Chapter 23: Estimating Volatilities and Correlations

Question 1

How is the variance rate calculated?

Answer 1

We define sigma_n as the volatility of a market variable on day n, as estimated at the end of day n - 1. The square of this volatility is the variance rate.

Notation:

S_i = the value of the market variable at the end of day i.
u_i = the percentage change in the market variable between the end of day i - 1 and the end of day i.
u^bar = the mean of the mu_i's and is assumed to be 0.
m = number of observations of mu_i.

The formulas for calculating u_i and the variance rate are shown below.

Question 2

ARCH(m) model: How is a simple weighting scheme added to the variance rate formula?

Answer 2

The problem with the simple variance rate formula is that it gives equal weight to all observations of u_i (i.e. the percentage change in the market variable between the end of day i - 1 and the end of day i). Since our objective is to estimate the volatility today it makes sense to give more weight to recent data.

Also, we assume that there is a long-run average variance rate and that this should be give some weight.

Notation:

alpha_i = the weight assigned to each observation of mu_i.
V_L = long-run variance rate.
gamma = the weight assigned to V_L.

Defining omega = gamma * V_L this gives rise to the equation seen below.

Question 3

Explain the EWMA model

Answer 3

The EWMA model is an extension of the ARCH(m) model where the weights (alpha_i) assigned to the daily percentage change in the asset from the end of day i - 1 and the end of day i (u_i) decreases exponentially as we move back through time.

Specifically, alpha_i+1 = lambda * alpha_i where lambda is the smoothing constant between 0 and 1.

The estimate, sigma_n, of the volatility of a variable for day n (made at the end of day n - 1) is calculated from sigma_n-1 (the estimate that was made at the end of day n - 2 of the volatility for day n - 1) and u_n-1 (the most recent daily percentage change in the variable).

The formula for calculating the variance rate using the EWMA model is shown below:

Question 4

Explain the GARCH(1,1) model

Answer 4

In GARCH(1,1), the variance rate is calculated from a long-run average variance rate, V_L, as well as from sigma_n-1 and u_n-1. Weights of gamma, alpha and beta are assigned to these parameters respectively - note that these weights must sum to unity (1),

The GARCH(1,1) model is to prefer over the EWMA model. This is because variance rates tend to be mean reverting and the GARCH(1,1) model captures this, but the EWMA model does not.

The formula for calculating the variance rate using the GARCH(1,1) model is shown below.

Question 5

How can VaR be calculated using a GARCH(1,1) model?

Answer 5

In the standard VaR we assume that assets returns are normally distributed with mean (mu) and constant variance.

Instead we now assume that each asset return u_i are normally distributed with mean (mu) and time-varying variance. This will cause the VaR measure to change over time as well because it depends on the distribution.

Question 6

Explain how the 1-day VaR, using the GARCH(1,1), is calculated

Answer 6

See below

Question 7

How is the volatility estimate updated using the EWMA model?

Answer 7

See below formula

Question 8

How is the volatility estimate updated using the GARCh(1,1) model?

Answer 8

See below