Position change means **any activity that changes the value of positions**, including open, close or liquidate positions. Position change cause changes in naked positions.

The hAMM system has a special **Position Change Fee** (PCF) mechanism rewarding position changes that decrease naked positions, and discouraging position changes that increase naked positions.

When traders increase the naked positions of a certain derivative, they will pay extra PCF to the system, and on the flip side, receive PCF from the system.

The existence of PCF will encourage arbitrageurs to earn risk-free profit as follows: open a position under Derify protocol that is opposite to the naked position, then open a hedged position in an external market.

Such action is called "**naked position arbitrage**", which effectively takes naked position to external markets, and decreases the risk exposure of the system.

The existence of arbitrageurs will keep the value of naked position $N$ close to zero, thus restricting naked position.

**Liquidity Depth Factor** (LDF) $D$ is a function to position pool $P$, the formula of which is as follows:

$D=\boldsymbol{f}(P)=\min{(\kappa*P,\ \psi)}=\min{\Big(\kappa*(X_{all}-Y_{all}),\ \psi\Big)}$

When the liquidity depth is limited, $D$ equals **risk sensitivity coefficient** $\kappa$ multiplied by position pool $P$.

When the liquidity depth reached a certain threshold ($\kappa*P>\psi$), $D$ always equals to **external liquidity depth simulator** $\psi$.

Risk sensitivity coefficient $\kappa$ is a pre-set constant. A smaller $\kappa$ means the higher position pool $P$ is needed to maintain the liquidity depth, i.e. the system is more sensitive.

External liquidity depth simulator $\psi$ is another constant, determined by the liquidity depth of external markets.

$\kappa$ and $\psi$ are changeable and can be adjusted via DAO community voting.

**PCF rates** $R$ is used to calculate the PCF, the formula of which is as follows:

$R=\dfrac{N}{D}=\dfrac{X_c+Y_c}{\min{\Big(\kappa*(X_{all}-Y_{all}),\ \psi\Big)}}$

$X$

*is always positive and*$Y$*is always negative.*

The sign of $R$ is determined by $N$.

When $N$ is positive:

naked position is long

PCF is positive

opening long position/closing short position shall pay PCF payment

opening short position/closing long position shall receive PCF rewards

When $N$ is negative:

naked position is short

PCF is negative

opening short position/closing long position shall pay PCF payment

opening long position/closing short position shall receive PCF rewards

$R$ is an instantaneous state of the system.

PCF cost $c$ for a transaction, means the PCF payable upon the transaction which shift the system PCF rates from $R$ to $R'$. Assuming the position changed is $\Delta p$, $c$ is calculated as follows:

$c=\begin{cases}
\Delta p*\Big(\dfrac{R+R'}{2}+\rho\Big),&R+R'>0\\
\\
0,&R+R'=0\\
\\
\Delta p*\Big(\dfrac{R+R'}{2}-\rho\Big),&R+R'<0
\end{cases}$

$\Delta p$ is positive if the position change is an increase, and negative if it decrease. If a original position is long, the sign for $\Delta p$ remains the same. If a original position is short, the sign for $\Delta p$ changes to the opposite (i.e., if the original position is short, increasing original position means $\Delta p$ is negative).

When $c$ is positive, trader pay PCF payment. When $c$ is negative, trader receive PCF rewards.

$\rho$ is the **coefficient of external arbitrage cost** set to cover the cost that arbitrageurs have in external markets, including slippage or trading fees. Generally, $\rho$ is set at 0.1%, which should cover most of the normal costs, but this constant is also changeable and can be adjusted via DAO community voting.