- The sample size n At (the number n of return observations)

- The scaling interval A/Scaie. e.g., 1 if annualized volatility is desired

Choose the exponent p of Equation 3.8 (often 2 or 1 as discussed below) and the basic time scale of the computation. Instead of physical time, a business time omitting the weekends may be used.

Compute realized volatility according to Equation 3.8.

Scale the result by applying Equation 3.9. If Afscale = 1 year, this means annualization.

If the volatility has to be expressed in percentage, multiply the result by

A traditionally popular choice of realized volatility is the annualized volatility of daily returns on a yearly sample (using p = 2): At = 1 working day, n % 250, sample size nAt = 250 working days =a year, ArSCaie = 1 year. It should be noted that a business time scale with approximately 250 working days per year is used in this example instead of physical time.

The following examples illustrate the computation of realized volatility. Suppose we have regularly spaced 10-min returns from January 1, 1998 to December 31,1999. This two-year sample has a total of n = 105120 return observations. From this data, we want to compute realized volatility in three forms:

1. Volatility of 10-min returns on May 12, 1999, scaled to one day. The return interval is At = 10 min, n = 144, and therefore, the sample size is nAt = 1440 min = 1 day. v is computed from 144 returns according to Equation 3.8. To obtain the desired scaling, Equation 3.9 is used to

compute located = \/ day/10 min v = Vl44u = 12u.

2. Annualized volatility of 10-min returns over the whole sample where / = 10 min. All return observations, i.e., n = 105120 are used to calculate v according to Equation 3.8. The sample size is nAt = 1051200 min = 2 years. To obtain annualization according to Equation 3.10, Uann = yi year/10 min v = V52596 v 229.3 v is computed. This reflects the fact that an average year contains about 365.25 days and thus 52596 10-min intervals. Note that physical time is used in this particular example and weekends are not omitted.

3. Annualized volatility of 20-min returns over the whole sample. This is analogous to the example above, except for the different time interval At = 20 min. The 20-min returns Rj can be obtained by taking the sum of two 10-min returns: R\ = r\ + ri, Ri = + 4, and so on.4 The number of Rj observations is half that of the 10-min returns: n = 52560. The sample size is n At = 52560 x 20 min = 2 years, as above, v is computed

4 An alternative scheme to obtain 20-min returns from 10-min returns would be as follows: R\ = r\ + r2< Rl - r2 + r3> 3 = r3 + r4 so on- This scheme leads to overlapping 20-min returns and will be treated in Section 3.2.8.

100%.

from Equation 3.8 from the 20-min returns Rj instead of rj. To obtain annualization according to Equation 3.10, uann = yj\ year/20 min v = V26298 v % 162.2 v is computed. One average year contains 26298 20-min intervals, hence the annualization factor.

All of these examples are static and based on calculating realized volatility values at a fixed time t-,. Of course, we can treat realized volatility as a time series and compute it for a sequence of time points: f,, t,+i, r,+2, • • • • For each of these realized volatility computations, the sample is shifted by one interval /.

The concept of historical or realized volatility is rather old. We find it already in Taylor (1986)5 and in early high-frequency studies such as Muller etal. (1990). We distinguish three main types of volatility:

Realized volatility, also called historical volatility: determined by past observations by a formula such as Equation 3.8.

Model volatility: a virtual variable in a theoretical model such as GARCH or stochastic volatility (but there may be means to estimate this variable from the data).

Implied volatility: a volatility forecast computed from market prices of derivatives such as options (see e.g., Cox and Rubinstein, 1985), based on a model of the underlying process such as the log-normal random walk assumed by Black and Scholes (1973).

The term "realized volatility" has recently been popularized by Andersen et al. (2000) and others. By exploring realized volatility, Andersen et al. (2000) show that this is more than a conveniently measured quantity; it can also be used for process modeling.

An alternative definition of realized volatility is

v(ti)

n - 1

r(At; U-n+j)--

r{At;ti-n+k)

.k=\

(3.11)

For p = 2, this is the standard deviation of the returns about the sample mean. This definition is popular in portfolio analysis where the risk is measured in terms of deviations of the return from the average. In most other applications such as risk management and in the examples of this book, Equation 3.8 is preferred to Equation 3.11. The two definitions essentially differ only in the presence of a strong linear drift (i.e., if the returns have an expectation far from zero).

5 There, absolute values of returns and squared returns were explicitly introduced as proxies of volatility in autocorrelation studies. Note that these quantities are special cases of Equation 3.8 with = 1 and p = I (absolute return) or p = 2 (squared return).

Realized volatility v(tj) is based on a homogeneous series of returns as defined by Equation 3.7 and is a homogeneous time series in its own right. As an alternative, we can also compute a homogeneous series of realized volatility based on overlapping returns (see Section 3.2.8) or directly compute volatility from an inhomogeneous series with the help of convolution operators (see Section 3.3.11).

The parameters of Equation 3.8 have to be carefully chosen. A large exponent p gives more weight to the tails of the distribution. If p is too large, the realized volatility may have an asymptotically infinite expectation if returns have a heavy-tailed density function. In practice, p should stay below the tail index of the distribution, which is empirically estimated to be around 3.5 for typical high-frequency FX data (as explained in Section 5.4). The fourth moment of the return distribution often diverges. Moreover, there are studies where realized volatility appears in the squared form (as in autocorrelation studies of volatility). There, p should be limited to the half of the tail index. The empirical autocorrelation of squared returns is of little relevance. Instead, autocorrelation studies can be made with absolute values of returns (p = 1), as already done by Taylor (1986), Miiller et al. (1990) and, Granger and Ding (1995).

The choice of At and n is also important. Given a constant total sample size T = nAt, Andersen et al. (2001) recommend choosing At as small as possible. This means a large number n of return observations and thus high precision and significance. However, realized volatility results become biased if At is chosen to be too small, as found by Andersen et al. (2000). Therefore, the best choice of At is somewhere between 15 min and 2 hr, depending on the market and the data type. The bias has several implications, among them the negative short-term autocorrelation found for some financial data (see Section 5.2.1). Corsi et al. (2001) propose a bias-corrected realized volatility with At around 5 min, in order to maintain the high precision gained due to a large n. Interval overlapping is a further method to make realized volatility more precise by a limited amount (see Section 3.2.8).

A more fundamental question has to be discussed. Does a realized volatility with a constant T = nAt essentially stay the same if the time resolution, At, is varied? "Coarse" realized volatility (with large At) predicts the value of "fine" volatility (with small At) better than the other way around, as discussed in Section 7.4.1. This lead-lag effect indicates that the dynamics of volatility are complex, and realized volatility with one choice of At is not a perfect substitute for realized volatility with another value of At.

The relative merits of realized, modeled, and implied volatility are discussed at several places in this book. For low-frequency (including daily) data, models such as GARCH (Bollerslev, 1986) and option markets may yield volatility estimates that are as good or better than realized volatility. For high-frequency, intraday data, realized volatility is superior. Intraday data cannot be described by one homogeneous GARCH model because of the seasonality and heterogeneity of the markets, as shown by Guillaume et al. (1994) and Gencay et al. (2001c, 2002).