Root T rule

The root-T rule describes a method for scaling a diffusion process over time.

The following remarks refer only to a special application in financial mathematics . Other areas of application are also conceivable.

Application in financial mathematics

Time scaling of volatilities.

Assuming that the volatility remains constant over time, volatilities of constant interest rates from different periods of time or holding periods can be converted to one another using a relationship known as the (square) root t-rule.

${\displaystyle {\text{Vola}}_{\text{neu}}={\text{Vola}}_{\text{alt}}\cdot {\sqrt {T_{\text{neu}}/T_{\text{alt}}}}}$

${\displaystyle T_{\text{neu}}}$and are the holding periods or periods of time. ${\displaystyle T_{\text{alt}}}$

Example:

${\displaystyle {\text{Vola}}_{\text{30 Kalendertage}}={\text{Vola}}_{\text{1 Handelstag}}\cdot {\sqrt {T_{\text{22 Handelstage}}/T_{\text{1 Handelstag}}}}={\text{Vola}}_{\text{1 Handelstag}}\cdot {\sqrt {22}}}$

d. H. if the daily volatility were 2%, the monthly volatility would be 2% multiplied by the root 22, i.e. 9.38%. 22 trading days per month were assumed.

The assumption of normal distribution is not a mandatory prerequisite for this, but it should be noted that, for example, in the case of skewed, i.e. asymmetrical, distributions , the change in the holding period also changes the expected value . If the “holding period” is doubled, for example a Poisson distribution , which is equivalent to a convolution , its variance doubles. As in the example above, the standard deviation (probability theory) increases in the square root of it. However, the expected value has also doubled. A reasonably symmetrical distribution should therefore be a prerequisite for the application of this rule.

Since the time series of prices on the financial markets (share prices, foreign exchange rates, etc.) generally do not show constant volatility ( i.e. are heteroscedastic ), the T-rule here can lead to considerable errors.

Individual evidence

1. ↑ Danielsson, Jon and Zigrand, Jean-Pierre (2005): On Time-scaling of Risk and the Square-root-of-time rule
2. ↑ Diebold, Francis X. et al. (1998) Scale Models, Risk, 11, 104-107. (Revised and abridged version of "Converting 1-Day Volatility to h-Day Volatility: Scaling by Root-h is Worse than You Think," Wharton Financial Institutions Center, Working Paper 97-34.)