代写FN3142 Quantitative Finance EXAM 2023帮做R编程

日期：2025-11-22 11:05

FN3142 Quantitative Finance

EXAM 2023

Question 1

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

(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. 40 marks

(b)  Now assume that x t  is conditionally normally distributed N(0, σt(2)) where σt(2) = w +

β σt(2)-1  + αxt(2)-1 . Write down the log-likelihood function for this model given a sample of data (x1; x2; …; xT). 40 marks

(c)   Describe and explain how we can obtain estimates of (w, α, β) for the GARCH(1,1) model and discuss any issues that arise. 20 marks

Total = 100 marks

Question 2

Suppose that for a given set of data VaR forecasts are calculated with historical simulation and GARCH methods.

(a)  Show how to construct a sequence of ‘hit’ variables Hitt(H)s  and Hitt(GARC)H  for testing the accuracy of the VaR forecasts. 40   marks

(b)  The following regression was run (standard errors are in parentheses below the parameter estimates):

Hitt(H)s  = 0.095 + μt

(0.025)

Hitt(GARC)H  = -0.2825 + μt

(0.35)

Explain how the above information can be used to test the accuracy of the VaR forecasts from these two models. 40 marks

(c)   Describe a method based on the chi-squared statistic that can be used to test for the serial correlation in hits.20 marks

Total = 100 marks

Question 3

(a)  Describe how one can test forecast optimality with a Mincer-Zarnowitz regression? 40 marks

(b)  Consider a forecast ytα of a variable, Yt. You have 100 observations of  ytα and  Yt  and you run the following regression:

The following results are obtained:

	Estimate	std error	t-statistic
β0	-0.008	0.0052	-2.3329
β1	1.6135	1.0399	0.1468

(i)           What can be inferred from this output? 20 marks

(ii)          What hypothesis do you need to test in relation to a Mincer-Zarnowitz regression and what is your test and conclusion? 40 marks

Total = 100 marks

a)   What is the “efficient market hypothesis”? 30 marks

b)   Discuss two of the modifications/extensions/refinements of the original definition of the efficient market hypothesis. 40 marks

c)   How does “collective data snooping” relate to the efficient market hypothesis? 30 marks

Total = 100 marks