Section I Study Notes are available (and have been sent to preorder customers). Section II will be ready Friday, June 2nd. The first two readings are about volatility. Our second movie tutorial (”How to Estimate Volatility,” already released) covers the AIMs. Each of the parametric approaches to estimating volatility assume past is prologue: an historical series of squared returns is the starting point for estimating current volatility.
Both authors (Hull and Allen) permit a series of squared returns because they assume the average return is zero. Normally, a variance is the average of a series of squared differences (i.e., where variance is the sum of the squared differences between the actual return and the series’ average return), but if we can assume the average return is zero, then we’ve simply got a series of squared returns.
The easiest thing to do with this series is to average the squared returns
That’s an unweighted approach, or if you like, it implicitly gives equal weight to each historical u^2 in the series.
This equation above is unweighted: all (squared) returns in the historical series get the same implicit weight. We improve on that by assigning greater weight to more recent squared returns. If the weights happen to decrease in constant ratio to each other?Ce.g., as in the RiskMetricsTM approach where the day(n-1) variance is given a weight of 6% and the day(n-2) is 94% of the day(n-1) weight; and the day(n-3) weight is 94% of the day (n-2) weight; and so on?Cthen the exponentially weighted moving average (EWMA) simplifies into a recursive formula:
Which in words isn’t so bad: today’s estimate of variance is a function of yesterday’s variance plus (+) yesterday’s squared return. The weights, by definition, must sum to one (”unity”) and lambda is the persistence parameter or smoothing constant. A higher lambda (i.e., nearer to one) signifies a series that hugs itself, or is “sticky to itself.” When Allen warns that in practice “volatility clusters,” that is a better way of saying “lambda is large enough to matter”
GARCH(1,1) is a more general form of EWMA (conversely: EWMA is a restricted form of GARCH) because GARCH(1,1) adds “reversion to the mean.”; To get from EWMA to GARCH(1,1), we just add the first term below for mean reversion:
The additional term is the weighted average long-run variance (VL). By including this, we are saying, “the volatility series?Cto some degree?Cexperiences a gravitational pull toward its long-run variance. In addition to being a function of yesterday’s variance and yesterday’s squared return.”; Gamma is needed to assign a weight the long-run average variance. Where EWMA had only two weights, GARCH (1,1) has three: alpha, beta, gamma. (In the Hull reading, omega replaces the first term above, so omega = gamma x long-run variance).
GARCH(p,q) is very flexible, but now we can understand the “1,1″ - GARCH(1,1) is informed by only “1″ historical squared return and only “1″ variance.
Both authors (Hull and Allen) permit a series of squared returns because they assume the average return is zero. Normally, a variance is the average of a series of squared differences (i.e., where variance is the sum of the squared differences between the actual return and the series’ average return), but if we can assume the average return is zero, then we’ve simply got a series of squared returns.
The easiest thing to do with this series is to average the squared returns
That’s an unweighted approach, or if you like, it implicitly gives equal weight to each historical u^2 in the series.
This equation above is unweighted: all (squared) returns in the historical series get the same implicit weight. We improve on that by assigning greater weight to more recent squared returns. If the weights happen to decrease in constant ratio to each other?Ce.g., as in the RiskMetricsTM approach where the day(n-1) variance is given a weight of 6% and the day(n-2) is 94% of the day(n-1) weight; and the day(n-3) weight is 94% of the day (n-2) weight; and so on?Cthen the exponentially weighted moving average (EWMA) simplifies into a recursive formula:
Which in words isn’t so bad: today’s estimate of variance is a function of yesterday’s variance plus (+) yesterday’s squared return. The weights, by definition, must sum to one (”unity”) and lambda is the persistence parameter or smoothing constant. A higher lambda (i.e., nearer to one) signifies a series that hugs itself, or is “sticky to itself.” When Allen warns that in practice “volatility clusters,” that is a better way of saying “lambda is large enough to matter”
GARCH(1,1) is a more general form of EWMA (conversely: EWMA is a restricted form of GARCH) because GARCH(1,1) adds “reversion to the mean.”; To get from EWMA to GARCH(1,1), we just add the first term below for mean reversion:
The additional term is the weighted average long-run variance (VL). By including this, we are saying, “the volatility series?Cto some degree?Cexperiences a gravitational pull toward its long-run variance. In addition to being a function of yesterday’s variance and yesterday’s squared return.”; Gamma is needed to assign a weight the long-run average variance. Where EWMA had only two weights, GARCH (1,1) has three: alpha, beta, gamma. (In the Hull reading, omega replaces the first term above, so omega = gamma x long-run variance).
GARCH(p,q) is very flexible, but now we can understand the “1,1″ - GARCH(1,1) is informed by only “1″ historical squared return and only “1″ variance.
