1. Estimating Volatilities and Correlations
Chapter 21

2. Outline Standard Approach to Estimating Volatility Weighting Scheme
ARCH(m) Model
EWMA Model
GARCH model
Maximum Likelihood Methods

3. Standard Approach to Estimating Volatility
Define sn as the volatility per day between day n-1 and day n, as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define ui= ln(Si/Si-1)

4. Simplifications Usually Made
Define ui as (Si −Si-1)/Si-1
Assume that the mean value of ui is zero
Replace m−1 by m
This gives

5. Weighting Scheme
Instead of assigning equal weights to the observations we can set

6. ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate, VL:

7. EWMA Model (1/3)
In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time, and the decline rate is
This leads to

8. EWMA Model (2/3)

9. EWMA Model (3/3)

10. Attractions of EWMA Relatively little data needs to be stored
We need only remember the current estimate of the variance rate and the most recent observation on the market variable
Tracks volatility changes
RiskMetrics uses l = 0.94 for daily volatility forecasting

11. GARCH (1,1)
In GARCH (1,1) we assign some weight to the long-run average variance rate
Since weights must sum to 1
g + a + b =1

12. GARCH (1,1)
Setting w = gV the GARCH (1,1) model is
and

13. GARCH (p,q)

14. The weight
The weights decline exponentially at rate β The GARCH(1,1) is similar to EWMA except that assign some weight to the long-run volatility

15. Mean Reversion
The GARCH(1,1) model recognizes that over time the variance tends to get pulled back to a long-run avarage level
The GARCH(1,1) is equivalent to a model where the variance V follows the stochastic process

16. Prove(1/2)
moment function

17. Prove(2/2)

18. Example
Suppose
The long-run variance rate is so that the long-run volatility per day is 1.4%

19. Example continued
Suppose that the current estimate of the volatility is 1.6% per day and the most recent percentage change in the market variable is 1%.
The new variance rate is
The new volatility is 1.53% per day

20. Maximum Likelihood Methods
In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring

21. Example 1
We sample 10 stocks at random on a certain day and find that the price of one of them declined and the price of others remaind same or increased what is the best estimate of the probability of a price decline?
The probability of the event happening on one particular trial and not on the others is
We maximize this to obtain a maximum likelihood estimate. Result: p=0.1

22. Example 2(1/2)
Estimate the variance of observations from a normal distribution with mean zero

23. Example 2(2/2)

24. Thank you for listening

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