# Position closing

Once a position is open, a trader could choose to close it before expiry. If that's the case, the protocol needs to exit the lending position, which at expiry would have represented an amount $L$(principal + interest), and repay the owed debt, which at expiry would have represented an amount $D$(principal + interest).

## Pricing

The table below presents the price at which a trader could close a long position at a price $P_{C,L}$ or a short position at a price $P_{C,S}$(other notations have been introduced in theoretical pricing).

Side | Price to close a position |
---|---|

Long | $P_{C,L}= \dfrac{S_{S}}{{(1+r_{B,b})}^T} + D \bigg( 1 - \dfrac{1}{{(1+r_{Q,l})}^T} \bigg)$ |

Short | $P_{C,S}= \dfrac{S_{L}}{{(1+r_{B,l})}^T} + L \bigg( 1 - \dfrac{1}{{(1+r_{Q,b})}^T} \bigg)$ |

## Example

Let's consider the close of the long and short positions presented in the numerical example in position opening:

A trader wants to immediately close the long position with an open price $P_{O,L}= 100.59 \: DAI$. Given one could borrow ETH at a yearly fixed rate of $r_{B,b}=3.10\%$ā and lend DAI at a yearly fixed rate $r_{Q,l}=9.90\%$, and given the total debt to reimburse at expiry is $D=50.59 \: DAI$, the price at which a trader could immediately close the position is:

$P_{C,L}= \dfrac{99.90}{{(1+0.0310)}^{0.25}} + 50.59 * \bigg( 1 - \dfrac{1}{{(1+0.0990)}^{0.25}} \bigg) = 100.32 \:DAI$

A trader wants to immediately close the short position with an entry price $P_{O,S} =102.70 \: DAI$. Given one could lend ETH at a yearly fixed rate of $r_{B,l}=2.90\%$ā and borrow DAI at a yearly fixed rate $r_{Q,b}=10.10\%$, and given the total money to get back from lending (principal + interest) is $L=152.70 \:DAI$, the price at which a trader could immediately close the position is:

$P_{C,S}= \dfrac{100.10}{{(1+0.0290)}^{0.25}} + 152.70 * \bigg( 1 - \dfrac{1}{{(1+0.1010)}^{0.25}} \bigg) = 103.02 \:DAI$

## Demonstration

### Long

Let's consider a trader who wants to immediately close a long position of $1 \:ETH$ (numerical applications rely on the above example):

1. The protocol gets back the base currency which was lent, $\dfrac{1}{{(1+r_{B,b})}^T}$, i.e. $0.9924 \:ETH$.

2. This base currency is swapped back to the quote currency, $\dfrac{S_{S}}{{(1+r_{B,b})}^T}$, i.e. $99.14 \:DAI$.

3. The protocol buys back the debt $D$, today worth $\dfrac{D}{{(1+r_{Q,l})}^T}$, i.e. $49.41 \: DAI$.

4. For closing the position earlier, the trader will get back money on the debt, $D - \dfrac{D}{{(1+r_{Q,l})}^T}$ or $D \bigg( 1 - \dfrac{1}{{(1+r_{Q,l})}^T} \bigg)$, i.e. $1.18 \: DAI$.

5. Hence the money the trader gets back for closing the long position is $\dfrac{S_{S}}{{(1+r_{B,b})}^T} + D \bigg( 1 - \dfrac{1}{{(1+r_{Q,l})}^T} \bigg)$, i.e $100.32 \:DAI$.

### Short

Let's consider a trader who wants to immediately close a short position of $1 \:ETH$ (numerical applications rely on the above example):

1. The protocol needs $\dfrac{1}{{(1+r_{B,l})}^T} \: ETH$to close the debt, i.e. $0.9929 \:ETH$.

2. Hence the protocol needs $\dfrac{S_{L}}{{(1+r_{B,l})}^T} \: DAI$to close the debt, i.e. $99.39 \: DAI$.

3. On the other hand, the protocol gets back $\dfrac{L}{{(1+r_{Q,b})}^T} \: DAI$from lending, i.e. $149.07 \: DAI$.

4. The money lost in lending for closing the position earlier is $L-\dfrac{L}{{(1+r_{Q,b})}^T} \: DAI$or $L \bigg(1-\dfrac{1}{{(1+r_{Q,b})}^T} \bigg) \: DAI$, i.e. $3.63 \: DAI$.

5. Hence the money the trader gets back for closing the short position is $\dfrac{S_{L}}{{(1+r_{B,l})}^T} + L \bigg( 1 - \dfrac{1}{{(1+r_{Q,b})}^T} \bigg)$, i.e. $103.02\:DAI$.

Last updated