代做FN3142 Quantitative Finance 2015代做留学生SQL 程序

日期：2025-12-02 07:46

FN3142

Quantitative Finance

Question 1

(a) (25 points) Assume that daily returns evolve as

Derive the GARCH(2,2) conditional variance model from the following ARMA(2,2) model for the squared residual:

where ηt+1 is a Gaussian white noise process.

(b) (25 points) Under the assumption of covariance stationarity, derive the unconditional variance of the GARCH(2,2) model in (a).

(c) (25 points) Show that the k steps ahead forecast can be written recursively as follows:

where V is the unconditional variance.

(d) (25 points) Explain how you can estimate the constant mean-GARCH(2,2) model by maximum likelihood, and write also the expression for the conditional likelihood cor-responding to a sequence of observations (r1,r2,...,rT ). Recall that the probability density function of a normally distributed random variable with mean µ and variance σ2 is

Question 2

(a) You hold two different corporate bonds (bond A and bond B), each with a face value of 1m $. The issuing firms have a 2% probability of defaulting on the bonds, and both the default events and the recovery values are independent of each other. Without default, the notional value is repaid, while in case of default, the recovery value is uniformly distributed between 0 and the notional value.

(a1) (20 points) Find the 1% VaR for bond A or bond B and report your calculations. (HINT: remember that the CDF of a uniformly distributed random variable on [a,b] is F(x) = (x−a)/(b−a).)

(a2) (60 points) Taking into account that the PDF of the sum of two independent uniformly distributed random variables on [a,b] is:

explain how you would find the 1% VaR for a portfolio combining the two bonds (A + B). Report your calculations without finding the actual value.

(b) (20 points) The α% expected shortfall is defined as the expected loss given that the loss exceeds the α% VaR. Find the 1% expected shortfall for bond A and report your calculations.

Question 3

(a) (10 points) What are the two main problems that a researcher encounters in multi-variate volatility modeling?

(b) (45 points) Describe in detail the Constant Conditional Correlations - GARCH(1,1) model of multivariate volatility, making use of matrix notation. Discuss benefits and drawbacks of this model.

(c) (45 points) Suggest a two-stage estimation method for this model based on Maximum Likelihood. In particular, write explicitly one of the likelihood functions that you maximize at the first stage, for an individual returns series i being one of the N assets. For simplicity, assume that the one period mean of returns is the constant µi. Recall that the probability density function of a normally distributed random variable with mean µ and variance σ2 is

Question 4

Your aim is to compare two forecasts and of some financial variable Yt. To this aim, consider some loss functions

(a) (50 points) Describe the Diebold-Mariano test to compare the forecasts in case the data are serially independent. Provide a numerical example where the loss function is the squared forecast error:

(b) (50 points) Explain how you modify the procedure in (a) when the data is serially dependent. Describe the Newey-West estimator that has been proposed for this case.