Treynor–Black Weights and Redundant Option Exposures

Summary

The document examines why a simplified Treynor–Black weighting expression appears to favor a call option over its underlying stock. In the example, the option’s alpha is scaled by delta while its variance is scaled by delta squared, which can make the alpha-to-variance ratio look larger. The question is whether that outcome reflects an economic preference or a limitation of the formula.

The answer says the expression applies under an independence assumption for idiosyncratic risk. A call option is locally a leveraged or de-leveraged exposure to the stock, so its risk is not independent of the underlying’s risk. From the portfolio optimization perspective described, the option is locally redundant, and the simplified expression cannot be applied by treating the two instruments as independent. The document provides a conceptual correction rather than a full derivation; its local relationship does not capture broader nonlinear option behavior or changing exposures.

Key ideas

• The simplified Treynor–Black weights rely on an assumption about independent idiosyncratic risks.
• A call option’s local exposure to its underlying is scaled by delta.
• The option and stock therefore have related risk rather than independent risk.
• Treating the call as an independent asset can make the simplified weighting formula misleading.
• The redundancy argument is local and does not describe all nonlinear option behavior.

Tags

# Is there economic/intuitive reason why the treynor-black model favour low delta instruments?


# Is there economic/intuitive reason why the treynor-black model favour low delta instruments?












In the treynor-black model optimal instrument weights are proportional to: $w_i = \frac{\frac{\alpha_i}{\sigma_i^2}}{\sum_j \frac{\alpha_j}{\sigma_j^2} }$.

Let Instrument 1 be a stock with $\alpha_1$ and $\sigma_1^2$ and let Instrument 2 be a call option with 50 delta on instrument 1. Then for Instrument 2 we have $\alpha_2 = \delta* \alpha_1$ and $\sigma_2^2 = \delta^2 \sigma_1^2$.

Simply plugging in these parameters will show that the model gives a higher weight to instrument 2. Is there any reason (intuitive) reason why the model would favour the call option over the stock? Or is this purely an "artefact" because the model was designed with stocks in mind?

## Answer by mbison (score 2)

https://quant.stackexchange.com/a/75273

So after thinking a bit about it I found the solution:

The formula in the treynor-black model only applies if the covariance matrix of the idiosyncratic risk is independent. The introduction of a call option, is locally simply a (de)leveraged version of the stock.

From a optimization perspective the call option is (locally) redundant. Therefore we can not simply plug in the values in the formula.

So it was a case of using the correct formula for the wrong situaton.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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