Decision and Risk Exercise 6 & 7

Decision and Risk – Exercise 6

Question 1

 Consider a model for log-returns of a particular stock:

Yt = µ + ut

ut = σtεt           εt N (0, 1)

Write down the general form of the GARCH(1, 1) model for the conditional variance of Yt and explain what common feature of financial data such a model is designed to capture.

Question 2

 The log-returns of a particular stock have a N (0, σ2) distribution, where the variance σ2 obeys the following ARCH(2) model:

σ2 = α0 + α1Y 2  + α2Y 2

t                                     t1                     t2

Suppose the log-returns over the past 2 days have been Y1 = 0.1, Y2 =

0.3. Write down an integral expression to compute the 99% Value-at- Risk on day 3, assuming that the ARCH parameters have known values α0 = 0.3, α1 = 0.1, and α2 = 0.2. You do not need to evaluate the integral.

  • Now suppose that although α0 and α1 are known and equal to the above values, α2 is Bayesian inference has been performed based on some earlier historical data y (which also includes data points Y1 = 0.1, Y2 = 0.3),  and  the  resulting  posterior  distribution  for  α2  is  α2 N (0.1, 0.2). Write down an integral expression to compute the 99% Value- at-Risk on day 3 based on this posterior distribution. You do not need to evaluate any parts of the integral.

Question 3

 Let Yt denote log-returns which can be described as follows (note that Y 2                 =

u2  ): t1

Yt = ut

t1

ut = σtεt          εt  ∼  N (0, 1)

σ2 = α0 + α1u2

t                                     t1

  1. Compute the conditional expectation of Yt given past information Ωt1, E(Yt|Ωt1).
  2. Compute the conditional variance of Yt given past information Ωt1, Var(Yt|Ωt1).
  3. Compute the unconditional expectation of Yt, E(Yt).
  4. Compute the unconditional variance of Yt, Var(Yt).

SOLUTION FOR DECISION AND RISK – EXERCISE 6

Decision and Risk – Exercise 7

Question 1

Let Yt denote the inter-event time. The following historical record shows the number of years between successive floods:

Y1 = 3, Y2 = 4, Y3 = 1, Y4 = 1

The flood risk expert suspects that there might have been a change in the average flood frequency after the second observation. If this is the case, then it would imply that a change point has occurred at t = 2, so that observations Y1, Y2 come from Exponential(λ1) distribution, and Y3, Y4 from Exponential(λ2) distribution.

Define the two models:

M0 : There has been no change point in the flood frequency.

M1 : There was a change point at t = 2.

Assuming that both models are equally likely, and using Gamma(1, 1) prior in model M0, and for both segments in model M1, compute the posterior probabil- ities p(M0|Y1 = 3, . . . Y4 = 1) and p(M1|Y1 = 3, . . . Y4 = 1) for models M0 and M1 respectively. Hence decide whether the data supports the expert’s belief. Consider only one possible change point specified in model M1.