Risk Exposure
When
X+Y0X+Y\neq0
, the system has risk exposure because of the constant price fluctuation.
At any given moment
tt
, we have the state of a derivative
StS_t
, the sum of total long positions
XtX_t
, the sum of total short positions
YtY_t
and the current price index
ItI_t
. Assuming that we have a new price
It+1I_{t+1}
at the moment
t+1t+1
, then the risk exposure
VexposureV{exposure}
can be calculated as follows:
Vexposure=((Xt+Yt)(It+1It))V_{exposure}=\sum\Big((X_t+Y_t)*(I_{t+1}-I_t)\Big)
  • XtX_t
    is always positive and
    YtY_t
    is always negative.
Consider that
It+1ItI_{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
Xt+YtX_t+Y_t
.
Obviously, if
Xt+Yt=0X_t+Y_t=0
, the risk exposure will not increase at moment
tt
.
Thus, the hAMM system reaches a balanced state (i.e.
X+Y=0X+Y=0
), which is so called "risk exposure closed".

Naked Position

At any moment, the value of
X+YX+Y
means the "unhedged" or "naked" position
NN
of a certain derivative at that moment:
N=X+YN=X+Y
When
N=0N=0
, the sum of total long positions equal to the sum of total short positions, and there are no naked positions.
When
N>0N>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<0N<0
, the sum of total short positions are greater than the sum of total long positions, and the naked positions are short positions.
VexposureV_{exposure}
can be simplified as follows:
Vexposure=(Nt(It+1It))V_{exposure}=\sum\Big(N_t*(I_{t+1}-I_t)\Big)
It can be concluded that:
When
N=0N=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
N0N\neq0
, if the sign of
It+1ItI_{t+1}-I_t
and
NN
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=0N=X+Y=0
The key to achieve the balance of hAMM system is to restrict naked position.
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