T; where T > t. The Black-Scholes option price is given by
where
d1 =
ln(Pt=K)+?rf +0:5?2?(T ?t) ?pT?t ;
ct =Pt?(d1)?Ke?rf(T?t)?(d2);
(2.2)
ln(Pt=K)+?rf ?0:5?2?(T ?t) ?pT?t
p =d1?? T?t;
d2 =
rf denotes the annual risk free rate of interest and ? (?) denotes the cumulative distribution
function (cdf) of the standard normal distribution.
b. Explain what is meant by a European call option having strike price K and maturity date T; where T > t.
- Question 2 continues on next page -
[4 marks]
Question 2 (continued)
c. What assumptions are required to derive the expression in (2.2)?
d. Explain how the Black-Scholes option pricing formula in (2.2) may be derived from con- sidering a portfolio comprised of ?1 European call option contracts and ?2 units of the corresponding underlying asset, and having an overall portfolio value at time t, denoted by Vt: Comment on how the derivation sheds light on why the Black-Scholes option price ct does not depend upon the underlying rate of return associated with the underlying asset, Pt, and why instead options are (at least theoretically) priced at the prevailing ërisk- neutralí rate.
[8 marks]
e. Explain what is meant by Black-Scholes implied volatility, and describe how it may be obtained from each of the following:
i. A single European call option price, denoted by cm;
ii. A collection of n observed European call option prices, denoted by cmi, for i =
1; 2; :::; n.
Detail the additional assumptions (if any) required to obtain these implied volatility quan- tities.
- End of Question 2 -
[4 marks]
[8 marks]
Question 3 (25 marks)
Consider estimation of the Capital Asset Pricing Model of Sharpe (1964) and Lintner
(1965), using the linear regression model given by
(Rt?Ft)= + (Mt?Ft)+"t; fort=1;2;:::;T;
where Rt denotes the percentage return of a Önancial portfolio in period t, Ft denotes the risk-free rate in period t, and Mt denotes the market return in period t.
(3.1)
Note: You may Önd it helpful to refer to yt as the excess portfolio return (Rt ? Ft), and refer to xt as the excess market return (Mt ? Ft), for all t.
a. Explain in detail why the Ordinary Least Squares (OLS) estimator, given by
?b?PT 2
bOLS; OLS =arg min [(Rt ?Ft)? ? (Mt ?Ft)] (3.2)
2R; 2Rt=1
may be viewed as a Generalised Method of Moments (GMM) estimator for the coef- Öcient vector ( ; ), of the linear regression model in (3.1). In particular, comment on the moment assumptions commonly used to produce ?bOLS ; bOLS ? as the GMM estimator of ( ; ).
[8 marks]
b. Explain why the GMM estimator in part a. will be the unique GMM estimator for this situation, and will satisfy the speciÖed sample moment conditions exactly.
- Question 3 continues on next page -
[4 marks]
Question 3 (continued)
Next consider Öxing = 0 in (3.1). Since the vector of excess market returns, (Mt ? Ft), is stochastic, there is concern that perhaps E ["t (Mt ? Ft)] 6= 0. However, two other instrumental variables(IVs)relevanttotheportfolioexcessreturns,denotedasw10 =(w1;1;w2;1;:::;wT;1) andw20 =(w1;2;w2;2;:::;wT;2)areavailable,whereitseemsreasonabletoassume
E[wt;k"t]=0andE[wt;k(Mt?Ft)]6=0; fork=1;2:
c. Explain what is a GMM estimator in this setting, and how (in general terms) one may be obtained for a given (2 ? 2) weighting matrix, AT . (Note you are not required to formally derive the algebraic form of the GMM estimator!)
[7 marks]
d. Explain how, and the sense in which, the GMM estimator associated with the weighting matrix AT can be made e¢ cient.
- End of Question 3 -
*** END OF EXAMINATION ***
[6 marks]
