Does delta neutral portfolio mean you add up deltas of all positions and the sum should be zero? Is this true? Also, in a FX portfolio consisting of FX calls puts and Fwds, if FWD delta is given for each how do you make the portfolio delta neutral? Do you just add up the fwd deltas for all and depending on the sum, buy or sell a FWD to bring the sum of delta to zero? Is this the right approach?


You are right in saying that to check whether your position is $\Delta$-neutral, you have to check the $\Delta$s of its constituents. That's a general statement that applies to positions that you are not rebalancing too fast, see e.g. this recent question.

In general, each Greek measures a particular risk/exposure of your position to a market condition that may change. For example, $\Delta$ ($\Theta$, $\rho$) tells you how much your position will change if the underlying level (time, interest rate) changes given that all other things being equal. The latter condition makes it possible to express Greeks as partial derivatives of the portfolio's value with respect to relevant variables, e.g. $\Delta = \frac{\partial V}{\partial S}$. Since partial derivatives are linear, if you hold $m$ at the money calls calls and $n$ stocks $$ \frac\partial{\partial S}(mC + nS) = m\Delta_C + n \approx m/2 + n $$ where $\Delta_C \approx 1/2$ is the $\Delta$ of one at the money call. In that case, to be $\Delta$-neutral, you would hold $n = -m/2$ stocks against your at the money call position.

As such, to compute exposure of your entire position, it is enough to compute separately exposures of each leg in your position. For this reason, the approach you've described in the OP is correct. Just note that $\Delta$ changes as well, and hence to stay $\Delta$ neutral you will need to do additional trades while you proceed to hold the option position. This concept is called continuous hedging. The changes of $\Delta$ are described by higher-order Greeks such as Gamma, Charm, DdelV etc.

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