# Position Change Fee

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 a state of the system.PCF cost

$c$

for a transaction, means the PCF payable upon the transaction which shift the system naked position from $N$

to $N'$

，and PCF rates from $R$

to $R'$

. Assuming the position changed is $\Delta p$

, it is calculated as follows:$c=\begin{cases}
-\Big||N'|-|N|\Big|*\Big(\dfrac{|R|+|R'|}{2}+\rho\Big),&|R|>|R'|\\
\\
0,&|R|=|R'|\\
\\
\Big||N'|-|N|\Big|*\Big(\dfrac{|R|+|R'|}{2}+\rho\Big),&|R|<|R'|
\end{cases}$

The actual PCF fee is not correlated to the size and direction of

$\Delta p$

, but only related to the change of naked position ($\Big||N'|-|N|\Big|$

) and PCF rates ($|R|$

和$|R'|$

).When

$c$

is positive, the trader pays a fee, when $c$

is negative, the trader gets rewards.

$\rho$

is **external arbitrage cost coefficient**, to encourage external arbitrageurs by covering their slippage and trading cost when they doing the arbitrage. Generally,$\rho$

= 0.1% may cover most of the arbitrage cost. $\rho$

is changeable and can be adjusted via DAO community voting.Last modified 1mo ago