# How to perform risk budgeting for non-linear portfolios?

I am using this question to compute optimal weights following a risk budgeting approach. The problem is I am using non-linear portfolios (options,equity,fixed income,fx).

What I am looking for is that each asset class contributes the same amount of risk to the portfolio, and I am sure I can't use the regular approach if I have derivatives in my portfolio.

• It is a very interesting and perhaps unsolved problem. Jul 31, 2015 at 12:35
• – John
Sep 29, 2015 at 19:43
• Perhaps setting expected shortfall (percentage) equal across asset classes would be an interesting approach? This would capture the non-linear aspects of derivatives. Monte carlo methods could be invoked if necessary. Dec 28, 2015 at 13:37
• I agree expected shortfall would be a good way to measure the risk I think I would use something like this. But this approach doesn't consider the correlations beetween assets classes Dec 28, 2015 at 13:41
• Valid point. This approach could include correlation if expected shortfall were to be measured at the portfolio level, then marginal expected shortfall could be set equal across asset classes. Dec 28, 2015 at 13:58

An approach to consider is:

1. Computing the total return streams of all the instruments in the portfolio
2. Calculate the risk parameters using 1
3. Weight appropriately (Equal risk contribution, min variance etc)
• I understand that your approach would only work for a linear portfolio, because I cant measure the risk contribution of an option to my portfolio, only based on actual returns stream. Also consider that most derivatives dont have a long data history Aug 4, 2015 at 18:18
• the approach works for any asset class that is marked to market. Generally speaking you will need to create implied histories which can be tricky for options Aug 4, 2015 at 18:24

Just brainstorming here, could you possibly approach risk of an option from a probabilistic perspective?

Because the price of the option ($S - X$, where $S$ is lognormally distributed) is lognormally distributed with the same standard deviation as $S$ (aside from being truncated at 0 and having the probability go to infinity as $S$ decreases or $X$ increases, which would pose issues) we can assume that the volatility of the distribution of $P$ is most sensitive to changes in $\sigma_S$, not $S$. Therefore, based on the historical distribution of $\sigma_S$, could you not compute the implied distribution of values of the price of the option?

Essentially run a low-iteration Monte Carlo sampling from the historical distribution of volatility, then use the output of option prices to estimate a distribution of returns for $P$, and therefore the risk. Just find the optimal volatility by running volatility for different periods and finding which one most closely matches the current implied vol.

I understand that the math to support this is completely absent and there is likely a huge flaw in the assumptions made, but it may be a solution. Just choose a holding period and only calculate for that one $t$, or iterate across all $t$ and have a dynamic volatility that would require automatic rebalancing.