FN3142 ZA
Quantitative Finance
Question 1
(a) What is the“efficient markets hypothesis”? [30 marks]
(b) Suppose we are at time t, and we are interested in the efficiency of the market of a given
stock. Let Ωt(w) denote the weak-form. efficient markets information set at time t, Ωt(ss) denote the semi strong-form. efficient markets information set at time t, and Ωt(s) denote the strong-form.
efficient markets information set at time t. To which information set, if any, do the following variables belong? Explain. [70 marks]
1. The stock price today.
2. The 3-month US Treasury bill rate today.
3. The inflation rate last year.
4. Next year’s expenditures just approved by the company’s board of directors (and not announced yet).
5. The value today of a put option on the stock that has a six-month maturity.
6. The value of the stock at time t + 3.
7. The number of shares AQR Capital Management (a hedge fund) purchased today of the stock.
Question 2
(a) Show that a stationary GARCH(1,1) model can be re-written as a function of the unconditional variance and the deviations of the lagged conditional variance and lagged squared residual from the unconditional variance. [20 marks]
Hint: a GARCH (1,1) model can be written in the form.
σt(2)+1 = W + βσt(2) + Qεt(2) ;
where W, Q, and β are constants, and εt is zero-mean white noise with conditional variance σt(2) .
(b) Derive the two-step ahead predicted variance for a GARCH(1,1), denoted by σt(2)+2.t and defined as Et [εt(2)+2], as a function of the parameters of the model and the one-step ahead forecast. [40 marks]
(c) Derive the three-step ahead predicted variance for a GARCH(1,1) and con-jecture the general expression for a h-step ahead forecast. Give an example of a financial application that may require using ah-step ahead forecast. [40 marks]
Question 3
(a) Define the concept of “trade duration” in financial markets and explain brieflywhy this
concept is economically useful. What features do trade durations typically exhibit and how can we model these features? [25 marks]
(b) Describe the Engle and Russell (1998) autoregressive conditional duration (ACD) model. [25 marks]
(c) Compare the conditions for covariance stationarity, identification and positiv-ity of the duration series for the ACD(1,1) to those for the GARCH(1,1). [25 marks]
(d) Illustrate the relationship between the log-likelihood of the ACD(1,1) model and the estimation of a GARCH(1,1) model using the normal likelihood function. [25 marks]
Question 4
Consider two stochastic processes: (i) Xt , about which we do not know anything for now, and (ii) vt , which is a zero-mean white noise with
Moreover, we know that v and X are uncorrelated at all leads and lags, that is, E[Xtvt-j ] = 0
for all j integers. Let an observed series Zt bedefined as
Zt = Xt + vt.
(a) What does covariance stationarity mean? [15 marks]
(b) Prove that the process Zt is covariance stationary if and only if Xt is covariance stationary. [20 marks]
Now assume that Xt is an MA(1) process:
Xt = ut + δut-1 ,
where ut is zero-mean white noise with
(c) Calculate the autocovariances of the process Xt and show that it is covariance stationary.
[20 marks]
(d) Calculate the autocovariances of Zt and show that they are zero beyond one lag.
[20 marks]
(e) Is it possible to represent the process Zt as an MA(1) process? In particular, assume that we write
Zt = εt + θεt-1 , (1)
where εt isa zero-mean white noise with variance σε(2) . What would be the required restrictions
on θ and σε(2) so that Ztis an MA(1) process? [25 marks]
Hint: based on the assumption (1), derive the autocovariances of Z as a function of θ and σε(2), and compare them to your results in (d) .