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Numerical Integration over Logarithmic Time Horizons in Market Shock Scaling

Article Quant Q&A · Author: dqd
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Summary

The document asks how to compute a Scale of Market Shocks measure that combines volatility estimates across multiple time horizons. Its integral is taken with respect to the logarithm of the horizon, so numerical integration should use log-horizon coordinates: transform each horizon to its logarithm, evaluate the volatility and smoothing weight there, and integrate across that variable. The smoothing function is normalized so its integral over log horizon equals one, making it a unit-weight average across scales; its constant is determined by that normalization.

The text identifies the method’s ingredients and the numerical implementation question, but gives no worked calculation or empirical results. A practical implementation must account for the chosen horizon range and discretization, since finite sampling and truncation affect the numerical integral and the resulting normalization. The discussion is about interpreting and implementing the published definition, not evidence that the shock measure predicts market outcomes.

Key ideas

  • The market shock measure aggregates volatility estimates across time horizons using logarithmic horizon as the integration variable.
  • Numerical integration can be performed after converting each horizon to its logarithm.
  • The smoothing function’s constant is chosen so its integral over log horizon equals one.
  • The numerical result depends on the sampled horizons and the range used for integration.

Tags

VolatilityStatisticsRisk management
Full text
# Computation of an integral containing d ln x (Scale of Market Shocks)


# Computation of an integral containing d ln x (Scale of Market Shocks)












I am trying to implement a Scale of Market Shocks method (SMS) which was presented in a 1999 working document by Olsen & Associates named Introducing a Scale of Market Shocks and later refined in a paper named Measuring Shock in Financial Markets. SMS is based on Operators on Inhomogeneous Time Series.

Now, I was able to figure out everything except for the main part, that is defined in the latter document on page 3 as:

$$S(t) = \gamma\int d~\text{ln}~\tau~~\mu(\text{ln}~\tau)~~\nu[\tau](t)$$

Here, $\tau$ is a time horizon, for which the price volatility $\nu$ is measured. And $\mu$ is just a smoothing function.

The thing I am not able to comprehend is $d~\text{ln}~\tau$. With $dx$, I would use a numerical method, such as scipy.integrate.romb, to approximate the value, but here, I have no idea of how to write a code for this kind of integration.

Furthermore, the authors mention this pattern again when discussing the smoothing function, which is defined as:

$$\mu(\text{ln}~\tau) = c~e^{-x}\left(1 + x + {x^2 \over 2}\right)\text{, where}~~~x = \left|\alpha~\text{ln}\left({\tau \over \tau_{center}}\right)\right|$$

They say: “The constant $c$ is adjusted so that $\mu$ is a unit measure $\int d~\text{ln}~\tau~~\mu(\text{ln}~\tau) = 1$.” I do not even understand what that sentence means, let alone how to transform such a remark to a working code.

So my questions are: how can I calculate the before-mentioned integral for $S(t)$? And how to use that knowledge in order to obtain a value of the constant $c$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.