Two-scale risk and margin

Confidence is quick to lose and slow to regain, so many prices fall faster than they rise. Give the upside its own scale a and the downside its own scale b, and the quantiles, the long and short margin and the skew follow in closed form. A single Gaussian at the average scale cannot tell the two sides apart. The default is the book’s oil example.

Start from

Upside scale a (%)

Downside scale b (%)

Confidence level

Brent example from the book: upside 18%, downside 24%, as log-returns over the horizon. Illustrative, not market data.

Margin for a long position

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Margin for a short position

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The two-scale distribution

Density of the log-return, against a Gaussian at the average scale. Dashed lines mark the two-scale quantiles.

Chance of a move at least this large

Tail probability on a log scale, for falls (left) and rises (right).

The formulas

The density is a half-bell of scale a above the peak and of scale b below, the Gaussian when a = b. Its distribution function and quantiles are

The peak, median and mean separate; the mean is , and the variance is . Every measure of shape depends only on the ratio of the two scales, the circangle’s k′: the modulus of the two scales is the market’s skew.

Working backwards, an observed mean and variance give a and b directly, a far simpler calibration than fitting a five-parameter model to a surface of quotes. Worked figures are in Circangle: π is Obsolete, Part 4.

Use of the model is acceptance that it is for educational purposes only. It is not a margin, capital or risk methodology. See the disclaimer.