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Given the daily price data of equity and government-bond futures contracts, how can I identify the leverage effect (when prices move down, the volatility of prices increases), and see if it is statistically significant on the given time period?

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    $\begingroup$ Is this what you mean by Leverage Effect (or do you mean something else)? "The leverage effect refers to the observed tendency of an asset's volatility to be negatively correlated with the asset's returns. Typically, rising asset prices are accompanied by declining volatility, and vice versa." $\endgroup$ – noob2 Sep 16 '18 at 15:13
  • $\begingroup$ Yes, that is what I mean by the leverage effect. $\endgroup$ – jroy Sep 16 '18 at 18:46
  • $\begingroup$ This somewhat implies that normal vol might be more appropriate than lognormal. $\endgroup$ – will Oct 17 '18 at 5:49
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A simple way to show the Leverage Effect (not necessarily the only way) is:

Collect daily futures price data for at least 5 years. Use Adjusted historical data (sometimes called continuous contract data), which incorporates the effect of changing or rolling from one futures contract to the next.

For each calendar month in the period compute two numbers:

  • The daily volatility for this month, which is the square root of average value of $(\ln(P_t/P_{t-1}))^2$ for t ranging over all days of the month. (This will be the average of about 20 numbers, since there are about 20 (or 21) trading days each month.)

  • The return in the prior month. For example if the current month is June, then the prior month is May so we would compute the return from the last trading day of April to the last trading day of May.

Plot these 60 (or more) pairs of numbers, with prior month return on the x axis and volatility on the y axis. There should be a negative (downward sloping) relationship visible. Fit a regression line and test whether the slope of the regression line is significant.

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