# Price improvement

Contango protocol provides a better pricing than the theoretical formulas presented in the theoretical pricing section. This price improvement comes from the capital efficiency of using the trader's margin.

## Definition

We define the price improvement as:
• $pi_L=(P_{th,L} - P_{O,L} )/P_{O,L}$
for a long position
• $pi_S=(P_{O,S} - P_{th,S} )/P_{th,S}$
for a short position.
Knowing the theoretical pricing and using the pricing formulas with the collaterisation ratio to open a position, we find the following expressions for the price improvement:
Side
Price improvement
Long
$pi_L={CR*[{(1+r_{Q ,b })}^T -1]}$
Short
$pi_S=\dfrac{CR*[{(1+r_{Q ,l })}^T-1]}{1-CR*[{(1+r_{Q ,l })}^T -1]}$

## Implications

Several interesting facts could be inferred from the above expressions:
• The price improvement is always positive, i.e. the pricing of the protocol is more advantageous than the theoretical pricing for a trader opening long and short positions.
• The higher the collateriation ratio
$CR$
, the better the price improvement.
• The higher the time to expiry
$T$
, the better the price improvement.
• For a long position, the higher the interest rate on the quote currency
$r_{Q ,b }$
the better is the price improvement: compared to the theoretical formula, the protocol borrows less money and hence owes less debt.
• For a short position, the higher the interest rate on the quote currency
$r_{Q ,l }$
the better is the price improvement: compared to the theoretical formula, the protocol lends more money and hence makes an extra profit.
• In the case of a long fully collaterised position (CR=100%), the price improvement is equal to the interest rate to borrow the quote currency. Making the assumption that the borrowing and lending rates are equal, the price improvement is equivalent to lend the margin at a fixed rate.

## Examples

Here are different CR scenarios to open a long and short position for
$1 \: ETH$
, where the spot price is
$ETHDAI=10000$
, with the corresponding price improvements:

### Long

Collaterisation ratio (CR)
25%
50%
100%
Theoretical price
101.81
101.81
101.81
Pricing with collateral
101.19
100.58
99.39
Price improvement
0.61%
1.22%
2.43%

### Short

Collaterisation ratio (CR)
25%
50%
100%
Theoretical price
101.51
101.51
101.51
Pricing with collateral
102.12
102.73
103.99
Price improvement
0.60%
1.21%
2.45%

## Simulations

Here are some simulations to visualise the price improvement on long positions depending on the borrowing rate
$r_{Q,b}$
. Similar results can be found for price improvements on short positions. 