# 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. – noob2 Jul 31 '15 at 12:35
• – John Sep 29 '15 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. – user25064 Dec 28 '15 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 – FernandoG Dec 28 '15 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. – user25064 Dec 28 '15 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 – FernandoG Aug 4 '15 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 – Kyle Balkissoon Aug 4 '15 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.