# Risk Exposure

When $X+Y\neq0$, the system has risk exposure because of the constant price fluctuation.

At any given moment $t$, we have the state of a derivative $S_t$, the sum of total long positions $X_t$, the sum of total short positions $Y_t$ and the current price index $I_t$. Assuming that we have a new price $I_{t+1}$ at the moment $t+1$, then the risk exposure $V{exposure}$ can be calculated as follows:

$V_{exposure}=\sum\Big((X_t+Y_t)*(I_{t+1}-I_t)\Big)$
• $X_t$ is always positive and $Y_t$ is always negative.

Consider that $I_{t+1}-I_t$ from external markets is unpredictable and cannot be controlled, the only way to reduce the risk exposure is to restrict the value of $X_t+Y_t$.

Obviously, if $X_t+Y_t=0$, the risk exposure will not increase at moment $t$.

Thus, the hAMM system reaches a balanced state (i.e. $X+Y=0$), which is so called "risk exposure closed".

# Naked Position

At any moment, the value of $X+Y$ means the "unhedged" or "naked" position $N$ of a certain derivative at that moment:

$N=X+Y$

When $N=0$, the sum of total long positions equal to the sum of total short positions, and there are no naked positions.

When $N>0$, the sum of total long positions are greater than the sum of total short positions, and the naked positions are long positions.

When $N<0$, the sum of total short positions are greater than the sum of total long positions, and the naked positions are short positions.

$V_{exposure}$ can be simplified as follows:

$V_{exposure}=\sum\Big(N_t*(I_{t+1}-I_t)\Big)$

It can be concluded that:

When $N=0$, no matter how the index was fluctuated, the net profit/loss of all positions equals zero, and the net loss of the system will not increase.

When $N\neq0$, if the sign of $I_{t+1}-I_t$ and $N$ are the same (i.e., naked positions and the index fluctuation has the same direction), the net loss of the system may increase.

Thus, the balanced state of hAMM system equals to "zero naked position":

$N=X+Y=0$

The key to achieve the balance of hAMM system is to restrict naked position.