Comparative statistics of Garman-Klass, Parkinson, Roger-Satchell and bridge estimators

Comparative statistical properties of Parkinson, Garman-Klass, Roger-Satchell and bridge oscillation estimators are discussed. Point and interval estimations, related with mentioned estimators are considered


Examples of volatility estimators
Consider dependence on time t of the price P(t) of some financial instrument. As a rule, at discussing of volatility, one consider its logarithm Let us point out one of the conventional volatility V(T) definition, which we are using in this paper: It is the variance of the log-price increment Y(t, T) = X(t + T) − X(t) within given time interval duration T.

PUBLIC INTEREST STATEMENT
The volatility estimation is an important problem in finance. The level of volatility is a signal to trade for speculators. It is more important factor than a direction of trend. A cause of existence of some estimators of volatility is the principal differences of calculation of parameters, efficient of the estimation, what is depended from market situation. A goal of that paper is to compare parameters of main volatility estimators.

while G&K estimator given by expression
Here C := Y (t, T) is the close value of the log-price increment. Recall else R&S estimator, equal to Besides mentioned well-known estimators, we discuss bridge oscillation estimator. Below, we call it shortly by bridge estimator. Before to define it, recall bridge Z(t,t′) stochastic process definition. It is equal to Let high and low of the bridge be introduced: Accordingly, mentioned above bridge volatility estimator given by The value of the factor κ will be calculated later.

Geometric Brownian motion
One of the conventional models of price stochastic behavior is geometric Brownian motion (see Cont & Tankov, 2004;Jeanblanc, Yor, & Chesney, 2009;Saichev, Malevergne, & Sornette, 2010). In particular, it is used in theoretical justification of G&K, PARK and R&S estimators. Below we discuss statistics of mentioned volatility estimators in frame of geometric Brownian motion model. Namely, we assume that increment of the log-price is of the form Here μ is the drift of the price, while B(t) is the standard Brownian motion B(t) ~ N(0,t). Factor σ 2 is the intensity of the Brownian motion.
Recall, Brownian motion posses by self-similar property where and below sign ~ means identity in law.
Using pointed out self-similar property, one can ensure that Henceforth we call process x(τ,γ) by canonical Brownian motion, while factor γ by canonical drift. Using relations (3), (4), (8) and (11), one find that (4) We have used above canonical estimators: It is worthwhile to note that the closer expected values of canonical estimators ̂p( ),̂g( ),̂r and̂b to unity, the less biased corresponding original volatility estimators.
Analogously, the smaller variances of canonical estimators the more efficient original volatility esti- Notice additionally that canonical drift γ of the canonical Brownian motion x(τ,γ) (11) is, as a rule, unknown. Nevertheless, to get some idea about dependence on drift μ of bias and efficiency of = sup

Comparative efficiency of PARK and bridge estimators
Resting on, given at Appendix A, analytical formulas for probability density functions (pdfs) of random variables (13)and (14), we explore in this section some statistical properties of canonical PARK estimator ̂p( ) and bridge one ̂b (12).
Let unbiasedness of canonical PARK estimator be checked. To make it, let mean square of oscillation d = h−l of the canonical Brownian motion x(τ,γ) at the zero canonical drift (γ = 0) be calculated with the help of pdf q x (δ) (A7). After simple calculations obtain From here and from expression (12) of canonical PARK estimator ̂p( ) one can see that the following expression is true Let us find now the factor κ at expressions (8) and (12). To make it, calculate first of all the mean square of the bridge oscillation. Due to expression (A9) Accordingly, unbiased canonical bridge estimator has the form The great advantage of the bridge estimator is its unbiasedness for any drift. This remarkable property of the pointed out estimator is the consequence of the fact that bridge Z(t,t′) (6) and its (16)  As the next step, we calculate variance of canonical bridge estimator ̂b(17). Sought variance is equal to After substitution here, following from (A9), relation obtain Comparing equalities (18) and (19), one can see that variance of bridge estimator approximately twice smaller than variance of PARK estimator.
Recall, variance of bridge estimator does not depend on drift. On the contrary, variance of PARK estimator essentially depends on the drift. One can see it in Figure 3, where depicted plot of dependence, on canonical drift γ, of canonical PARK estimator variance.
Notice that bias of some estimator is insignificant only if it is much smaller than rms of corresponding estimator, i.e. the relative bias is small: Plot of canonical PARK estimator relative bias, as function of canonical drift γ depicted in Figure 4.

Given at Appendix analytical expressions (A6), (A7) and (A9) for canonical Brownian motion and canonical bridge random oscillations pdfs
Similarly, pdf of canonical bridge estimator is equal to Here q b (δ) (A9) is the pdf of canonical bridge oscillation. Plots of canonical PARK estimator pdf, for γ = 0, and pdf of canonical bridge estimator are depicted in Figure 5. In Figure 6 pdfs of canonical PARK estimator, for γ = 1, and pdf of canonical bridge estimator are compared. It is seen in both figures that pdf of canonical bridge estimator is better concentrated around its expected value E ̂b = 1 than canonical PARK estimator pdf.
Consider typical interval estimation: Let V is some volatility estimator, equal to Here ̂ is corresponding canonical estimator, while V(T) is the measured volatility. One needs to find probability that unknown (random) volatility (T) is not more than N times exceeds known (measured) volatility estimated value V . It follows from (23) that following inequalities are equivalent: Last means in turn that sought probability F(N) is expressed through pdf of canonical estimator ̂ by the following way: here W(x) is the pdf of canonical estimator ̂.
Plots of probabilities F(N) (24) dependence on the level N, for PARK estimator (in the case of zero drift μ = 0) and for bridge volatility estimator are given in Figure 7.

Comparative statistics of canonical estimators
Above, we explored in detail statistical properties of two, PARK and bridge estimators. Here, we compare their statistics and statistics of other well-known volatility estimators: G&K and R&S one. Despite the previous chapters, where we have used known analytical expressions for pdfs of canonical PARK and the bridge estimators, below we use predominantly results of numerical simulations.
Namely, we produce M ≫ 1 numerical simulations of random sequences where {∈ n } are iid Gaussian variables ~ N(0,1). Notice that stochastic process x n (γ) of discrete argument n rather accurately approximates, for large N ≫ 1, paths of canonical Brownian motion x(τ,γ) (11). Knowing M iid sequences {x n (γ)} one can find corresponding iid samples of pointed out above canonical estimators. Everywhere below we take number of iid samples M and discretization number N equal to Plots in Figure 8 demonstrate rather convincingly accuracy of numerical simulations. In Figure 9 are given two hundred samples of canonical G&K and bridge estimators, ensuring "by naked eye" that canonical bridge estimator is more efficient than G&K one. In Figure 10 are given, obtained by numerical simulations, plots of canonical G&K, PARK, R&S and bridge estimators mean values, illustrating bias of G&K and PARK estimators for nonzero canonical drift γ = 06, and actual absence of bias for bridge and R&S estimators Figure 11. Eventually, in Figure 12 are given plots of probabilities that true volatility V (T) is larger than half of corresponding estimator value and less than twice of it: It is seen that for any γ mentioned probability is essentially larger for bridge estimator, than for G&K, R&S and PARK estimators.