# What is the delta of a portfolio invested in different stocks?

I understand that if I have a portfolio invested in stock A and options on stock A, the delta of my portfolio is going to be the weighted sum of the delta of the stock (=1) and of the option.

Now if I have a portfolio invested in stocks A and B and in options on these stocks, does it make sense to compute a global delta of the portfolio as the weighted sum of all the deltas? Or do we have to compute a delta that relates to A and a delta that relates to B?

• What are you trying to achieve with this number ("portfolio delta")? Jan 7, 2018 at 9:18
• Nothing, I just want to know if it still makes sense to talk about the delta of a portfolio when there are different stocks and options on these stocks in it. Jan 7, 2018 at 16:24
• What do you mean by "sense"? Sense in what way? Jan 7, 2018 at 16:48
• I think your question doesn't make sense as it stands. A better question would be about the delta of an option with different underlyings (e.g. a basket option). Jan 7, 2018 at 16:58
• I believe you did not understand my question. If I have a portfolio with Z shares of A, W shares of B, X options on stock A and Y options on stock B, it does not make sense to talk about the delta of the portfolio right ? It makes sense to talk about the *delta of the portfolio related to A or related to B right ? Jan 7, 2018 at 17:25

Strictly speaking, you cannot aggregate (i.e. sum) deltas. However, equity traders often provide their net exposure in currency units, which is a useful number. The same reasoning is possible with equity options: You can compute the 'delta equivalent position', i.e. delta times number of contracts (times multiplier) for each stock. Taking the delta equivalent position times the stock price gives you a hypothetical exposure in currency units, and adding these exposures up gives you a total exposure in currency units.

You can consider a multivariable delta if your security $V$ depends on two stocks $A$ and $B$: the gradient of $V$ is $$\nabla V=\left\langle \frac{\partial V}{\partial A}, \frac{\partial V}{\partial B}\right\rangle.$$ If you want a single number, there are indeed Greeks for multi-asset options.

• I think this doesn't answer the question because the OP wanted to have a delta of an portfolio and not on a security which depends on a portfolio (see my comments above). Additionally: do you have a source for your definition of a multivariable delta? Also: could you give an example how to use that definition in practice? Jan 8, 2018 at 7:09
• @vonjd One thing you could do in practice is use the well-known fact that the gradient points in the direction of greatest increase. So you can find out what relative changes in $A$ and $B$ will lead to greatest increase in $V$ (like, an increase of 5 dollars in $A$ for each 1 dollar increase in $B$). Jan 8, 2018 at 7:42
• @vonjd I should have mentioned that in my answer $V$ represents the total value of all stocks and options held. So $\partial V/\partial A$ is, by linearity of partial derivatives, a "weighted" sum of the deltas of the various assets. Jan 8, 2018 at 7:55
• A problem here is that the movements of different stocks are not independent (orthogonal). Jan 8, 2018 at 13:26

Since Delta is an options construct only, your question needs to be restated. I am assuming your are trying to find the risk of your portfolio against a market correlation. You can use Beta to combine your portfolio as long as you are trying to correlate your portfolio against SPY.