代做FN3142 Quantitative Finance 2014代做Python编程

日期：2025-12-02 07:46

FN3142

Quantitative Finance

Question 1

The component GARCH model is defined by the following equations

where Rt represents the return for period t.

Show that this model is equivalent to a GARCH(2,2) model.

100 marks

Question 2

Assume daily returns that are normally distributed with constant mean and variance

(a) Derive the formula for the Value-at-Risk at the α critical level and 1-day horizon.

25 marks

(b) The expected shortfall at critical level α and horizon K can be defined as

Derive the formula for calculating the 1-day expected shortfall at crit-ical level α.

25 marks

(c) Prove that the difference of the 1-day Value-at-Risk and the 1-day expected shortfall as a proportion of the 1-day Value-at-Risk converges to zero when α goes to zero.

50 marks

Question 3

(a) Can an economic forecast be ‘optimal’ but have poor forecasting power (in terms of a low R2 from a regression of the forecasted variable on a constant and the forecast)? If so, give an example or explain why not.

30 marks

(b) Can an economic forecast fail to be ‘optimal’ and still forecast well (in terms of a high R2 from a regression of the forecasted variable on a constant and the forecast)? If so, give an example or explain why not.

40 marks

(c) Use your answers from the above two questions to discuss statis-tical ‘optimality’ of forecasts and its relation to the practical ‘quality’ of forecasts.

30 marks

Question 4

Recall that the probability density function of a normally distributed random variable, with mean µ and variance σ2 is

(a) Assuming µ = 0 derive the maximum likelihood estimator of σ2; given a sample of iid data (x1; x2; . . . ; xT)

30 marks

(b) Now assume that xt is conditionally normally distributed as N(0, ) where

Write down the log-likelihood function for this model given a sample of data (x1; x2; . . . ; xT).

30 marks

(c) Describe how we can obtain estimates of (ω, α, β) for the GARCH(1,1) model and discuss any issues that may arise.

40 marks