CF4103 Financial Time Series Analysis
Suggested Solutions of Tutorial 10 (Semester 2/06-07)
Questions and Answers
1. Explain the exponentially weighted moving average (EWMA)
model for estimating volatility from historical data.
Solution. Define ui as (Si − Si−1)/Si−1, where Si is the
value of a market variable on day i. In the EWMA model,
the variance rate of the market variable (i.e., the square of its
volatility) calculated for day n is a weighted average of the
u2n−i’s (i = 1, 2, 3, · · ·). For some constant λ (0 < λ < 1)
the weight given to u2n−i−1 is λ times the weight given to
u2n−i. The volatility estimated for day n, σn, is related to
the volatility estimated for day n − 1, σn−1, by
σn2 = λσn−1
2
+ (1 − λ)u2n−1.
This formula shows that the EWMA model has one very
attractive property. To calculate the volatility estimate for
day n, it is sufficient to know the volatility estimate for day
n − 1 and un−1.
1
, 2. What is the difference between the exponentially weighted
moving average model and the GARCH(1, 1) model for up-
dating volatilities?
Solution. The EWMA model produces a forecast of the
daily variance rate for day n which is a weighted average
of (i) the forecast for day n − 1, and (ii) the square of the
proportional change on day n − 1. The GARCH(1, 1) model
produces a forecast of the daily variance for day n which is
a weighted average of (i) the forecast for day n − 1, (ii) The
square of the proportional change on day n − 1. and (iii)
a long run average variance rate. GARCH (1, 1) adapts the
EWMA model by giving some weight to a long run average
variance rate. Whereas the EWMA has no mean reversion.
GARCH(1, 1) is consistent with a mean-reverting variance
rate model.
3. The most recent estimate of the daily volatility of an asset
is 1.5% and the price of the asset at the close of trading
yesterday was $30.00. The parameter λ in the EWMA model
is 0.94. Suppose that the price of the asset at the close of
trading today is $30.50. How will this cause the volatility to
be updated by the EWMA model?
Solution. In the case σn−1 = 0.015 and un = 0.5/30 =
2
Suggested Solutions of Tutorial 10 (Semester 2/06-07)
Questions and Answers
1. Explain the exponentially weighted moving average (EWMA)
model for estimating volatility from historical data.
Solution. Define ui as (Si − Si−1)/Si−1, where Si is the
value of a market variable on day i. In the EWMA model,
the variance rate of the market variable (i.e., the square of its
volatility) calculated for day n is a weighted average of the
u2n−i’s (i = 1, 2, 3, · · ·). For some constant λ (0 < λ < 1)
the weight given to u2n−i−1 is λ times the weight given to
u2n−i. The volatility estimated for day n, σn, is related to
the volatility estimated for day n − 1, σn−1, by
σn2 = λσn−1
2
+ (1 − λ)u2n−1.
This formula shows that the EWMA model has one very
attractive property. To calculate the volatility estimate for
day n, it is sufficient to know the volatility estimate for day
n − 1 and un−1.
1
, 2. What is the difference between the exponentially weighted
moving average model and the GARCH(1, 1) model for up-
dating volatilities?
Solution. The EWMA model produces a forecast of the
daily variance rate for day n which is a weighted average
of (i) the forecast for day n − 1, and (ii) the square of the
proportional change on day n − 1. The GARCH(1, 1) model
produces a forecast of the daily variance for day n which is
a weighted average of (i) the forecast for day n − 1, (ii) The
square of the proportional change on day n − 1. and (iii)
a long run average variance rate. GARCH (1, 1) adapts the
EWMA model by giving some weight to a long run average
variance rate. Whereas the EWMA has no mean reversion.
GARCH(1, 1) is consistent with a mean-reverting variance
rate model.
3. The most recent estimate of the daily volatility of an asset
is 1.5% and the price of the asset at the close of trading
yesterday was $30.00. The parameter λ in the EWMA model
is 0.94. Suppose that the price of the asset at the close of
trading today is $30.50. How will this cause the volatility to
be updated by the EWMA model?
Solution. In the case σn−1 = 0.015 and un = 0.5/30 =
2
