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I'm looking for a statistical test to understand the relationship (if any) between the model volatilities of a stochastic process, and the occurrence of 'break', defined as the instance when an empirical price breaks past a 95% confidence interval, created by Monte-Carlo.

Please see the graph below.

Monte-Carlo Simulated 97.5 and 2.5 percentiles, and the incidence of breaks

  • Blue graph is the empircal price series (left-axis)
  • Yellow graph is the 97.5 percentile of the Monte-Carlo distribution (left-axis)
  • Green graph is the 2.5 percentile of the Monte-Carlo distribution (left-axis)
  • Orange dots are instances when empirical price > 97.5 percentile; a break (left-axis)
  • Green dots are instances when empirical price < 2.5 percentile; a break (left-axis)
  • Red graph is the model volatilities feeding into the Monte-Calro simulation; volatilities are calibrated every 3 months (right-axis)

In an ideal state, the Monte-Carlo simulation perfectly contains the number of breaks to 5% of the number of observations (hence the 95% confidence interval). In this instance, there are more. I want to know if the breaks occur at times where the model volatility is too low. As such, I want to determine the relationship between low model volatilities and incidences of breaks.

The first thought is to perform an OLS regression. I'm having trouble selecting the correct regressor and regressand. Once this has been selected appropriately, I'll perform hypothesis testing on a coefficient.

Edit: Or a Chi-Sqaured Test?

Thank you

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